Experimental Game Theory and Its Application in Sociology...

Journal of Applied Mathematics

Experimental Game Theory and Its Application in Sociology and Political Science

Guest Editors Arthur Schram Vincent Buskens Klarita Geumlrxhani and Jens Groszliger

Experimental Game Theory and Its Application

in Sociology and Political Science




Guest Editors Arthur Schram Vincent BuskensKlarita Geumlrxhani and Jens Groszliger

Copyright copy 2015 Hindawi Publishing Corporation All rights reserved

is is a special issue published in ldquoJournal of Applied Mathematicsrdquo All articles are open access articles distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the originalwork is properly cited

Editorial Board

Saeid Abbasbandy IranMina B Abd-El-Malek EgyptMohamed A Abdou EgyptSubhas Abel IndiaJanos Abonyi HungarySergei Alexandrov RussiaM Montaz Ali South AfricaMohammad R Aliha IranCarlos J S Alves PortugalMohamad Alwash USAGholam R Amin OmanIgor Andrianov GermanyBoris Andrievsky RussiaWhye-Teong Ang SingaporeAbul-Fazal M Arif KSASabri Arik TurkeyAli R Ashra IranAllaberen Ashyralyev TurkeyFrancesco Aymerich ItalyJalel Azaiez CanadaSeungik Baek USAHamid Bahai UKOlivier Bahn CanadaAntonio Bandera SpainJean-Pierre Barbot FranceMostafa Barigou UKRoberto Barrio SpainHenri Baudrand FranceAlfredo Bellen ItalyVasile Berinde RomaniaJafar Biazar IranAnjan Biswas KSAAbdellah Bnouhachem MoroccoGabriele Bonanno ItalyWalter Briec FranceJames Robert Buchanan USAHumberto Bustince SpainXiao Chuan Cai USAPiermarco Cannarsa ItalyYijia Cao ChinaJinde Cao ChinaZhenfu Cao ChinaBruno Carpentieri NetherlandsAna Carpio SpainAlexandre Carvalho Brazil

Song Cen ChinaTai-Ping Chang TaiwanShih-sen Chang ChinaWei-Der Chang TaiwanShuenn-Yih Chang TaiwanKripasindhu Chaudhuri IndiaYuming Chen CanadaJianbing Chen ChinaXinkai Chen JapanRushan Chen ChinaKe Chen UKZhang Chen ChinaZhi-Zhong Chen JapanRu-Dong Chen ChinaHui Cheng ChinaChin-Hsiang Cheng TaiwanEric Cheng Hong KongChing-Hsue Cheng TaiwanQi Cheng USAJin Cheng ChinaFrancisco Chiclana UKJen-Tzung Chien TaiwanChongdu Cho KoreaHan H Choi Republic of KoreaMd S Chowdhury MalaysiaYu-Ming Chu ChinaHung-Yuan Chung TaiwanAngelo Ciaramella ItalyPedro J Coelho PortugalCarlos Conca ChileVitor Costa PortugalLirong Cui ChinaJie Cui USALivija Cveticanin SerbiaGaoliang Dai GermanyHua Dai ChinaBinxiang Dai ChinaMichael Defoort FranceHilmi Demiray TurkeyMingcong Deng JapanYoujun Deng FranceOrazio Descalzi ChileRaaele Di Gregorio ItalyKai Diethelm GermanyDaniele Dini UK

Urmila Diwekar USAVit Dolejsi Czech RepublicBoQing Dong ChinaRodrigo W dos Santos BrazilWenbin Dou ChinaRafael Escarela-Perez MexicoMagdy A Ezzat EgyptMeng Fan ChinaYa Ping Fang ChinaIstvaacuten Faragoacute HungaryDidier Felbacq FranceRicardo Femat MexicoAntonio J M Ferreira PortugalGeorge Fikioris GreeceMichel Fliess FranceMarco A Fontelos SpainDimitris Fotakis SerbiaTomonari Furukawa USAMaria Gandarias SpainXiao-wei Gao ChinaHuijun Gao ChinaXin-Lin Gao USANaiping Gao ChinaLaura Gardini ItalyWinston Garira South AfricaLeonidas N Gergidis GreeceBernard J Geurts NetherlandsSandip Ghosal USADibakar Ghosh IndiaPablo Gonzaacutelez-Vera SpainAlexander N Gorban UKKeshlan S Govinder South AfricaSaid R Grace EgyptJose L Gracia SpainMaurizio Grasselli ItalyZhi-Hong Guan ChinaNicola Guglielmi ItalyFreacutedeacuteric Guichard CanadaKerim Guney TurkeyShu-Xiang Guo ChinaVijay Gupta IndiaSaman K Halgamuge AustraliaRidha Hambli FranceAbdelmagid S Hamouda QatarBo Han China

Maoan Han ChinaPierre Hansen CanadaFerenc Hartung HungaryXiao-Qiao He ChinaYuqing He ChinaNicolae Herisanu RomaniaOnesimo Hernandez-Lerma MexicoLuis Javier Herrera SpainJ C D Hoenderkamp Netherlandsomas Houmlhne GermanyWei-Chiang Hong TaiwanSun-Yuan Hsieh TaiwanYing Hu FranceNing Hu JapanJianguo Huang ChinaDan Huang ChinaTing-Zhu Huang ChinaZhilong L Huang ChinaZhenkun Huang ChinaAsier Ibeas SpainMustafa Inc TurkeyGerardo Iovane ItalyAnuar Ishak MalaysiaTakeshi Iwamoto JapanGeorge Jaiani GeorgiaGunHee Jang KoreaZhongxiao Jia ChinaDaqing Jiang ChinaHaijun Jiang ChinaJianjun Jiao ChinaZhen Jin ChinaXing Jin ChinaLucas Joacutedar SpainZlatko Jovanoski AustraliaTadeusz Kaczorek PolandDongsheng Kang ChinaSangmo Kang USAIdo Kanter IsraelAbdul Hamid Kara South AfricaDimitrios A Karras GreeceIhsan Kaya TurkeyDogan Kaya TurkeyChaudry M Khalique South AfricaWaqar A Khan PakistanKhalil Khanafer USAHyunsung Kim KoreaYounjea Kim Republic of KoreaJong Hae Kim Republic of Korea

Svein Kleiven SwedenKazutake Komori JapanVassilis Kostopoulos GreeceJisheng Kou ChinaRoberto A Kraenkel BrazilKannan Krithivasan IndiaVadim A Krysko RussiaJin L Kuang SingaporeGennady M Kulikov RussiaVenugopal Kumaran IndiaMiroslaw Lachowicz PolandHak-Keung Lam UKTak-Wah Lam Hong KongHeung-Fai Lam Hong KongLuciano Lamberti ItalyPeter G L Leach CyprusUsik Lee Republic of KoreaJaehong Lee KoreaWen-Chuan Lee TaiwanMyung-Gyu Lee KoreaJinsong Leng ChinaXiang Li ChinaQingdu Li ChinaWenlong Li Hong KongShuai Li Hong KongLixiang Li ChinaWan-Tong Li ChinaShiyong Li ChinaQin Li ChinaHua Li SingaporeYan Liang ChinaJin Liang ChinaDongfang Liang UKChing-Jong Liao TaiwanTeh-Lu Liao TaiwanYong-Cheng Lin ChinaGao Lin ChinaYiping Lin ChinaChong Lin ChinaIgnacio Lira ChileSheng Liu AustraliaZhengrong Liu ChinaZhuangjian Liu SingaporeMeiqin Liu ChinaPeter Liu TaiwanPeide Liu ChinaPengfei Liu ChinaChein-Shan Liu Taiwan

Yansheng Liu ChinaZhijun Liu ChinaKang Liu USAWeiqing Liu ChinaTao Liu ChinaShutian Liu ChinaFei Liu ChinaChongxin Liu ChinaJose L Loacutepez SpainShiping Lu ChinaHongbing Lu ChinaBenzhuo Lu ChinaHao Lu USAYuan Lu ChinaHenry Horng-Shing Lu TaiwanGilles Lubineau KSAZhen Luo ChinaLifeng Ma ChinaChangfeng Ma ChinaLi Ma ChinaRuyun Ma ChinaShuping Ma ChinaKrzysztof Magnucki PolandNazim I Mahmudov TurkeyOluwole D Makinde South AfricaFlavio Manenti ItalyFrancisco J Marcellaacuten SpainVasile Marinca RomaniaGiuseppe Marino ItalyGuiomar Martiacuten-Herraacuten SpainCarlos Martiacuten-Vide SpainAlessandro Marzani ItalyNikos E Mastorakis BulgariaNicola Mastronardi ItalyPanayotis Takis Mathiopouloss GreeceGianluca Mazzini ItalyMichael McAleer NetherlandsStephane Metens FranceMichael Meylan AustraliaEfren Mezura-Montes MexicoFan Min ChinaAlain Miranville FranceRam N Mohapatra USASyed A Mohiuddine KSAGisele Mophou FranceJoseacute Morais PortugalCristinel Mortici RomaniaEmmanuel Moulay France

Jaime E Munoz Rivera BrazilJavier Murillo SpainRoberto Natalini ItalySrinivasan Natesan IndiaTatsushi Nishi JapanMustapha Nourelfath CanadaAndreas ampquotOchsner AustraliaWlodzimierz Ogryczak PolandRoger Ohayon FranceJavier Oliver SpainSoontorn Oraintara USADonal OrsquoRegan IrelandMartin Ostoja-Starzewski USATurgut ampquotOziş TurkeyClaudio Padra ArgentinaVincent Pagneux FranceReinaldo Martinez Palhares BrazilQuanke Pan ChinaEndre Pap SerbiaSehie Park KoreaManuel Pastor SpainGiuseppe Pellicane South AfricaFrancesco Pellicano ItalyJuan Manuel Pentildea SpainJian-Wen Peng ChinaRicardo Perera SpainMalgorzata Peszynska USAAllan C Peterson USAAndrew Pickering SpainSomyot PlubtiengailandHector Pomares SpainRoland Potthast UKKerehalli V Prasad IndiaRadu-Emil Precup RomaniaMario Primicerio ItalyMorteza Rafei NetherlandsP Rattanadecho ailandLaura Rebollo-Neira UKMahmoud M Reda Taha USARoberto Renograve ItalyBruno G M Robert FranceJ A Rodriguez-Velaacutezquez SpainIgnacio Rojas SpainCarla Roque PortugalDebasish Roy IndiaImre J Rudas HungaryAbbas Saadatmandi IranKunihiko Sadakane Japan

Samir H Saker EgyptR Sakthivel Republic of KoreaJuan J Salazar Gonzaacutelez SpainMiguel A F Sanjuan SpainBogdan Sasu RomaniaRichard Saurel FranceWolfgang Schmidt GermanyMehmet Sezer TurkeyNaseer Shahzad KSAPengjian Shang ChinaM Shariyat IranHui-Shen Shen ChinaJian Hua Shen ChinaYong Shi ChinaYasuhide Shindo JapanPatrick Siarry FranceFernando Simotildees Portugaleodore E Simos GreeceGeorgios Sirakoulis GreeceRobert Smith CanadaFrancesco Soldovieri ItalyAbdel-Maksoud A Soliman EgyptXinyu Song ChinaQiankun Song ChinaYuri N Sotskov BelarusIvanka Stamova USANiclas Stroumlmberg SwedenRay KL Su Hong KongHousheng Su ChinaChengjun Sun Hong KongJitao Sun ChinaWenyu Sun ChinaW Y Szeto Hong KongToshio Tagawa JapanYing Tan ChinaSan-Yi Tang ChinaXianHua Tang ChinaNasser-Eddine Tatar KSAZhidong Teng ChinaEngang Tian ChinaAlexander Timokha NorwayAydin Tiryaki TurkeyYiying Tong USAHossein Torkaman IranMariano Torrisi ItalyJuan J Trujillo SpainJung-Fa Tsai TaiwanGeorge Tsiatas Greece

Charalampos Tsitouras GreeceAntonia Tulino USAStefano Ubertini ItalyJeong S Ume KoreaAlexandrina Untaroiu USASergey Utyuzhnikov UKKuppalapalle Vajravelu USAAlvaro Valencia ChileOlaf van der Sluis NetherlandsErik Van Vleck USAEzio Venturino ItalyJesus Vigo-Aguiar SpainMichael N Vrahatis GreeceMichael Vynnycky SwedenXiang Wang ChinaYaonan Wang ChinaShuming Wang SingaporeYouqing Wang ChinaDongqing Wang ChinaPeiguang Wang ChinaYuh-Rau Wang TaiwanHeng Wang SingaporeBaolin Wang ChinaMingxin Wang ChinaQing-WenWang ChinaGuangchen Wang ChinaJunzo Watada JapanWei Wei ChinaJunjie Wei ChinaJinjia Wei ChinaGuoliang Wei ChinaLi Weili ChinaMartin Weiser GermanyFrank Werner GermanyMan Leung Wong Hong KongXiangjun Wu ChinaCheng Wu ChinaShi-Liang Wu ChinaWei Wu ChinaQingbiao Wu ChinaYuqiang Wu ChinaMin-Hsien Wu TaiwanXian Wu ChinaTiecheng Xia ChinaYonghui Xia ChinaXian Xian-Yong ChinaXuejun Xie ChinaGongnan Xie China

Daoyi Xu ChinaYuesheng Xu USAZhiqiang Xu ChinaWei Xu ChinaFuzhen Xuan ChinaGregory S Yablonsky USAChao Yan USAChao Yang ChinaChunyu Yang ChinaGuowei Yang ChinaBin Yang ChinaXiao-Jun Yang ChinaDar-Li Yang TaiwanNa Yang ChinaChao-Tung Yang TaiwanZhichun Yang ChinaHer-Terng Yau TaiwanWei-Chang Yeh TaiwanGuan H Yeoh Australia

Chih-Wei Yi TaiwanSimin Yu ChinaBo Yu ChinaXiaohui Yuan ChinaGonglin Yuan ChinaJinyun Yuan BrazilRafal Zdunek PolandAshraf Zenkour EgyptGuisheng Zhai JapanJianming Zhan ChinaMeng Zhan ChinaZhengqiu Zhang ChinaJiangang Zhang ChinaKe Zhang ChinaLong Zhang ChinaSheng Zhang ChinaJifeng Zhang ChinaHeping Zhang ChinaHenggui Zhang UK

Liang Zhang ChinaJingxin Zhang AustraliaZhihua Zhang ChinaShan Zhao USAChongbin Zhao AustraliaBaosheng Zhao ChinaYun-Bo Zhao ChinaRenat Zhdanov USADong Zheng USAHuaichun Zhou ChinaBin Zhou ChinaShangbo Zhou ChinaHuamin Zhou ChinaQuanxin Zhu ChinaXinqun Zhu AustraliaWilliam Zhu ChinaGoangseup Zi KoreaZhiqiang Zuo China

Contents

Experimental Gameeory and Its Application in Sociology and Political Science Arthur SchramVincent Buskens Klarita Geumlrxhani and Jens GroszligerVolume 2015 Article ID 280789 2 pages

e Strategic Role of Nonbinding Communication Luis A Palacio Alexandra Corteacutes-Aguilar andManuel Muntildeoz-HerreraVolume 2015 Article ID 910614 11 pages

Intermediaries in Trust Indirect Reciprocity Incentives and Norms Giangiacomo Bravo FlaminioSquazzoni and Kaacuteroly TakaacutecsVolume 2015 Article ID 234528 12 pages

Altruism Noise and the Paradox of Voter Turnout An Experimental Study Sarah A TulmanVolume 2015 Article ID 972753 22 pages

Preference for Eciency or Confusion A Note on a Boundedly Rational Equilibrium Approach to

Individual Contributions in a Public Good Game Luca Corazzini and Marcelo TyszlerVolume 2015 Article ID 961930 8 pages

Competition Income Distribution and the Middle Class An Experimental Study Bernhard KittelFabian Paetzel and Stefan TraubVolume 2015 Article ID 303912 15 pages

EditorialExperimental Game Theory and Its Application inSociology and Political Science

Arthur Schram12 Vincent Buskens3 Klarita Geumlrxhani4 and Jens Groszliger5

1CREED University of Amsterdam Plantage Muidergracht 14 1018 TV Amsterdam Netherlands2Robert Schuman Centre for Advanced Studies European University Institute Fiesole Italy3Department of SociologyICS Utrecht University Padualaan 14 3584 CH Utrecht Netherlands4Department of Political and Social Sciences European University Institute Via Roccettini 9 San Domenico di Fiesole50014 Fiesole Italy5Department of Political Science Florida State University Bellamy Hall 531 Tallahassee FL 32306-2230 USA

Correspondence should be addressed to Arthur Schram schramuvanl

Received 13 August 2015 Accepted 16 August 2015

Copyright copy 2015 Arthur Schram et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Game theory laboratory experiments and field experimentsare common and powerful tools in many social sciences[1] However applications in Sociology and Political Scienceremain scarce and scattered [2] Yet the combination of gametheory with controlled experiments provides a powerful toolto better understand social and political processes for exam-ple [3ndash5]Themathematical structure offered by game theoryand the control offered by an experimental environmentallow the researcher to isolate sociological andor politicalphenomena to study their development and their effectsThe relationship between game theory and experiments istwofold On the one hand game theory provides solid groundon which to design an experiment and a formal benchmarkthat serves as a measuring rod for a structured analysisof observed behavior On the other hand experiments canbe used to test equilibrium predictions and to pinpointshortcomings of theory as well as point to directions in whichthe theory can be adapted

The aim of the special issue is to encourage originalresearch that seeks to study sociological or political phenom-ena using laboratory experiments that are based on game the-oretical benchmarks and that seek mathematical modeling

of game theoretical arguments to inspire experiments in thefields of Sociology and Political Science and vice versa

In a research article of the special issue G Bravo et alexperimentally study whether intermediaries can positivelyinfluence cooperation between a trustor and a trustee inan investment or trust game Another article by L APalacio et al develops a game theoretical foundation forexperimental investigations of the strategic role in gameswithnonbinding communication In another article L Corazziniand M Tyszler employ quantal response equilibrium (QRE)to find out the extent of confusion and efficiency motivesof laboratory participants in their decisions to contribute topublic good The article by S A Tulman utilizes QRE (ienoisy decision-making) and altruism-motivated players toinvestigate the ldquoparadox of voter turnoutrdquo in a participationgame experiment Finally in another article B Kittel et alpresent a laboratory study in which they examine the roleof the middle class on income distribution within the frame-work of a contest game

We hope that the selection of articles in this special issuewill help to inspire scholars in Sociology and Political Scienceto add mathematics to their tool box and adopt game theoryand experimentation in their research methodology

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 280789 2 pageshttpdxdoiorg1011552015280789

2 Journal of Applied Mathematics

Acknowledgment

We sincerely thank all the authors and reviewers who con-tributed greatly to the success of the special issue

Arthur SchramVincent BuskensKlarita Gerxhani

Jens Groszliger

References

[1] A Falk and J J Heckman ldquoLab experiments are a major sourceof knowledge in the social sciencesrdquo Science vol 326 no 5952pp 535ndash538 2009

[2] M Jackson and D R Cox ldquoThe principles of experimentaldesign and their application in sociologyrdquo Annual Review ofSociology vol 39 pp 27ndash49 2013

[3] R Corten and V Buskens ldquoCo-evolution of conventions andnetworks an experimental studyrdquo Social Networks vol 32 no1 pp 4ndash15 2010

[4] K Gerxhani J Brandts and A Schram ldquoThe emergenceof employer information networks in an experimental labormarketrdquo Social Networks vol 35 no 4 pp 541ndash560 2013

[5] J Groszliger and A Schram ldquoPublic opinion polls voter turnoutand welfare an experimental studyrdquo American Journal of Polit-ical Science vol 54 no 3 pp 700ndash717 2010

Research ArticleThe Strategic Role of Nonbinding Communication

Luis A Palacio1 Alexandra Corteacutes-Aguilar1 and Manuel Muntildeoz-Herrera2

1Escuela de Economıa y Administracion Universidad Industrial de Santander Calle 9 con 27 Bucaramanga Colombia2ICS Faculty of Behavioural Social Sciences University of Groningen Grote Rozenstraat 31 9712 TG Groningen Netherlands

Correspondence should be addressed to Manuel Munoz-Herrera emmunozherrerarugnl

Received 1 December 2014 Revised 27 March 2015 Accepted 28 March 2015

Academic Editor Jens Groszliger

Copyright copy 2015 Luis A Palacio et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper studies the conditions that improve bargaining power using threats and promises We develop a model of strategiccommunication based on the conflict game with perfect information in which a noisy commitment message is sent by a better-informed sender to a receiverwho takes an action that determines thewelfare of bothOurmodel captures different levels of aligned-preferences for which classical games such as stag hunt hawk-dove and prisonerrsquos dilemma are particular cases We characterisethe Bayesian perfect equilibrium with nonbinding messages under truth-telling beliefs and senderrsquos bargaining power assumptionsThrough our equilibrium selection we show that the less conflict the game has the more informative the equilibrium signal is andless credibility is necessary to implement it

1 Introduction

Bargaining power refers to the relative ability that a player hasin order to exert influence upon others to improve her ownwellbeing It is related also to idiosyncratic characteristicssuch as patience so that a player turns the final outcome intoher favour if she has better outside options or if she is morepatient [1] In addition Schelling [2] described bargainingpower as the chance to cheat and bluff the ability to setthe best price for oneself For instance when the union saysto the management in a firm ldquowe will go on strike if youdo not meet our demandsrdquo or when a nation announcesthat any military provocation will be responded with nuclearweapons it is clear that communication has been used with astrategic purpose to gain bargaining power

In bargaining theory strategic moves are actions takenprior to playing a subsequent game with the aim of changingthe available strategies information structure or payofffunctionsThe aim is to change the opponentrsquos beliefsmakingit credible that the position is unchangeable Following Selten[3] the formal notion of credibility is subgame perfectness(Schelling developed the notion of credibility as the outcomethat survives iterated elimination of weakly dominated strate-gies We know that in the context of generic extensive-formgames with complete and perfect information this procedure

does indeed work (see [4])) Nevertheless we argue that ifa message is subgame perfect then it is neither a threat nora promise Consider the following example a union says tomanagement ldquoIf you increase our salaries we will be grate-fulrdquo In such case credibility is not in doubt but we couldhardly call this a promise or a threat Schelling [2] denotesfully credible messages as warnings and we follow thisdifferentiation to threats and promises

Commitment theory was proposed by Schelling [2] (fora general revision of Schellingrsquos contribution to economictheory see Dixit [4] and Myerson [5]) who introduced atactical approach for communication and credibility insidegame theory Hirshliefer [6 7] and Klein and OrsquoFlaherty [8]worked on the analysis and characterisation of strategicmoves in the standard game theory framework In thesame way Crawford and Sobel [9] formally showed thatan informed agent could reveal his information in order toinduce the uninformed agent to make a specific choice

There are three principal reasons for modelling pre-play communication information disclosure (signalling)coordination goals (cheap-talk) and strategic influence (inSchellingrsquos sense) Following Farrell [10] and Farrell andRabin [11] the main problem in modelling nonbinding mes-sages is the ldquobabbling equilibriumrdquo where statements meannothing However they showed that cheap talk can convey



information in a general signalling environment displayinga particular equilibrium in which statements are meaningfulIn this line Rabin [12] developed credible message profileslooking for a meaningful communication equilibrium incheap-talk games

Our paper contributes to the strategic communicationliterature in three ways First we propose a particularcharacterisation of warnings threats and promises in theconflict game with perfect information as mutually exclusivecategories For this aim we first define a sequential protocolin the 2 times 2 conflict game originally proposed by Baliga andSjostrom [13] This benchmark game is useful because it isa stylised model that captures different levels of aligned-preferences for which classical games such as stag hunthawk-dove and prisonerrsquos dilemma are particular cases

Second we model strategic moves with nonbindingmessages showing that choosing a particular message and itscredibility are related to the level of conflict In this way theconflict game with nonbinding messages captures a bargainingsituation where people talk about their intentions by simplyusing cheap talk More precisely we analyse a game wherea second player (the sender) can communicate her actionplan to the first mover (the receiver) (To avoid confusionand gender bias the sender will be denoted as ldquosherdquo and thereceiver as ldquoherdquo) In fact the sender must decide after sheobserves the receiverrsquos choice but the commitment messageis a preplay move

Third we introduce a simple parameterisation that canbe used as a baseline for experimental research By means ofthismodel it is possible to study how in a bargaining environ-ment information and communication influence the powerone of the parts may have In other words this addressesthe following the logic supporting Nash equilibrium is thateach player is thinking given what the other does whatis the best he could do Players in a first stance cannotinfluence othersrsquo behaviour On the contrary Schelling [2]argues that players may consider what they can do as apreplay move to influence (ie manipulate) the behaviourof their counterpart and turn their payoffs in their favourTherefore our behavioural model provides a frameworkwhere it is possible to (experimentally) study the strategicuse of communication in order to influence others underdifferent levels of conflict

We analyse conceptually the importance of three essentialelements of commitment theory (i) the choice of a responserule (ii) the announcement about future actions and (iii) thecredibility of messages We answer the following questionswhat is the motivation behind threats and promises and canbinding messages improve the senderrsquos bargaining power Inthis paper threats and promises are defined as a secondmoverself-serving announcement committing in advance how shewill play in all conceivable eventualities as long as it specifiesat least one action that is not her best response (see [4 7])With this definition we argue that bindingmessages improvethe senderrsquos bargaining power in the perfect informationconflict game even when it is clear that by assuming bindingmessages we avoid the problem of credibility

The next step is to show that credibility is related tothe probability that the sender fulfills the action specified in

Table 1 The 2 times 2 conflict game

Player 2119889 ℎ

Player 1 119889 1 1 119909 119910

ℎ 119910 119909 025 025

the nonbinding message For this we highlight that playersshare a common language and the literal meaning mustbe used to evaluate whether a message is credible or notHence the receiver has to believe in the literal meaning ofannouncements if and only if it is highly probable to face thetruth Technically we capture this intuition in two axiomstruth-telling beliefs and the senderrsquos bargaining power We askare nonbinding messages a mechanism to improve the senderrsquosbargaining power and how much credibility is necessary for astrategic move to be successful In equilibrium we can provethat nonbinding messages will convey private informationwhen the conflict is low On the other hand if the conflictis high there are too strong incentives to lie and cheap talkbecomes meaningless However even in the worse situationthe nonbinding messages can transmit some meaning inequilibrium if the players focus on the possibility of fulfillingthreats and promises

The paper is organised as follows In Section 2 the conflictgame is described In Section 3 the conditioned messageswill be analysed and the definitions of threats and promisesare presented Section 4 presents the model with nonbind-ing messages showing the importance of response rulesmessages and credibility to improve the senderrsquos bargainingpower Finally Section 5 concludes

2 The 2 times 2 Conflict Game

The 2 times 2 conflict game is a noncooperative symmetricenvironment There are two decision makers in the set ofplayers119873 = 1 2 (In this level of simplicity playersrsquo identityis not relevant but since the purpose is to model Schellingrsquosstrategic moves in the following sections player 2 is going tobe a sender of commitment messages) Players must choosean action 119904

119894isin 119878119894= 119889 ℎ where 119889 represents being dove

(peaceful negotiator) and ℎ being hawk (aggressive negotia-tor)The utility function 119906

119894(1199041 1199042) for player 119894 is defined by the

payoffs matrix in Table 1 where rows correspond to player 1and columns correspond to player 2

Note that both mutual cooperation and mutual defectionlead to equal payoffs and the combination of strategies (119889 119889)is always Pareto optimal In the same way the combinationof strategies (ℎ ℎ) is not optimal and can only be understoodas the disagreement point Assuming that 119910 ge 119909 payoffs areunequal when a player behaves aggressively and the othercooperates given that the player who plays aggressively hasan advantage over hisher opponent In addition we willassume that 119909 = 025 and 119910 = 1 to avoid the multiplicityof irrelevant equilibria Therefore it will always be preferredthat the opponent chooses 119889 To have a parameterisation thatserves as a baseline for experimental design it is desirableto fix 119909 isin [0 05] and 119910 isin [05 15] within these intervals

Journal of Applied Mathematics 3

Table 2 Nash equilibria in the conflict game

(119904lowast

1 119904lowast

2) (119880

lowast

1 119880lowast

2) Pareto optimal

C1 (119889 119889) (1 1) Yes

C2 (119889 119889) (1 1) Yes(ℎ ℎ) (025 025) No

C3 (119889 ℎ) (119909 119910) Yes(ℎ 119889) (119910 119909) Yes

C4 (ℎ ℎ) (025 025) No

because if they are modelled as random variables withuniform distribution we would have four games with thesame probability of occurring

Under these assumptions the 2 times 2 conflict game hasfour particular cases that according to Hirshliefer [6] can beordered by their level of conflict or affinity in preferences

(1) Level of conflict 1 (C1) if 119910 lt 1 and 119909 gt 025 thereis no conflict in this game because cooperating is adominant strategy

(2) Level of conflict 2 (C2) if 119910 lt 1 and 119909 lt 025 thisis the so-called stag hunt game which formalises theidea that lack of trust may lead to disagreements

(3) Level of conflict 3 (C3) if 119910 gt 1 and 119909 gt 025depending on the history used to contextualise it thisgame is known as either hawk-dove or chicken gameBoth anticipation and dissuasion are modelled herewhere fear of consequences makes one of the partsgive up

(4) Level of conflict 4 (C4) if119910 gt 1 and119909 lt 025 this is theclassic prisoners dilemma where individual incentiveslead to an inefficient allocation of resources

Based on the system of incentives it is possible to explainwhy these games are ordered according to their level ofconflict from lowest to highest (see Table 2) In the C1game the playersrsquo preferences are well aligned and there isno coordination problem because the Nash equilibrium isunique in dominant strategies Therefore a rational playerwill always choose to cooperate 119889 which will lead to theoutcome that is Pareto optimal In the C2 game mutualcooperation (119889 119889) is a Nash equilibrium but it is not uniquein pure strategies The problem lies in coordinating on eithera Pareto dominant equilibrium (119889 119889) or a risk dominantequilibrium (ℎ ℎ) In other words negotiating as a doveimplies a higher risk and will only take place if a playerbelieves that the adversary will do the sameThis is the reasonwhy it is possible to state that lack of trust between the partiesmay lead to the disagreement point

The C3 game portrays an environment with higher levelsof conflict since there are two equilibriawith unequal payoffsIn other words players face two problems a distributive and acoordination one If only one of the players chooses to behaveaggressively this will turn the result in hisher favour but itis impossible to predict who will be aggressive and who willcooperate In this 2times2 environment there is no clear criterionto predict the final outcome and therefore the behaviour

Table 3 The conflict game illustrative cases

(a) C1 119910 minus 119909 = 0

2119889 ℎ

1 119889 1 1 05 05ℎ 05 05 025 025

(b) C2 119910 minus 119909 = 05

2119889 ℎ

1 119889 1 1 0 05ℎ 05 0 025 025

(c) C3 119910 minus 119909 = 1

2119889 ℎ

1 119889 1 1 05 15ℎ 15 05 025 025

(d) C4 119910 minus 119909 = 15

2119889 ℎ

1 119889 1 1 0 15ℎ 15 0 025 025

The last game is the classical social dilemma about thelimitations of rational behaviour to allocate resources effi-ciently The C4 game is classified as the most conflictive onebecause the players are facedwith a context where the rationalchoice clearly predicts that the disagreement point will bereached Additionally we will argue along this document thatchanging incentives to achieve mutual cooperation is not asimple task in this bargaining environment

Until this moment we have used equilibrium unicity andits optimality to argue that the games are ordered by theirlevel of conflict However it is possible to understand thedifference in payoffs (119910 minus 119909) as a proxy of the level of conflictIn other words the difference in payoffs between the playerwho takes the advantage by playing aggressively and theplayer who is exploited for cooperating is large we can statethat the incentives lead players to a preference to behaveaggressively (see the illustrative cases in Table 3)

3 Response Rules and Commitment Messages

We consider now the conflict gamewith a sequential decisionmaking protocol The idea is to capture a richer set ofstrategies that allows us to model threats and promises asself-serving messages In addition the set of conditionedstrategies include the possibility of implementing ordinarycommitment because a simple unconditional message isalways available for the sender

Schelling [2] distinguishes between two different typesof strategic moves ordinary commitments and threats Anordinary commitment is the possibility of playing first


d h

d h d h

1

1

x

y

y

x

025

025

P1

P2P2

u1

u2

Figure 1 The conflict game with perfect information

announcing that a decision has already been made and itis impossible to be changed which forces the opponent tomake the final choice On the other hand threats are secondplayer moves where she convincingly pledges to respond tothe opponentrsquos choice in a specified contingent way (see [7])

31TheConflict Gamewith Perfect Information Suppose thatplayer 1moves first and player 2 observes the action made byplayer 1 and makes his choice In theoretical terms this is aswitch from the 2 times 2 strategic game to the extensive gamewith perfect information in Figure 1 A strategy for player 2 isa function that assigns an action 119904

2isin 119889 ℎ to each possible

action of player 1 1199041isin 119889 ℎ Thus the set of strategies for

player 2 is 1198782= 119889119889 119889ℎ ℎ119889 ℎℎ where 119904

2= 11990421198891199042ℎrepresents

a possible reaction rule such that the first component 1199042119889

denotes the action that will be carried out if player 1 plays119889 and the second component 119904

2ℎis the action in case that 1

plays ℎ The set of strategies for player 1 is 1198781= 119889 ℎ

In this sequential game with perfect information a strat-egy profile is (119904

1 1199042) Therefore the utility function 119906

119894(1199041 1199042)

is defined by 119906119894(119889 11990421198891199042ℎ) = 119906

119894(119889 1199042119889) and 119906

119894(ℎ 11990421198891199042ℎ) =

119906119894(ℎ 1199042ℎ) based on the 2 times 2 payoff matrix presented before

As the set of strategy profiles becomes wider the predictionsbased on the Nash equilibrium are less relevant Thus inthe conflict game with perfect information the applicableequilibrium concept is the subgame perfect Nash equilibrium(SPNE)

Definition 1 (SPNE) The strategy profile (119904lowast1 119904lowast

2) is a SPNE

in the conflict game with perfect information if and only if1199062(1199041 119904lowast

2) ge 1199062(1199041 1199042) for every 119904

2isin 1198782and for every 119904

1isin 1198781

and 1199061(119904lowast

1 119904lowast

2) ge 1199061(1199041 119904lowast

2) for every 119904

1isin 1198781

The strategy 119904lowast2= 119904lowast

2119889119904lowast

2ℎrepresents the best response

for player 2 in every subgame In the same way the strategy119904lowast

1is the best response for player 1 when player 2 chooses119904lowast

2 By definition and using the payoffs assumptions it is

clear that the strategy 119904lowast2= 119909lowast

2119889119904lowast

2ℎis the unique weakly

dominant strategy for player 2 and in consequence thereason for player 1 to forecast his counterpartrsquos behaviourbased on the common knowledge of rationality The forecast

Table 4 SPNE in the conflict game with perfect information

(119904lowast

1 119904lowast

2) (119906

lowast

1 119906lowast

2) Pareto optimal

C1 (119889 119889119889) (1 1) YesC2 (119889 119889ℎ) (1 1) YesC3 (ℎ ℎ119889) (119910 119909) YesC4 (ℎ ℎℎ) (025 025) No

possibility leads to a first mover advantage as we can see inProposition 2

Proposition 2 (first mover advantage) If (119904lowast1 119904lowast

2) is a SPNE in

the conflict game with perfect information then 1199061(119904lowast

1 119904lowast

2) = 119909

and 1199062(119904lowast

1 119904lowast

2) = 119910

The intuition behind Proposition 2 is that there is anadvantage related to the opportunity of playing first which isthe idea behind the ordinary commitment In consequencethe equilibrium that is reached is that in favour of Player 1because he always obtains at least as much as his opponentThis is true except for the C4 game because the level ofconflict is so high that regardless of what player 1 chooseshe cannot improve his position The SPNE for each game ispresented in Table 4

We can see that the possibility to play a response ruleis not enough to increase player 2rsquos bargaining power Forthis reason we now consider the case where player 2 has thepossibility to announce the reaction rule she is going to playbefore player 1makes his decision

32 Threats and Promises as Binding Messages FollowingSchelling [14] the senderrsquos bargaining power increases if sheis able to send a message about the action she is going toplay since with premeditation other alternatives have beenrejected For the receiver it must be clear that this is theunique relevant option This strategic move can be imple-mented if it is possible to send bindingmessages about secondmoverrsquos future actions With this kind of communication weare going to show that there always exists a message thatallows player 2 to reach an outcome at least as good as theoutcome in the SPNE By notation 119898

2isin 1198782is a conditioned

message where 1198982= 11989821198891198982ℎ From now on player 2

represents the sender and player 1 the receiver

Definition 3 (commitment message) 119898lowast2isin 1198782is a commit-

ment message if and only if 1199062(119904lowast

1119898 119898lowast

2) ge 119906

2(119904lowast

1 119904lowast

2) where

1199061(119904lowast

1119898 119898lowast

2) ge 119906

1(1199041 119898lowast

2) for every 119904

1isin 1198781 It means 119904lowast

1119898is

player 1 best response given119898lowast2

The idea behind commitment messages is that player 2wants to achieve an outcome at least as good as the onewithout communication given the receiverrsquos best responseThis condition only looks for compatibility of incentivessince the receiver also makes his decisions in a rational wayFollowing closely the formulations discussed in Schelling[14] Klein and OrsquoFlaherty [8] and Hirshliefer [7] we classifythe commitment messages in three mutually exclusive cate-gories warnings threats and promises


Table 5 Commitment messages

Warning (119906lowast

1 119906lowast

2) Threat (119906

1 1199062) Promise (119906

1 1199062)

C1 (119889 119889119889) (1 1) (119889 119889ℎ) (1 1)C2 (119889 119889ℎ) (1 1) (119889 119889119889) (1 1)C3 (ℎ ℎ119889) (119910 119909) (119889 ℎℎ) (119909 119910) (119889 119889ℎ) (1 1)C4 (ℎ ℎℎ) (025 025) (119889 119889ℎ) (1 1)

Definition 4 (warnings threats and promises) (1) The com-mitment message119898lowast

2isin 1198782is a warning if and only if119898lowast

2= 119904lowast

2

(2) The commitment message 119898lowast2isin 1198782is a threat if and

only if 1199062(119889119898lowast

2) = 1199062(119889 119904lowast

2) and 119906

2(ℎ119898lowast

2) lt 1199062(ℎ 119904lowast

2)

(3)The commitment message119898lowast2isin 1198782is a promise if and

only if 1199062(119889119898lowast

2) lt 1199062(119889 119904lowast

2)

The purpose of a warning commitment is to confirm thatthe sender will play her best response after every possibleaction of the receiver Schelling does not consider warningsas strategic moves but we prefer to use it in this waybecause the important characteristic of warnings is their fullcredibility condition If agents want to avoidmiscoordinationrelated to the common knowledge of rationality they couldcommunicate it and believe it as well On the contrarycredibility is an inherent problem in threats and promisesThe second and third points in Definition 4 show that atleast one action in the message is not the best response afterobserving the receiverrsquos choice In threats the sender does nothave any incentive to implement the punishment when thereceiver plays hawk In promises the sender does not haveany incentive to fulfill the agreement when the receiver playsdove

The strategic goal in the conflict game is to deter the oppo-nent of choosing hawk because by assumption 119906

119894(119904119894 119889) gt

119906119894(119904119894 ℎ)This is exactly the purpose of these bindingmessages

as shown in Proposition 5

Proposition 5 (second mover advantage) If 2is a threat or

a promise in the conflict game with perfect information then119904lowast

1= 119889

The intuition behind Proposition 5 is that in Schellingrsquosterms if a player has the possibility to announce her inten-tions she will use threats or promises to gain an advantageover the first mover That is player 2 uses these messagesbecause if believed by player 1 she can make him cooperate

Proposition 6 specifies for which cases player 2 influencesplayer 1rsquos choices by means of threats and promises That isin which cases when player 1 has no incentives to cooperatemessages can prompt a change in his behaviour

Proposition 6 (message effectivity) There exists a commit-ment message 119898lowast

2such that 119906

2(119904lowast

1119898 119898lowast

2) gt 119906

2(119904lowast

1 119904lowast

2) if and

only if 119910 gt 1

Therefore threats and promises provide amaterial advan-tage upon the adversary only in cases with high conflict (egC3 and C4) Thus the condition 119910 gt 1 is not satisfiedin C1 and C2 cases where the level of conflict is low

The implication is that mutual cooperation is achieved inequilibrium and this outcome is the highest for both playersThe use of messages under these incentives only needs toconfirm the senderrsquos rational choice If player 2 plays 119898lowast =119904lowast

2 receiver can anticipate this rational behaviour which is

completely credibleThis is exactly the essence of the subgameperfect Nash equilibrium proposed by Selten [3]

An essential element of commitments is to determineunderwhat conditions the receivermust take into account thecontent of a message given that the communication purposeis to change the rivalrsquos expectations The characteristic of awarning is to choose the weakly dominant strategy but forthreats or promises at least one action is not a best responseProposition 6 shows that in the C3 and C4 cases the senderrsquosoutcome is strictly higher if she can announce that she doesnot follow the subgame perfect strategy We summarise thesefindings in Table 5

Up to this point we have considered the first two elementsof commitment theory We started by illustrating that themessages sent announce the intention the sender has toexecute a plan of action (ie the choice of a response rule)Subsequently we described for which cases messages areeffective (ie self-serving announcements) Now we inquireabout the credibility of these strategic moves because if thesender is announcing that she is going to play in an oppositeway to the game incentives this message does not change thereceiverrsquos beliefs The message is not enough to increase thebargaining power It is necessary that the specified action isactually the one that will be played or at least that the senderbelieves it The objective in the next section is to stress thecredibility condition It is clear that binding messages implya degree of commitment at a 100 level but this conditionis very restrictive and it is not a useful way to analyse areal bargaining situation We are going to prove that for asuccessful strategic move the degree of commitment must behigh enough although it is not necessary to tell the truth witha probability equal to 1

4 The Conflict Game withNonbinding Messages

The credibility problem is related to how likely it is that themessage sent coincides with the actions chosen The senderannounces her way of playing but it could be a bluff Inother words the receiver can believe in the message if it ishighly probable that the sender is telling the truth In orderto model this problem the game now proceeds as follows Inthe first stage Nature assigns a type to player 2 following aprobability distribution The senderrsquos type is her action planher way of playing in case of observing each of the possible


receiverrsquos action In the second stage player 2 observes hertype and sends a signal to player 1The signal is the disclosureof her plan and it can be seen as a noisy message because itis nonbinding In the last stage player 1 after receiving thesignal information chooses an actionThis choice determinesthe playersrsquo payoffs together with the actual type of player 2

Following the intuition behind crediblemessage profile inRabin [12] a commitment announcement can be consideredcredible if it fulfills the following conditions (i) Whenthe receiver believes the literal meanings of the statementsthe types sending the messages obtain their best possiblepayoff hence those types will send these messages (ii) Thestatements are truthful enough The enough comes from thefact that some types might lie to player 1 by pooling with acommitmentmessage and the receiver knows it However theprobability of facing a lie is small enough that it does not affectplayer 1rsquos optimal response

The objective of this section is to formalise these ideasusing our benchmark conflict game The strategic credibilityproblem is intrinsically dynamic and it makes sense ifwe consider threats and promises as nonbinding messagesBearing these considerations in mind from now on themessages are used to announce the senderrsquos intentions butthey are cheap talk Clearly negotiators talk and in most ofthe cases it is free but we show that this fact does not implythat cheap talk is meaningless or irrelevant

41 The Signalling Conflict Game Consider a setup in whichplayer 2moves first player 1 observes a message from player2 but not her type They choose as follows In the firststage Nature assigns a type 120579

2to player 2 as a function that

assigns an action 1199042isin 119889 ℎ to each action 119904

1isin 119889 ℎ

Player 2rsquos type set is Θ2= 1198782= 119889119889 119889ℎ ℎ119889 ℎℎ where

1205792= 11990421198891199042ℎ Nature chooses the senderrsquos type following a

probability distribution where 119901(1205792) gt 0 is the probability

to choose the type 1205792 and sum

1205792isinΘ2

119901(1205792) = 1 In the second

stage player 2 observes her own type and chooses a message1198982isin Θ2 At the final stage player 1 observes this message

and chooses an action from his set of strategies 1198781= 119889 ℎ

The most important characteristic of this conflict game withnonbinding messages is that communication cannot changethe final outcomeThough strategies aremore complex in thiscase the 2 times 2 payoff matrix in the conflict game is always theway to determine the final payoffs

In order to characterise the utility function we need somenotation A message profile 119898

2= (119898119889119889 119898119889ℎ 119898ℎ119889 119898ℎℎ) is a

function that assigns a message 1198982isin Θ2to each type 120579

2isin

Θ2 The first component 119898

119889119889isin 1198782is the message chosen in

case of observing the type 1205792= 119889119889 the second component

119898119889ℎisin 1198782is the message chosen in case of observing the type

1205792= 119889ℎ and so on By notation 119898

1205792

= 11990421198891199042ℎ

is a specificmessage sent by a player with type 120579

2 and119898

2= (1198981205792

119898minus1205792

) isa generic message profile with emphasis on the message sentby the player with type 120579

2

There is imperfect information because the receiver canobserve the message but the senderrsquos type is not observ-able Thus the receiver has four different information setsdepending on the message he faces A receiverrsquos strategy1199041119898= (119904

1119889119889 1199041119889ℎ 1199041ℎ119889 1199041ℎℎ) is a function that assigns

an action 1199041isin 1198781to each message 119898

2isin 1205792 where 119904

1119889119889is

the action chosen after observing the message 1198982= 119889119889 and

so on In addition 1199041119898= (1199041119898 1199041(minus119898)) is a receiverrsquos generic

strategy with emphasis on the message he faced In this casethe subindex 119898 is the way to highlight that the receiverrsquosstrategies are a profile of single actions Therefore in theconflict game with nonbinding messages the utility functionis 119906119894(11990411198981205792

1199041(minus1198981205792) 1198981205792

119898minus1205792

) = 119906119894(1199041 1199042) for 119904

11198981205792

= 1199041and

1205792= 1199042

In this specification messages are payoff irrelevant andwhat matters is the senderrsquos type For this reason it isnecessary to define the receiverrsquos beliefs about who is thesender when he observes a specific message The receiverrsquosbelief 120572

1205792|1198982

ge 0 is the conditional probability of obtainingthe message from a sender of type 120579

2 given that he observed

the message1198982 Naturally sum

1205792isinΘ2

1205721205792|1198982

= 1All the elements of the conflict game with nonbinding

messages are summarised in Figure 2 The most salient char-acteristics are the four information sets in which the receivermust choose and that messages are independent of payoffsFor instance the upper left path (blue) describes each possibledecision for the sender of type 119889119889 In the first place Naturechooses the senderrsquos type in this case 120579

2= 119889119889 In the

next node 119889119889 must choose a message from the 4 possiblereaction rules We say that 119889119889 is telling the truth if shechooses 119898

119889119889= 119889119889 leading to the information set at the

top We intentionally plot the game in a star shape in orderto highlight the receiverrsquos information sets At the end thereceiver chooses between 119889 and ℎ and cheap talk implies thatthere are 4 feasible payoffs

The signalling conflict game has a great multiplicity ofNash equilibria For this particular setting a characterisationof this set is not our aim Our interest lies on the character-isation of the communication equilibrium For this reasonthe appropriate concept in this case is the perfect Bayesianequilibrium

Definition 7 (PBE) A perfect Bayesian equilibrium is asenderrsquos message profile 119898lowast

2= (119898

lowast

119889119889 119898lowast

119889ℎ 119898lowast

ℎ119889 119898lowast

ℎℎ) a

receiverrsquos strategy profile 119904lowast1119898= (119904lowast

1119889119889 119904lowast

1119889ℎ 119904lowast

1ℎ119889 119904lowast

1ℎℎ) and a

beliefs profile 120572lowast120579119904|119898119904

after observing each message 1198982 if the

following conditions are satisfied

(1) 119898lowast2is the argmax

1198981205792isinΘ2

1199061205792

(119904lowast

1119898 1198981205792

119898minus1205792

)

(2) 119904lowast1119898

is the argmax1199041205792

1isin1198781

sum1205792isinΘ2

1205721205792|1198982

sdot 1199061(11990411198981205792

1199041(minus1198981205792)

119898lowast

2)

(3) 120572lowast1205792|1198982

must be calculated following Bayesrsquo rule basedon the message profile 119898lowast

2 For all 120579

2who play the

message119898lowast2 the beliefs must be calculated as 120572

1205792|119898lowast

2

=

1199011205792

sum119901119898lowast

2

The conditions in this definition are incentive compatibil-ity for each player and Bayesian updatingThe first conditionrequires message 119898lowast

1205792

to be optimal for type 1205792 The second

requires strategy 119904lowast1119898

to be optimal given the beliefs profile120572lowast

1205792|1198982

For the last condition Bayesian updating the receiverrsquos


u1

u2

d

h

d

h

d

h

d

h

x

y

x

y x

y

x

y

N

1 1

1

1

11

1

1

1

1

1 1

x y

x y

025 025

025

025

025

025

025 025

120572hh120572dh

120572dh

120572dh

120572dh

120572hd

120572hd

120572hd

120572hd

120572dd

120572dd

120572dd

120572dd

120579hh

120579dh

120579hd

120579dd

mhh

mdh

mhd

mdd

d

h

d

h

d

h

d

h

d h d h d h d h

d hd hd hd h

120572hh

120572hh120572hh

u1 u2

y x

y x

1 1

1 1

x y

x y

025 025

025 025

u1 u2

y x

y x

u1

u2 x

y

x

y x

y

x

y

1

1

1

1

025

025

025

025

Figure 2 Conflict game with nonbinding messages

beliefs must be derived via Bayesrsquo rule for each observedmessage given the equilibrium message profile119898lowast

2

42 The Commitment Equilibrium Properties There are ingeneral several different equilibria in the conflict game withnonbinding messages The objective of this section is to showthat a particular equilibrium that satisfies the followingproperties leads to a coordination outcome given it is bothsalient and in favour of the sender In what follows we willpresent Axioms 1 and 2 which will be used to explain whichis the particular equilibrium that can be used as a theoreticalprediction in experimental games with different levels ofconflict

Axiom 1 (truth-telling beliefs) If the receiver faces a messagelowast

2= 1205792 then 120572

1205792|2

gt 0 If the message 1198982= 1205792is not part

of the messages profile119898lowast2 then 120572

1205792|1198982

= 1

Following Farrell and Rabin [11] we assume that people inreal life do not seem to lie as much or question each otherrsquosstatements as much as the game theoretic predictions stateAxiom 1 captures the intuition that for people it is natural totake seriously the literal meaning of a message This does notmean that they believe everything they hear It rather statesthat they use the meaning as a starting point and then assesscredibility which involves questioning in the form of ldquowhywould she want me to think that Does she have incentives toactually carry out what she saysrdquo

More precisely truth-telling beliefs emphasise that inequilibrium when the receiver faces a particular messageits literal meaning is that the sender has the intention ofplaying in this way Thus the probability of facing truth-telling messages must be greater than zero In the sameway when the sender does not choose a particular messageshe is signalling that there are no incentives to make thereceiver believe this given that the receiverrsquos best responseis ℎ Therefore we can assume that the receiver must fullybelieve in the message because both players understand thatthe purpose of the strategic move is to induce the receiverto play 119889 If the sender is signalling the opposite she isshowing her true type by mistake then the receiver believesher with probability 1 (see the column ldquobelief of truth-tellingrdquoin Table 6)

Axiom 2 (sendersrsquo bargaining power) If 119898lowast1205792

is part of themessages profile119898lowast

2 then 119904lowast

11198981205792

= 119889

Axiom 2 captures the use of communication as a meansto influence the receiver to play dove That is there is anequilibrium where the only messages sent are those thatinduce the receiver to cooperate In order to characterise acommunication equilibrium such as the one described abovewe first focus on the completely separating message profilewhen the sender is telling the truth Naturally119898

1205792

is a truth-telling message if and only if119898

1205792

= 1205792(see column ldquomessage

by typerdquo in Table 6) and given the message the receiverrsquos best


Table 6 Perfect Bayesian equilibria that satisfy Axioms 1 and 2

Message by type Player 1rsquos best resp Belief of truth-telling(119898lowast

119889119889 119898lowast

119889ℎ 119898lowast

ℎ119889 119898lowast

ℎℎ) (119904

lowast

1119889119889 119904lowast

1119889ℎ 119904lowast

1ℎ119889 119904lowast

1ℎℎ) (120572

lowast

119889119889|119889119889 120572lowast

119889ℎ|119889ℎ 120572lowast

ℎ119889|ℎ119889 120572lowast

ℎℎ|ℎℎ)

C1 (119889119889 119889ℎ 119889119889 ℎℎ) (119889 119889 ℎ 119889) (119901119889119889

(119901119889119889+ 119901ℎ119889) 1 1 1)

C2 (119889119889 119889ℎ 119889119889 119889ℎ) (119889 119889 ℎ ℎ) (119901119889119889

(119901119889119889+ 119901ℎ119889)119901119889ℎ

(119901119889ℎ+ 119901ℎℎ) 1 1)

C3 (119889ℎ 119889ℎ ℎℎ ℎℎ) (ℎ 119889 ℎ 119889) (1119901119889ℎ

(119901119889119889+ 119901119889ℎ) 1119901ℎℎ

(119901ℎ119889+ 119901ℎℎ))

C4 (119889ℎ 119889ℎ 119889ℎ 119889ℎ) (ℎ 119889 ℎ ℎ) (1119901119889ℎ

(119901119889119889+ 119901119889ℎ+ 119901ℎ119889+ 119901ℎℎ) 1 1)

response will be to cooperate (see column ldquoplayer 1rsquos bestresponserdquo in Table 6)

With this in mind it is possible to stress that a contri-bution of our behavioural model is to develop experimentaldesigns that aim to unravel the strategic use of communi-cation to influence (ie manipulate) othersrsquo behaviour Thatis the Nash equilibrium implies that players must take theother playersrsquo strategies as given and then they look for theirbest response However commitment theory in Schellingrsquossense implies an additional step where players recognise thatopponents are fully rational Based on this fact they evaluatedifferent techniques for turning the otherrsquos behaviour intotheir favour In our case the sender asks herself ldquoThis is theoutcome I would like from this game is there anything I cando to bring it aboutrdquo

Proposition 8 (there is always a liar) The completely truth-telling messages profile119898

2= (119889119889 119889ℎ ℎ119889 ℎℎ) cannot be part of

any PBE of the conflict game with nonbinding messages

Proposition 8 shows that the completely truth tellingmessage profile is not an equilibrium in the conflict gameThe problem lies in the sender type ℎ119889 because revealing heractual type is not incentive compatible and there always existsat least one successful message to induce the counterpart toplay dove For this reason we can ask whether there existssome message that induces the sender to reveal her actualtype but at the same time leads to a successful strategic moveDefinition 9 is the bridge between nonbinding messages andcommitment messages presented in the previous section

Definition 9 (self-committing message) Let lowast1205792

be a truth-telling message and 120572

1205792|lowast

2(1205792)= 1 lowast

1205792

is a self-com-mitting message if and only if 119906

1205792

(119904lowast

1119898 lowast

1205792

lowast

minus1205792

) ge 1199061205792

(119904lowast

1119898

1198981205792

lowast

minus1205792

) for every1198981205792

isin Θ2

We introduce the self-committing message propertybecause we want to stress that a strategic move is a two-stage process Not only is communication useful in revealinginformation but also it can be used to manipulate othersrsquobehaviour The sender of a message must consider how thereceiver would react if he believes it and if that behaviourworks in her favour she will not have incentives to lieA message is self-committing and if believed it creates

incentives for the sender to fulfill it [12] The idea behind athreat or a promise is to implement some risk for the opponentin order to influence him but this implies a risk for thesender too This fact has led to associating strategic moveswith slightly rational behaviours when actually in order tobe executed a very detailed evaluation of the consequences isneeded Proposition 10 and its corollary explain the relationbetween the conditioned messages and the incentives to tellthe truth

Proposition 10 (incentives to commit) Let 2=

lowast

1205792

be a commitment message in the conflict game with perfectinformation If 119904lowast

1119898lowast(1205792)= 119889 then lowast

1205792

is a self-committingmessage

Corollary to Proposition 10 If 2is a threat or a promise in

the conflict game with perfect information then lowast1205792

= 2is a

self-committing message

The intuition behind Proposition 10 and its corollary isthat if a message induces the other to cooperate then thesender has incentives to tell the truthMoreover as illustratedin Proposition 5 threats and promises always induce thecounterpart to cooperate therefore they endogenously givethe sender incentives to comply with what is announced

As we can see in the conflict game with perfect information(for an illustration see Table 5) in the C1 and C2 casesthe warning is the way to reach the best outcome If weconsider the possibility to send nonbinding messages whenthe senderrsquos type is equal to awarning strategy then revealingher type is self-committing The problem in the C3 and C4cases is more complex given the warning message is not self-committing and the way to improve the bargaining poweris using a threat or a promise This fact leads to a trade-offbetween choosing a weakly dominant strategy that fails toinduce the opponent to play dove and a strategy that improvesher bargaining power but implies a higher risk for both ofthem

The required elements for a perfect Bayesian equilibriumat each game are shown in Tables 6 and 7 It is importantto bear in mind that the beliefs that appear in Table 7 arenecessary conditions for implementing the PBE presented inTable 6 given that they satisfy truth-telling beliefs and senderrsquosbargaining power


Table 7 Beliefs that support the perfect Bayesian equilibrium

Warning Threat Promise

C1 120572lowast

119889119889|119889119889ge

120572lowast

ℎ119889|119889119889(119910 minus 119909)

(1 minus 119910)Truth

C2 120572lowast

119889ℎ|119889ℎge

120572lowast

ℎℎ|119889ℎ(025 minus 119909)

075120572lowast

119889119889|119889119889ge

120572lowast

ℎ119889|119889119889(119910 minus 119909)

(1 minus 119910)

C3 Lie 120572lowast

ℎℎ|ℎℎge

120572lowast

ℎ119889|ℎℎ(119910 minus 119909)

(119909 minus 025)120572lowast

119889ℎ|119889ℎge

120572lowast

119889119889|119889ℎ(119910 minus 1)

075

C4 Lie 120572lowast

119889ℎ|119889ℎge

120572lowast

119889119889|119889ℎ(119910 minus 1) + 120572

lowast

ℎ119889|119889ℎ(119910 minus 119909) + 120572

lowast

ℎℎ|119889ℎ(025 minus 119909)

075

The problem of which message must be chosen is assimple as follows in the next algorithm first the sender tellsthe truth If the truth-tellingmessage leads the receiver to playdove then she does not have any incentive to lie In the othercase she must find another message to induce the receiverto play dove If no message leads the receiver to play dovemessages will lack any purpose and she will be indifferentbetween them

Table 6 shows the messages the receiversrsquo strategies andtheir belief profiles in a particular equilibrium we argue isthe most salient As we showed above in the conflict game(see Table 5) the sender is always in favour of those messageswhere the receiverrsquos best response is dove In the C1 casethere are three different messages in the C2 and C3 casesthere are twomessages and the worst situation is the C4 casewhere every type of player sends the same message This factleads to a first result if the conflict is high there are verystrong incentives to lie and communication leads to a poolingequilibrium

In addition notice that Table 5 specifies which messageswill be used as commitment messages in the conflict gamewith binding communication illustrated in Figure 1 That isif credibility is exogenous the theoretical prediction would bethat suchmessages are sentThis means that messages are notrandomly sent but there is a clear intention behind them toinduce the receiver to choose the action most favourable forthe sender Now Table 7 presents the minimum probabilitythreshold that can make the strategic move successful Thatis if credibility is sufficiently high the message works andachieves its purpose in the conflict game with nonbindingcommunication illustrated in Figure 2

In Section 3 we assumed that the sender could communi-cate a completely credible message in order to influence hercounterpart The question is how robust is this equilibriumif we reduce the level of commitment Proposition 11 sum-marises the condition for the receiver to choose dove as theoptimal strategy It is the way for calculating the beliefs thatare shown in Table 7

Proposition 11 (incentives to cooperate) 119904lowast1119898(1205792)= 119889 if and

only if (1minus119910)120572119889119889|119898(120579

2)+(075)120572

119889ℎ|119898(1205792)+(119909minus119910)120572

ℎ119889|119898(1205792)+(119909minus

025)120572ℎℎ|119898(120579

2)ge 0

Based on Proposition 11 the second result is that cheaptalk always has meaning in equilibrium We consider thatthis equilibrium selection is relevant because the sender

focuses on the communication in the literal meanings of thestatements but understands that some level of credibility isnecessary to improve her bargaining power Table 7 sum-marises the true enough property of the statements Here thereceiver updates his beliefs in a rational way and he chooses toplay dove if and only if it is his expected best responseWe caninterpret the beliefs in Table 7 as a threshold because if thiscondition is satisfied the sender is successful in her intentionof manipulating the receiverrsquos behaviour Thus some level ofcredibility is necessary but not at a 100 level

It is clear that if the conflict is high the commitmentthreshold is also higher In C1 and C2 cases the sender mustcommit herself to implement the warning strategy whichis a weakly dominant strategy In the C3 case the strategicmovement implies a threat or a promise formulating anaggressive statement in order to deter the receiver frombehaving aggressively The worst situation is the C4 casewhere there is only one way to avoid the disagreementpoint to implement a promise The promise in this game isa commitment that avoids the possibility of exploiting theopponent because fear can destroy the agreement of mutualcooperation

In the scope of this paper threats are not only pun-ishments and promises are not only rewards There is acredibility problem because these strategic moves imply alack of freedom in order to avoid the rational self-servingbehaviour in a simple one step of thinking The paradoxis that this decision is rational if the sender understandsthat her move can influence other playersrsquo choices becausecommunication is the way to increase her bargaining powerThis implies a second level of thinking such as a forwardinduction reasoning

5 Conclusions

In this paper we propose a behavioural model followingSchellingrsquos tactical approach for the analysis of bargainingIn his Essay on Bargaining 1956 Schelling analyses situationswhere subjects watch and interpret each otherrsquos behavioureach one better acting taking into account the expectationsthat he creates This analysis shows that an opponent withrational beliefs expects the other to try to disorient himand he will ignore the movements he perceives as stagingsespecially played to win the game

The model presented here captures different levelsof conflict by means of a simple parameterisation In


a bilateral bargaining environment it analyses the strategicuse of binding and nonbinding communication Our findingsshow that when messages are binding there is a first moveradvantage This situation can be changed in favour of thesecond mover if the latter sends threats or promises in apreplay move On the other hand when players have thepossibility to send nonbinding messages their incentives tolie depend on the level of conflict When conflict is low thesender has strong incentives to tell the truth and cheap talkwill almost fully transmit private information When conflictis high the sender has strong incentives to bluff and lieTherefore in order to persuade the receiver to cooperate withher nonbinding messages the sender is required to provide aminimum level of credibility (not necessarily a 100)

In summary the equilibrium that satisfies truth-tellingbeliefs and senderrsquos bargaining power allows us to show thatthe less conflict the game has the more informative theequilibrium signal is and the less stronger the commitmentneeded to implement it is Our equilibrium selection is basedon the assumption that in reality people do not seem to lieas much or question each otherrsquos statements as much asrational choice theory predicts For this reason the conflictgame with nonbinding messages is a good environment to testdifferent game theoretical hypotheses because it is simpleenough to be implemented in the lab

With this in mind the strategic use of communicationin a conflict game as illustrated in our model is the rightway to build a bridge between two research programsthe theory on bargaining and that on social dilemmas AsBolton [15] suggested bargaining and dilemma games havebeen developed in experimental research as fairly separateliteratures For bargaining the debate has been centred onthe role of fairness and the nature of strategic reasoningFor dilemma games the debate has involved the relativeweights that should be given to strategic reputation buildingaltruism and reciprocity The benefit of the structure andpayoff scheme we propose is to study all these elements atthe same time Our model provides a simple frameworkto gather and interpret empirical information In this wayexperiments could indicatewhich parts of the theory aremostuseful to predict subjectsrsquo behaviour and at the same time wecan identify behavioural parameters that the theory does notreliably determine

Moreover the game presented here can be a very usefultool to design economic experiments that can lead to newevidence about bilateral bargaining and furthermore abouthuman behaviour in a wider sense On the one hand itcan contribute to a better understanding of altruism self-ishness and positive and negative reciprocity A model thatonly captures one of these elements will necessarily portrayan incomplete image On the other hand bargaining andcommunication are fundamental elements to understand thepower that one of the parts can have

In further research we are interested in exploring theemotional effects of cheating or being cheated on particularlyby considering the dilemma that takes place when theseemotional effects are compared to the possibility of obtainingmaterial advantages To do so it is possible to even considera simpler version of our model using a coarser type space

(eg only hawk and dove) This could illustrate the existingrelationship between the level of conflict and the incentivesto lie As the model predicts the higher the level of conflictthemore incentives players have to not cooperate but they arebetter off if the counterpart does cooperateTherefore playerswith type hawk would be more inclined to lie and disguisethemselves as cooperators By measuring the emotionalcomponent of lying and being lied to we will be able toshow that people do not only value the material outcomes ofbargaining but that the means used to achieve those ends arealso important to them

Appendix

Proof of Proposition 2 Suppose that 1199061(119904lowast

1 119904lowast

2) = 119909 and

1199062(119904lowast

1 119904lowast

2) = 119910 then 119910 ge 1 If 119904lowast

2= ℎ119889 then 119906

1(119889 ℎ119889) ge

1199061(ℎ ℎ119889) and 119909 ge 119910 but by assumption 119910 gt 119909 If 119904lowast

2= ℎℎ

then 1199061(119889 ℎℎ) ge 119906

1(ℎ ℎℎ) and 119909 ge 025 and at the same time

1199062(ℎ ℎℎ) ge 119906

2(ℎ ℎ119889) The only compatible case is 025 = 119909

but by assumption 025 = 119909 Therefore 1199061(119904lowast

1 119904lowast

2) = 119909 and

1199062(119904lowast

1 119904lowast

2) = 119910

Proof of Proposition 5 Let 2be a threat or a promise

Following Definitions 3 and 4 1199062(119904lowast

1 lowast

2) ge 119906

2(119904lowast

1 119904lowast

2)

Suppose that 119904lowast1= ℎ then there are twopossibilities lowast

2= 119904lowast

2

or 1199062(ℎ lowast

2) ge 1199062(119904lowast

1 119904lowast

2) If lowast

2= 119904lowast

2 then by definition lowast

2is

neither a threat nor a promise If 1199062(ℎ lowast

2) ge 1199062(119904lowast

1 119904lowast

2) then

119904lowast

1= 119889 or 119904lowast

1= ℎ If 119904lowast

1= 119889 by assumption 119906

2(ℎ lowast

2) lt

1199062(119889 119904lowast

2) If 119904lowast

1= ℎ and

2is a threat then 119906

2(ℎ lowast

2) lt

1199062(119904lowast

1 119904lowast

2) If 119904lowast

1= ℎ and

2is a promise it must fulfill

1199062(ℎ lowast

2) ge 119906

2(ℎ 119904lowast

2) and 119906

2(119889 lowast

2) lt 119906

2(119889 119904lowast

2) The C1

and C2 games are not under consideration because 119904lowast1= 119889

and for C3 y C4 cases there are no messages for which theseconditions are true at the same time Therefore 119904lowast

1= 119889

Proof of Proposition 6 Let us consider the message 1198982=

119889ℎ By Proposition 2 we know that 1199062(119904lowast

1 119904lowast

2) = 119910 and

by assumption 1199061(119889 119889ℎ) gt 119906

1(ℎ 119889ℎ) then 119898

2= 119889ℎ is a

commitment message because 1199062(119889 119889ℎ) = 1 ge 119906

2(119904lowast

1 119904lowast

2)

If 1199062(119889 119889ℎ) gt 119906

2(119904lowast

1 119904lowast

2) then 1 gt 119906

2(119904lowast

1 119904lowast

2) to satisfy this

condition and using Proposition 2 again we conclude that(119904lowast

1 119904lowast

2) = (ℎ ℎ119904

lowast

2ℎ) As 119904lowast

2= ℎ119904lowast

2ℎand it is part of the SPNE

then 1199062(119889 ℎ119904lowast

2ℎ) gt 1199062(119889 119889119904lowast

2ℎ) and therefore 119910 gt 1

The proof in the other direction is as follows Let 119910 gt 1then 119904lowast

2= ℎ119904lowast

2ℎ Using Proposition 2 we know that 119906

1(119904lowast

1 119904lowast

2) =

119909 therefore 119904lowast1= ℎ Now 119906

2(119904lowast

1 119904lowast

2) lt 1 As we show in

the first part 1198982= 119889ℎ is a commitment message such

that 1199062(119904lowast

1119898 119889ℎ) = 1 Therefore there exists a commitment

message such that 1199062(119904lowast

1119898 119898lowast

2) gt 1199062(119904lowast

1 119904lowast

2)

Proof of Proposition 8 Consider the sendersrsquo types 120579119889ℎ=

119889ℎ and 120579ℎ119889= ℎ119889 If 119898lowast

2is a completely truth-telling

message then 120572lowast119889ℎ|119889ℎ= 1 and 120572lowast

ℎ119889|ℎ119889= 1 By assumptions

1199061(119889 1199041(minus119889ℎ) 119889ℎ119898

minus119889ℎ) = 1 and 119906

1(ℎ 1199041(minus119889ℎ) 119889ℎ119898

minus119889ℎ) =

025 then 119904lowast1119889ℎ= 119889 In the sameway 119906

1(119889 1199041(minusℎ119889) ℎ119889119898

minusℎ119889) =

119909 and 1199061(ℎ 1199041(minusℎ119889) ℎ119889119898

minusℎ119889) = 119910 then 119904lowast

1ℎ119889= ℎ Therefore

the utility for the sender is 119906119889ℎ(119889 1199041(minus119889ℎ) 119889ℎ119898

minus119889ℎ) = 1

and 119906ℎ119889(119889 1199041(minusℎ119889) ℎ119889119898

minusℎ119889) = 119909 These conditions imply

that the sender type ℎ119889 has incentives to deviate and 1198982=

(119889119889 119889ℎ ℎ119889 ℎℎ) cannot be part of any PBE


Proof of Proposition 10 Let 2= lowast

1205792

be a commitmentmessage in the conflict game with perfect information and119904lowast

1119898lowast(1205792)= 119889 If lowast

1205792

= 2is not a self-committing mes-

sage then another message 1198981205792

must exist such that1199061205792

(119889 1199041(minus119898lowast(120579

2)) lowast

1205792

119898lowast

minus1205792

) lt 1199061205792

(119904lowast

1119898 1198981205792

119898lowast

minus1205792

) Given thepayoff assumptions 119906

1205792

(119889 1199041(minus119898lowast(120579

2)) lowast

1205792

119898lowast

minus1205792

) ge 1199061205792

(119904lowast

1119898

119898lowast

1205792

119898lowast

minus1205792

) for every 119898lowast1205792

isin Θ2 Therefore lowast

1205792

= 2is a self-

committing message

Proof of Corollary to Proposition 10 Theproof to the corollaryfollows fromPropositions 5 and 10 and thus it is omitted

Proof of Proposition 11 The expected utility for each receiverrsquosstrategy is as follows

1199061(1198891198981205792

) = 1120572119889119889|119898(120579

2)+ 1120572119889ℎ|119898(120579

2)+ 119909120572ℎ119889|119898(120579

2)+

119909120572ℎℎ|119898(120579

2)

1199061(ℎ1198981205792

) = 119910120572119889119889|119898(120579

2)+ 025120572

119889ℎ|119898(1205792)+ 119910120572ℎ119889|119898(120579

2)+

025120572ℎℎ|119898(120579

2)

therefore 1199061(1198891198981205792

) ge 1199061(ℎ1198981205792

) if and only if (1 minus119910)120572119889119889|119898(120579

2)+ (075)120572

119889ℎ|119898(1205792)+ (119909 minus 119910)120572

ℎ119889|119898(1205792)+ (119909 minus

025)120572ℎℎ|119898(120579

2)ge 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was elaborated during the authorsrsquo stay at theUniversity of Granada Spain The authors are grateful forthe help and valuable comments of Juan Lacomba FranciscoLagos Fernanda Rivas Aurora Garcıa Erik KimbroughSharlane Scheepers and the seminar participants at theVI International Meeting of Experimental and BehaviouralEconomics (IMEBE) and the Economic Science AssociationWorld Meeting 2010 (ESA) Financial support from theSpanish Ministry of Education and Science (Grant codeSEJ2009-11117ECON) the Proyecto de Excelencia (Junta deAndalucıa P07-SEJ-3261) and the Project VIE-1375 from theUniversidad Industrial de Santander (UIS) in BucaramangaColombia is also gratefully acknowledged

References

[1] A Rubinstein ldquoPerfect equilibrium in a bargaining modelrdquoEconometrica vol 50 no 1 pp 97ndash109 1982

[2] T SchellingThe Strategy of Conflict Harvard University PressCambridge Mass USA 1960

[3] R Selten ldquoReexamination of the perfectness concept for equi-librium points in extensive gamesrdquo International Journal ofGameTheory vol 4 no 1 pp 25ndash55 1975

[4] A Dixit ldquoThomas Schellingrsquos contributions to game theoryrdquoScandinavian Journal of Economics vol 108 no 2 pp 213ndash2292006

[5] R B Myerson ldquoLearning from schellingrsquos strategy of conflictrdquoJournal of Economic Literature vol 47 no 4 pp 1109ndash1125 2009

[6] J Hirshliefer On the Emotions as Guarantors of Threats andPromises MIT Press Cambridge Mass USA 1987

[7] J Hirshliefer ldquoGame-theoretic interpretations of commitmentrdquoin Evolution and the Capacity for Commitment RMNesse EdRussell Sage Foundation New York NY USA 2001

[8] D B Klein and B OrsquoFlaherty ldquoA game-theoretic renderingof promises and threatsrdquo Journal of Economic Behavior andOrganization vol 21 no 3 pp 295ndash314 1993

[9] V P Crawford and J Sobel ldquoStrategic information transmis-sionrdquo Econometrica vol 50 no 6 pp 1431ndash1451 1982

[10] J Farrell ldquoMeaning and credibility in cheap-talk gamesrdquoGamesand Economic Behavior vol 5 no 4 pp 514ndash531 1993

[11] J Farrell and M Rabin ldquoCheap talkrdquo Journal of EconomicPerspectives vol 10 no 3 pp 103ndash118 1996

[12] M Rabin ldquoCommunication between rational agentsrdquo Journal ofEconomic Theory vol 51 no 1 pp 144ndash170 1990

[13] S Baliga and T Sjostrom ldquoArms races and negotiationsrdquoReviewof Economic Studies vol 71 no 2 pp 351ndash369 2004

[14] T Schelling ldquoAn essay on bargainingrdquoThe American EconomicReview vol 46 pp 281ndash306 1956

[15] G E Bolton ldquoBargaining and dilemma games from laboratorydata towards theoretical synthesisrdquo Experimental Economicsvol 1 no 3 pp 257ndash281 1998

Research ArticleIntermediaries in Trust Indirect ReciprocityIncentives and Norms

Giangiacomo Bravo1 Flaminio Squazzoni2 and Kaacuteroly Takaacutecs3

1Department of Social Studies Linnaeus University Universitetsplatsen 1 35195 Vaxjo Sweden2Department of Economics and Management University of Brescia Via San Faustino 74B 25122 Brescia Italy3MTA TK Lendulet Research Center for Educational and Network Studies (RECENS) Hungarian Academy of SciencesOrszaghaz Utca 30 Budapest 1014 Hungary

Correspondence should be addressed to Giangiacomo Bravo giangiacomobravolnuse

Received 2 September 2014 Accepted 4 December 2014

Academic Editor Vincent Buskens

Copyright copy 2015 Giangiacomo Bravo et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Any trust situation involves a certain amount of risk for trustors that trustees could abuse In some cases intermediaries exist whoplay a crucial role in the exchange by providing reputational information To examine under what conditions intermediary opinioncould have a positive impact on cooperation we designed two experiments based on a modified version of the investment gamewhere intermediaries rated the behaviour of trustees under various incentive schemes and different role structures We found thatintermediaries can increase trust if there is room for indirect reciprocity between the involved parties We also found that the effectof monetary incentives and social norms cannot be clearly separable in these situations If properly designed monetary incentivesfor intermediaries can have a positive effect On the one hand when intermediary rewards are aligned with the trustorrsquos interestinvestments and returns tend to increase On the other hand fixed monetary incentives perform less than any other incentiveschemes and endogenous social norms in ensuring trust and fairness These findings should make us reconsider the mantra ofincentivization of social and public conventional policy

1 Introduction

A trust relationship is an exchange where at least two partiesinteract that is a trustor and a trustee and in which there is acertain amount of risk for the former If the trustor decides toplace trust the trustee can honour or abuse it If honouringtrust is costly as what happened in one-shot interactions andsometimes even in repeated exchanges the trustee will haveno rational incentive to be trustworthy Knowing this thetrustor is hardly likely to make the first move [1]

Understanding how trust can be established in such hos-tile situations is of paramount importance One of the mostinteresting sociological explanations suggests that social andeconomic exchanges are embedded in social contexts wherecertain norms and roles have evolved to mediate betweenindividual interests For instance intermediaries might act asadvisories and mediators between the parties involved and

reputation or gossip can help to spread information aboutunknown partners that helps trustors to take the risk of inter-action [1ndash3]

Recent experimental results have shown that individualscan overcome distrust and cooperate more frequently ifbehaviour in the exchange is observed by a third party [4ndash8] This happens even when the opinion of a third party hasno consequence on the payoffs of the individuals and repu-tational building strategies are ruled out [9] This indicatesthat in many real situations third parties can reduce infor-mation asymmetry and temptations of free riding inducemutual trust and ensure collective benefit This requiresunderstanding why and under which conditions informationfrom third parties should be trusted by trustors and whattype of incentives canmake judgements or recommendationsby third parties credible to the eyes of trustors Indeedfirst intermediariesrsquo opinion is often subjective Secondly



it is formulated on trusteersquos past behaviour in situationswhere the trustees could have strategically mimicked trust-whorty signals in view of benefits from future trustors Thistransforms the problem of trust in a ldquosecondary problemof trustrdquo [10] Here the challenge is to understand underwhat conditions in anonymous exchanges with unknownpartners intermediaries could be considered reliable andhow they should be motivated to provide information thatincreases trust

For instance let us consider certain important socialand economic interactions such as the trust relationshipsbetween employees and managers in big companies orbetween two potential partners for a relationship In thesecases intermediaries are called on to express their opinionon certain attributes or behaviour which are crucial to createtrust Sometimes they do so voluntarily without receiving anymaterial payoffs such as someone recommending a friend asa partner to another friend In other cases intermediaries arefinancially motivated professionals such as an HR managerrecommending an employee to be upgraded to hishercompany manager Therefore while in certain spheres ofsocial life the social function of intermediaries has beeninstitutionalized through material incentives and roles inother situations informal or voluntary norms have developed

The aim of our paper was to examine these trust problemsin triadic relations (ie between trustors intermediariesand trustees) to better understand conditions that couldtransform the intermediary opinion in a trust carrier Weconducted two experiments where subjects played amodifiedversion of the repeated investment game with intermediariesadded to the typical trustortrustee dyadic relation Wemanipulated incentives and roles of intermediaries to testtheir impact on cooperation in particularly hostile condi-tions For this we meant a situation where (i) intermediariesformulated their opinion on the trustee behaviour on a lim-ited set of information and (ii) their opinion was not publicand (iii) did not have long-term consequences on thematerialpayoffs of the trustees In this situation intermediaries hadonly a limited set of information (i) and bad standing was notso risky for trustees (ii-iii) Hence our experimental situationwas intentionally designed so that intermediary opinion waspoorly credible for trustors We examined the importance ofindirect reciprocity considerations and their interplay withmaterial incentives and social norms Furthermore we testeddifferent ways to align intermediaryrsquos incentives for examplerespectively with the trustorsrsquo or the trusteesrsquo interests andthe impact of rolesrsquo rotation

We compared results from two laboratory experimentsExperiment 1 first reported in [11] and Experiment 2previously unpublished The latter was designed to carefullyexamine the effect of indirect reciprocity on the behaviourof intermediariesmdashand more generally on trust at thesystem levelmdashsince this mechanism was crucial in the firstexperiment The rest of the paper introduces our researchbackground and hypotheses (Section 2) presents the designof the two experiments (Section 3) and their outcomes(Section 4) and finally discusses certain social and policyimplications of the results (Section 5)

2 Hypotheses

While dyadic embeddedness is important to explain trustand cooperation in situations involving stable relationshipsbetween two actors inmodern society trust is oftenmediatedby agents who facilitate the exchange between trustors andtrustees when they cannot rely on past experience [1 12 13]Important examples of this have been empirically found inthe development of trust between suppliers and customers ina variety of situations such as in electronic markets on theweb [14 15] and in theUS venture capitalmarket [16] In thesecases any chance for trust is undermined by informationasymmetry between the involved parties and does not simplyimply all-none choices such as trusting or not but requires apondered rational calculus of trust investment

Let us consider the investment game a typical frameworkto model trust problems [17] First Player A (the trustor)receives an initial endowment of 119889A points with a fixedexchange rate in real money Then A is called to decide theamount 119868 between 0 and 119889A to send to Player B (the trustee)keeping the part (119889A minus 119868) The amount sent by A is multipliedby 119898 gt 1 and added to Brsquos own endowment 119889B (Note thatin some investments games including [17] B players hadno endowment that is 119889B = 0 Trustor investments tendto be lower in experiments where 119889B gt 0 [18]) Then B iscalled to decide the share of the amount received (plus hisherendowment) to return toA As before the amount119877 returnedbyB can be between 0 and (119889B+119898119868)The amount not returnedrepresents Brsquos profit while 119877 is summed to the part kept by Ato form hisher final profit Therefore the playerrsquos profit is asfollows

119875A = 119889A minus 119868 + 119877

119875B = 119889B + 119898119868 minus 119877(1)

The structure of the game implies that the trustor can havean interest in investing on condition that the expected returnsare higher than hisher own investments that is if119877 = 119902(119889B+119898119868) gt 119868 where 119902 is the proportion returned by the trusteeHowever as the trustor can rationally presume that thetrustee has no interest in returning anything it is rationallyexpected that the trustor will not invest and trust will notbe placed giving rise to a suboptimal collective outcome of(119889A + 119889B) while the social optimum (119898119889A + 119889B) could havebeen reached with sufficient high levels of trust Howeverempirical evidence contradicts this prediction A recentreview of 162 experimental replications of the investmentgame indicated that on average trustors invested about 50(range 22ndash89) of their endowment and trustors returned37 (range 11ndash81) of their amount [18]

In this type of games the crucial challenge is to under-stand the mechanisms through which the trustor estimatesthe trustworthiness of the trustee by using available infor-mation Following Coleman [1] we can identify two possibleinformation sources for trustors (i) direct knowledge ofpast behaviour of the trustee and (ii) knowledge of trusteebehaviour obtained by a third party with positions andinterest differently aligned to those of the other partiesinvolved In both cases information on past behaviour of


the trustee may help the trustor in predicting the trustwor-thiness of the trustee creating in turn reputational incentivesfor the trustee to overcome any temptation of cheating in viewof future benefits

While it is widely acknowledged that knowing the pastbehaviour of a trustee can increase a trustorrsquos investmentand cooperation in dyadically embedded interaction [19ndash23]the case of triadic relations is more interesting as theserelationships can compensate for the lack of direct knowledgeand contacts between actors so enlarging the social circlesand the extent of the exchange [24] This requires us tounderstand the complex triangulation of the exchange andespecially the role of trust intermediaries who might haveeither analogous or different positions and interests to thetrustors When positions and interests between the trustorand the intermediary are aligned the intermediary can actin the trustorrsquos interest and the latter seriously considers theintermediaryrsquos opinion so that hisher decision will reflect theavailable reputational information When the intermediaryand the trustor do not have aligned interests the outcome ofthe exchange is heavily dependent on the motivations behindthe intermediaryrsquos action [1]

In order to represent triadic relations in trust situationswe modified the standard investment game framework byadding to the trustor (called Player A) and the trustee (calledPlayer B) a third player that is the intermediary (called PlayerC) We assumed that the intermediary could observe B pastbehaviour (ie the amount of returns sent to A comparedto the A investment) and was called to decide whether toprovide A with honest and accurate information or not Ifcredible to the eye of A information by C is expected tohelp A regulate hisher investment If this is so B Playershave a rational interest in building a good standing at theeye of C and so behave more fairly with A Therefore ifintermediaries are trusted both by the trustors and by thetrustees it is expected to generate higher levels of cooperationand fairness

When modelling intermediariesrsquo behaviour for the sakeof simplicity we assumed that C could only choose betweentwo levels of fairness providingAwith an accurate evaluationof B (=high fairness) or sharing inaccurate or deceptiveinformation (=low fairness) Adapting Frey and Oberholzer-Geersquos ldquomotivation crowding-outrdquo formalization [25] we alsoassumed that each intermediary chooses a level of fairness (119891)that maximizes hisher expected net benefit as follows

max [119901 (119891) (119887 + 119890) + 119889 (119890 119891) minus 119896] (2)

Since 119898 gt 1 any increase of investments leads to higherstakes to share and a higher level of fairness is beneficialto everybody In particular 119887 indicates the expected benefitfor C due to the aggregate fairness while 119890 indicates hisherexpected private earnings Given that all these figures areexpected the intermediariesrsquo benefits are weighted by theprobability 119901 to reach a given level of fairness 119891 The more Cplays fair the more heshe contributes to providing a contextfor fairness (1199011015840

119891gt 0)

It is worth noting that a crucial component of the modelis the intrinsic motivation of subjects (119889) This is expected

to increase with the overall level of fairness (1198891015840119891gt 0) and

to decline with private material earnings due to crowding-out effects (1198891015840

119890lt 0) Finally we assumed that a fixed small

but strictly positive cost 119896 of fair behaviour existed duethe cognitive requirements by C (eg information searchmemory and time) to perform a thoughtful evaluation of B

Assuming that intermediaries choose the level of fairness119891lowast maximising their expected benefit we derived the first

order condition as follows1199011015840

119891(119887 + 119890) + 119889

1015840

119891= 0 (3)

It is important to note that intermediaries could seea benefit 119887 from higher levels of fairness only if they areexpected to play as A or B in the future that is if roles arerotating If roles are fixed 119887 = 0 and the intermediaryrsquosdecision depends only on the personal earnings 119890 and theintrinsic motivation 119889 Given (3) ceteris paribus a situationwhere 119887 = 0 is expected to lead to a lower 119891lowast than whereintermediaries can derive benefits from the aggregate level offairness by playing other roles in the future

It is worth noting that this can be framed in terms ofindirect reciprocity [23 26 27] That is the idea of benefitingunknown trustors by punishing self-interested trustees tokeep the fairness standards high and benefit from the reci-procity of other reliable intermediaries when cast as trustors(ie 119887 gt 0)This can induce intermediaries to provide reliableinformation and investors to trust reputational informationThis concatenation of strategies cannot work in a systemwhere interaction roles are fixed given that intermediariescannot expect benefits from their evaluation as future trustors(ie 119887 = 0) In this case trustors cannot expect thatintermediaries provide a careful evaluation

This led us to formulate our first hypothesis as follows

Hypothesis 1 If roles of the interaction are fixed indirectreciprocitymotives cannotmotivate positive behaviour by theparties involved who will not see future benefits in keepingthe levels of fairness highThis implies that ceteris paribus thefixed role condition will decrease fairness and cooperation

The situation is different when the interests of interme-diaries and trustors are aligned In this case intermediariesreceive a direct payoff 119890 gt 0 by behaving fair Indeedthis interest alignment transforms the trust relationship in atypical principal-agent model where the intermediary (theagent) can behave in the interest of the trustor (the principal)In this case the rational choice theory predicts that monetaryincentives are crucial to motivate the intermediary to act onthe trustorrsquos behalf by guaranteeing that the self-interest ofthe former coincides with the objectives of the latter [28]The fact that the negative effect of monetary incentives onintrinsic motivations (1198891015840

119890lt 0) could be compensated by

higher levels of fairness (1198891015840119891gt 0) will lead to higher levels

of fairness and cooperation compared to the 119890 = 0 caseThis led us to formulate our second hypothesis as follows

Hypothesis 2 If the intermediary responds to monetaryincentives that are aligned with the trustorrsquos interests higherlevels of fairness will lead to higher cooperation in the system


In the symmetric case where monetary incentives arealigned with the trusteersquos interests intermediaries will receivea personal payoff from being unfair Note also that inthis case not only do the incentives crowd out intrinsicmotivations but also119889declines due to expected lower levels offairnessThis is why the intermediary is in a potential conflictof interest as heshe may be tempted to cheat the trustor byproviding opinions that benefit the trustee Knowing thisthe trustor could be induced to question the reliability ofthe intermediaryrsquos opinion and decide not to enter into theexchange or reduce hisher investment tominimal levelsThisis expected to erode the basis of cooperation

This led us to formulate our third hypothesis as follows

Hypothesis 3 When the intermediaryrsquos incentives are alignedwith the trusteersquos interests the levels of fairness will declineleading to lower cooperation in the system in comparisonwith the situation where no monetary incentives exist

A particular case is when intermediaries receive mon-etary incentives that are independent of their actions andthe level of fairness in the system as in the case of fixedrewards In this case 119890 will not enter the benefit calculus asexpressed by the first term of (2) as it will be earned in allcases while the incentive will decrease the intermediariesrsquointrinsic motivation because of 1198891015840

119890lt 0 leading to low levels

of fairness This will make the intermediaryrsquos opinion poorlycredible for both the trustor and the trustee

This led us to formulate our fourth hypothesis as follows

Hypothesis 4 With intermediaryrsquos incentives that are fixedand independent from the interests of both the trustor and thetrustee the levels of fairness and cooperation in the systemwill be lower than in case of no monetary incentives

Without any material interest in the exchange (119890 = 0)intermediaries intrinsic motivations are expected to increasewith 119891 As long as roles change over time being 119887 gt 0and 1199011015840

119891gt 0 subjects are expected to personally benefit

from higher levels of fairness in the system Furthermore theabsence ofmonetary incentives can transform the interactioninto a moral problem with intermediaries induced to punishmisbehaviour by trustees even more than in other incentiveschemes [29 30]

This led us to formulate our last hypothesis as follows

Hypothesis 5 With alternating roles and without monetaryincentives for intermediaries fairness will increase leading tohigh levels of cooperation in the system

3 Methods

To test our hypotheses we built two experiments basedon a modified version of the repeated investment gamedescribed above with 119889A = 119889B = 10MU and 119898 = 3 andwith the restriction of choices to integer amounts (see theAppendix for a detailed description of the experiments) Weextended the original dyadic game towards a third-partygame where we introduced intermediaries (Players C) not

directly involved in the transaction but asked to rate trusteesrsquobehaviour (Players B) for the benefit of the trustors (PlayersA) The opinion of Players C was formulated individuallyand was shared with both Players A and B involved in theexchange When selected as a C Player the subject wasmatched with one Player A and one Player B and privatelyinformed of the amount received and returned by the latterin the previous period Then heshe was asked to rate PlayerBrsquos behaviour as ldquonegativerdquo ldquoneutralrdquo or ldquopositiverdquo Hisheropinion was privately displayed to Player A before hisherinvestment decision The rest of the game worked as thestandard dyadic version described above Note that anycommunication between subjects was forbidden and subjectsplayed anonymously with possible partners from a large poolof subjects

In the first experiment game roles (ie trustor trusteeand intermediary) alternated regularly throughout the gamewith the same subject playing the same number of timesin the three roles with randomly determined partners [11]The second experiment followed the same design except thatroles were fixed throughout the gameThe aim of this secondexperiment was to rule out any (indirect) reciprocity motivefrom the intermediary behaviour as the intermediary nowcould not provide reliable information to expect good futureinformation in turn (in the trustorrsquos role)

More specifically in both experiments monetary incen-tives for intermediaries systematically varied across treat-ments according to our hypotheses In the No incentivetreatment intermediaries did not receive any rewards fortheir task also losing potential earnings as trustors or trusteeswhen selected to play in this role In the Fixed incentivetreatment intermediaries received a fixed payoff of 10MUequal to the trustor and trustee endowments In the Aincentive treatment intermediaries earningswere equal to thepayoff obtained by the trustors they advised In the B incentivetreatment intermediariesrsquo earnings were equal to the payoffobtained by the trustees they rated

4 Results

41 Experiment 1 (Alternating Roles) Our experiment pro-duced investments and returns comparable to previousexperiments in the Baseline [18] and consistent with previ-ous studies which introduced reputational motives [9 22]showed that the presence of the intermediaries dramaticallyimproved cooperation

A total of 136 subjects (50 female) participated in theexperiment held in late 2010 Both investments and returnswere higher when intermediaries were introduced withinvestments increasing from an average of 322MU in theBaseline up to 521MU in A incentive and returns rising from200 in the Baseline to 687 in No incentive (Figure 1) (Ourdataset may be accessed upon request to the correspondingauthor All statistical analyses were performed using the 1198772151 platform [31] Please note that the amounts exchangedin the first three periods of the game when intermediarieshad no previous information to evaluate and in the last threeperiods when trustees knew that no further ratingwould takeplace were not included in the analysis)


A investments

2

3

4

5

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(a)

B returns

1

2

3

4

5

6

7

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(b)

Cooperation index

045

05

055

06

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(c)

Figure 1 Experiment 2 results Average investments returns and CI by treatment Bars show the standard error of the means

Differences with the Baseline for both investments andreturns were significant at the 5 level for all treatmentsexcept Fixed incentive where the difference was significantonly for returns Significant differences also existed for PlayerB returns Both No incentive and A incentive led to higherreturns than Fixed incentive (Wilcoxon rank sum tests onindividual averages 119882 = 5310 119875 = 0002 one tailed and119882 = 1990 119875 = 0002 one tailed resp) There were nosignificant differences between No incentive and A incentive(119882 = 3850 119875 = 0365) Differences in investments were notconsiderable but still statistically significant at 5betweenNoincentive and Fixed incentive (119882 = 5080 119875 = 0006 onetailed) and between A incentive and Fixed incentive (119882 =1765 119875 = 0001 one tailed)

To compare the outcome of the different treatments bet-ter we built an indicator that considered both social welfareand fairness which are two aspects strictly linked with coop-eration due to the social dilemma nature of the investmentgame [32] Indeed in any investment game trust is a double-edged sword On the one hand it includes trustors who arebetter-off by placing trust and so can increase the welfare ofthe system by taking the risk of interaction On the otherhand it also includes trustees who can act in a more orless trustworthy way so contributing to giving rise to afair exchange Following [11] in order to look at these twosides of the this process we built an index that combinedattention to the social welfare which only depended on Ainvestments and fairness which depended on B returnswhich we called cooperation index (CI) The level of socialwelfare is an important indicator of the system efficiencyin the different treatments This is indicated by 119864 = 119868119889Awhere 119868 was A investment and 119889A was the endowment Thisindicator took 0 when A invested nothing and 1 when Ainvested the whole endowment Following previous research[33ndash36] fairness was calculated by comparing the differencein the payoffs to the sum of monetary gains 119865 = 1 minus [|119875A minus119875B|(119875A +119875B)] where 119875A and 119875B were the payoffs earned by A

and B Players respectivelyThis was 0when one of the playersobtained the whole amount at stake and the other receivednothing while it was 1 when both players obtained the samepayoff Note that no trade-off exists between maximising 119864and119865 given that for any level of investment (including zero)only the amount returned determined the fairness payoffThecooperation index was defined as CI = (119864 + 119865)2 This was0 when A Players invested nothing and B Players returnedall their endowments grew with the growth of A investmentsand a fairer distribution of final payoffs and became 1 whenAPlayers invested 119889A and B Players returned half of their totalendowment that is (119889B + 119898119868)2

The treatmentwith the highest CIwasNo incentive whichled to more fairness than any other treatment (Figure 1)Differences in the CI were statistically significant at 10 withA incentive and at 5 with all other treatments The highCI value in No incentive was especially important as in thiscase unlike A incentive intermediaries had no monetaryincentive to cooperate with trustorsThis would confirm thatthe lack of monetary incentives for intermediaries impliedhigher normative standards of behaviour for the other actorsinvolved

42 Experiment 2 (Fixed Roles) A total of 244 subjects (55females) participated in the second experiment which wasorganized in 2011 Results showed that trust and trustwor-thiness were generally lower than in the first experimentAverage investments ranged from 236MU in Fixed incentiveto 396MU in A incentive Returns ranged from 247MU inFixed incentive to 527MU in A incentive The cooperationindex ranged from 0507MU in the Baseline to 0557MU inA incentive (Figure 2)

The only treatment leading to investments significantlyhigher than the Baseline was A incentive (119882 = 82 119875 = 0049one tailed) while returnswere significantly higher (at the 10level) in both A incentive and B incentive (119882 = 88 119875 = 0071


A investments

1

2

3

4

5

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(a)

B returns

1

2

3

4

5

6

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(b)

Cooperation index

045

05

055

06

MU

Base

line

No

ince

ntiv

e

Fixe

d in

cent

ive

A in

cent

ive

B in

cent

ive

(c)


one tailed for both treatments) Finally the cooperation indexwas significantly higher than the Baseline in No incentive(119882 = 351 119875 = 0019 one tailed) A incentive (119882 = 299119875 = 0003 one tailed) and B incentive (119882 = 312 119875 = 0004)but not in Fixed incentive (119882 = 462 119875 = 0288 one tailed)

43 Comparing the Two Experiments A comparison betweenalternating versus fixed role treatments highlighted the effectsof indirect reciprocity (Table 1) Besides lower returns inthe Baselinemdasha result consistent with the existing literature[18]mdashinvestments and returns were generally higher in thealternating roles experiment This difference was especiallyrelevant in No incentive This was one of the best performingtreatments in the first experiment while in fixed role treat-ments investments and returns were similar to the Baseline(ie without intermediaries) On the other hand the twotreatments with sound monetary incentives led to a moremodest decrease in investment and especially in returnsNote also that B incentive performed similarly in the twoexperiments

In order to examine the effect of the different factorsinvolved in the exchange we estimated a random effectsmodel using dummies indicating the fixed role experi-ment each treatment the first and the last period andthe second half of the game as regressors (Table 2) Dueto the considerable interdependence between the observeddecisions random effects (multilevel) regression analysis wasperformed that considered the nestedness of observations atthe individual level Results showed that fixed roles caused anoverall decrease of trust but had less effect on trustworthinessA incentive had a stronger impact on trustee behaviour whileNo incentive had a stronger impact on CI that is on thefairness of the exchange All conditions except Fixed incen-tives led to higher trust and trustworthiness than the BaselineFinally all cooperation indicators declined during the gameconfirming the typical ldquoend effectrdquo found in previous studies[9 22 37]

If we consider the behaviour of the intermediaries wecan see that they generally played fairly asking B Playersto return significantly more to award a more positive rating(Table 3(a)) They were more demanding where trust washigher namely inNo incentive andA incentive of the alternat-ing roles experiment In both cases to award a positive ratingto trustees intermediaries asked trustees to return about one-third of the trusteesrsquo endowments On the other hand in lesscooperative treatments such asNo incentive or Fixed incentivein the fixed roles experiment they were less demanding (ieto one-sixth and even less) Moreover negative ratings weremore frequent in the most cooperative treatments Althoughonly descriptive these results suggest that a more rigorousreputation process took place in these conditions despite theoverall higher trustworthiness of trustees achieved in thesetreatments (Table 3(b))

It is also important to note that trustor investmentsreflected the behaviour of the intermediary Generallytrustors invested more when their opponents received pos-itive ratings less in cases of neutral ratings and even lessin cases of negative ratings (Table 4) It is worth notingthat also in this case there were no important differencesacross treatments Results suggest that trustors trusted inter-mediaries more when the latter gave positive ratings in Noincentives and A incentive in the alternating roles experimentIn these cases they invested on average 625MUand 704MUrespectivelyThey invested less in treatments where there waslow trust such as Fixed incentive in the fixed role conditionwhere average investment of trustors was 243MU evenwhentrustees had a positive rating

5 Discussion

Our experiments have highlighted the crucial role of inde-pendent judges in trust situations even when their opinionscould be viewed as subjective andor without serious long-term future consequences for the trustee We found that


Table 1 Overview of Experiments 1 and 2 results The ARFR column presents the ratio between the corresponding results in the alternatingand the fixed role experiment The last two columns present Wilcoxon rank sum tests of the null hypothesis that the result distribution is thesame in the two experiments

Treatment Altern roles Fixed roles ARFR WilcoxonMean SE Mean SE 119882 119875

Baseline

A investments 322 016 307 016 105 2075 0385B returns 200 016 354 026 056 1425 0079

B ret (prop) 009 001 016 001 056 1370 0059CI 050 000 051 000 098 3190 0118

No incentive


B ret (prop) 024 001 014 001 171 1890 0006CI 063 001 054 000 117 3960 0000

Fixed inc


B ret (prop) 017 001 010 001 170 1500 0153CI 054 001 052 000 104 2640 0319

A incentive


B ret (prop) 023 001 021 001 110 1340 0333CI 060 001 056 001 107 3650 0002

B incentive


B ret (prop) 019 001 018 001 106 1300 0387CI 058 001 055 000 105 3070 0071

Table 2Multilevel regression analysiswith individual-level randomeffects Significance codes lowastlowastlowast119875 lt 0001 lowastlowast119875 lt 001 lowast119875 lt 005dagger119875 lt 01

Dependent Investments Returns CI(Intercept) 3607

lowastlowastlowast 0154 0516lowastlowastlowast

Fixed roles minus0962lowastlowastlowast

minus0133 minus0030lowastlowastlowast

No incentive 1339lowastlowast

1509lowastlowast

0090lowastlowastlowast

Fixed incentive 0040 0684 0030lowast

A incentive 1538lowastlowastlowast



B incentive 1075lowastlowast

1012dagger


First period 0066 0954lowastlowastlowast

0024lowast

Last period minus0562lowastlowast

minus0848lowastlowastminus0031

lowastlowast

Periods 16ndash30 minus0313lowastlowastlowast

minus0420lowastlowastlowastminus0010

lowast

A investment 0853lowastlowastlowast

Random effects120590 (id) 1827 2553 0054120590 (residual) 2314 2971 0118Number of interactions 4070 4070 4070Number of individuals 222 222 222119865 215

lowastlowastlowast3207


lowastlowastlowast

somewhat counterintuitively the intermediaryrsquos opinions hada stronger effect when the intermediary had no materialinterest from the exchange If there was room for indirectreciprocity between the parties involved triadic relationshipscould provide a moral base to overcome the negative traps

of self-interest We also found that monetary incentives andmoral action were not clearly separable or substitutes [38]If properly designed monetary incentives for intermediarieshad a positive effect especially on trustorrsquos investmentsbut were less effective than social norms and reciprocity inensuring fair exchanges in terms of more equal distributionof payoffs between trustors and trustees

First the ldquoshadow of reciprocityrdquo implied that interme-diaries kept the standards of evaluation high to benefit goodinformation in turn by other intermediaries when acting astrustors This in turn induced trustees to be more reliableUnder evaluation trustees overestimated the future impactof their good standing and behaved more fairly On the otherhand having information about trustee behaviour inducedtrustors to predict higher cooperative responses by trusteesand increase their investment It is worth noting that recentwork showed that thismechanism is relatively independent ofthe quality of information For instance in an experiment onfinancial decisions in situations of uncertainty and informa-tion asymmetry similar to our experiment [39] showed thatthe availability of information from other subjects increasedrisky trust investment decisions of subjects independent ofthe quality of the information

It is worth noting that the motivation crowding-outmodel (2) was able to match our empirical data qualitativelyCoherently with H1 the comparison of the two experimentsconfirmed the paramount importance of indirect reciprocitymotives in trust situationsThe idea that indirect reciprocity isfundamental in human societies has been suggested by [40]


Table 3 Summary statistics by rating Average return proportionby rating (a) and rating distribution per treatment (b) AR standsfor ldquoalternating rolesrdquo and FR for ldquofixed rolesrdquo

(a) Return proportion

Treatment Negative Neutral PositiveNo incentive AR 014 022 034No incentive FR 013 015 014Fixed incentive AR 008 016 024Fixed incentive FR 011 009 017A incentive AR 015 022 033A incentive FR 018 016 028B incentive AR 014 020 022B incentive FR 019 015 018All treatments 015 016 022

(b) Rating distribution


Table 4 Average investments by treatment and rating


who argued that this is one of the most crucial forces inhuman evolution This has been found in different experi-mental settings [21 23] Our experiments suggest that ceterisparibus indirect reciprocity explains a significant part ofcooperation also in triadic relations (Tables 1 and 2) Thisindicates that evaluation systems where the roles of trustorstrustees and evaluators rotate could work better and moreefficiently than those in which roles are fixed

H2 was also confirmed by our data In both experimentsA incentive guaranteed high levels of cooperation as inter-mediary evaluations fostered both investments by trustorsand returns by trustees In this treatment trustees followedreputation building strategies and so returned more Fur-thermore trustors considered the opinions of intermediaries

credible and used them to discriminate between ldquogoodrdquo andldquobadrdquo opponents Therefore intermediaries were functionalto encapsulate mutual trust and cooperation

H3 was the only hypothesis not fully supported by ourdata On the one hand B incentive led to less cooperation thanA incentive On the other hand B incentive unexpectedly gavebetter results than the Baseline as intermediaries were lessdemanding than in A incentive to award positive ratings butstill discriminated between trustworthy and untrustworthyB Players (Table 3) This induced trustors to consider theopinion of reliable intermediaries and followed their ratingsindependent of the misalignment of mutual interests causedby the monetary incentives (Table 4) This would againconfirm that in condition of information asymmetry anyinformation on trustee is better than none as in the Baseline

The mismatch between monetary incentives and inter-mediaryrsquos behaviour is interesting Results indicated thatintermediaries did not predictably respond to incentives asthey played fairly in each treatment (Table 3) This can beexplained in terms of intrinsic motivations and the sense ofresponsibility that are typically associated with such ldquoneutralrdquopositionsMore specifically it is worth noting that willingnessto provide pertinent judgement by intermediaries was notsensitive to any variations in the incentive scheme Thiswould testify to the inherent moral dimension of this roleOn the other hand even the perception that their taskwas indirectly judged by the trustors who were informedof their ratings could have motivated the intermediariesto take their role seriously independent of the incentivesMore importantly the alternating roles protocol allowedintermediaries to follow indirect reciprocity strategies andso they kept the credibility and quality of their ratings highin order to benefit from reciprocity by other intermediarieswhen cast as trustors This explains the difference betweenthe two experiments

Consistent with H4 Fixed incentive led to less coopera-tion than other schemes A comparison betweenNo incentiveand Fixed incentive allows us to understand this point betterIn the alternating roles experiment while No incentive led tomore trust and cooperation Fixed incentive barely improvedthe Baseline that is when intermediaries were not presentThis does not make sense in a rational choice perspec-tive as in both cases the incentives of intermediaries wereambiguous On the other hand while intrinsic motivationswere crucial to induce intermediaries to formulate reliableratings in No incentive these aspects were crowded out bymonetary incentives in Fixed incentive This is consistentwith the motivation crowding theory [25 41] and with manyempirical studies that showed that incentivization policieswhich targeted self-interested individuals actually backfireby undermining individual ldquomoral sentimentsrdquo in a varietyof social and economic situations [29 38] In this sensesome possible extensions of our work could examine therelationship between crowding out effects on social normsand the presence of different incentive schemes in moredetail For instance it would be interesting to understandif different incentives for the intermediaries could influence


future interactions between trustors and trustees withoutintermediaries This could also help us to look at self-rein-forcing effects of incentives and social norms on trust

It is worth considering that more consistent monetaryincentives for intermediaries could increase their credibilityfor the other parties involved so motivating more coopera-tion Indeed a possible explanation of the low cooperationin Fixed incentive is that the magnitude of our monetaryincentives was sufficient to crowd out intrinsic motives ofsubjects without promoting reciprocal and self-interestedbehaviour as what happened in typical monetary markets[29] Further work is necessary to examine this hypothesisby testing fixed incentives of different magnitudes althoughit is worth considering that there are constraints in terms ofmagnitude of incentives that can be implemented also in realsituations

H5 was also confirmed by our results at least in the alter-nating roles experiment In this case No incentive was thebest treatment for cooperation On the other hand it didnot promote trust and especially trustworthiness when roleswere fixed As argued above by fixing the roles there wasno room for indirect reciprocity strategies The fact thatintermediaries could expect future benefits from their rolesand were subsequently cast as both trustors and trusteesinduced the parties involved to believemore in the credibilityof the intermediariesrsquo opinion Also in this case furtherempirical investigation is needed to compare social situationswhere intermediaries have a specialised role with situationswhere there is voluntarism and mixture of roles

These findings imply that the ldquomantrardquo of incentivizationpopularized by most economists as a means to solve trustand cooperation problems especially in economic and publicpolicy should be seriously reconsidered [42] Not only shouldincentives be properly designed to produce predictable out-comes and this is often difficult but also incentive-responsebehaviour of individuals is more heterogeneous and unpre-dictable than expected [43] If this occurred in a simplelaboratory game where individuals had perfect informationand the rules of the game were fully intelligible one shouldexpect even more heterogeneity of individual behaviour andunpredictability of social outcomes in real situations

Our results finally suggest that insisting on incentiviza-tion potentially crowds out other social norms-friendlyendogenous mechanisms such as reputation which couldensure socially and economically consistent results Whileincentives might induce higher investment risks (but only ifproperly designed) social norms can also help to achieve afairer distribution nurturing good behaviour which can beeven more endogenously sustainable in the long run [44]

Appendices

A Details of the Experiment Organization

All participants were recruited using the online systemORSEE [45]Theywere fully informed and gave their consent

when they voluntarily registered on ORSEE Data collectionfully complied with Italian law on personal data protection(DL 3062003 n 196) Under the applicable legal principleson healthy volunteersrsquo registries the study did not requireethical committee approval All interactionswere anonymousand took place through a computer network equipped withthe experimental software z-Tree [46]

Experiment 1 (Alternating Roles) A total of 136 subjects (50female) participated in the experiment held at the GECSexperimental lab (see httpwwwecounibsitgecs) in thelate 2010 Each experimental session took less than one hourand participants earned on average 1482 Euro including a5-Euro show-up fee

Twenty-eight subjects participated in a Baseline repeatedinvestment game (hereafter Baseline) set using the followingparameters 119889A = 119889B = 10 monetary units (MU) 119898 = 3EachMUwas worth 25 Euro Cents and subjects were paid incash immediately at the end of the experimentThe game wasrepeated 30 timeswith coupleswhowere randomly reshuffledin each period Playersrsquo roles regularly alternated throughoutthe gameThismeant that each subject played exactly 15 timesas A and 15 times as B

All the other treatments each played by 27 subjectsintroduced a third player into the game (Player C) in therole of intermediary Once C Players were introduced wevaried the monetary incentive schemes offered to them Inthe No incentive treatment intermediaries did not receiveany rewards for their task also losing potential earnings astrustors or trustees when selected to play in this role Inthe Fixed incentive treatment intermediaries received a fixedpayoff of 10MU equal to the trustor and trustee endowmentsIn the A incentive treatment intermediaries earnings wereequal to the payoff obtained by the trustors they advised Inthe B incentive treatment intermediariesrsquo earnings were equalto the payoff obtained by the trustees they rated

Experiment 2 (Fixed Roles) A total of 244 subjects (55female) participated in the second experiment which wasorganized at the GECS experimental lab in the late 2011Participants were recruited and played as in the first exper-iment Each experimental session took less than one hourand participants earned on average 1478 Euro including a5-Euro show-up fee

Any overlap with the first experiment participants wasavoided so that the two experiments could in principle beviewed as a single experiment with a between-subject designThe treatments were as before with the only difference thatroles remained fixed throughout the game Note that the factthat roles no longer alternated actually reduced the sampleof observations per role with a considerable consequenceespecially on the treatments involving three parties (ie allbut the Baseline) To overcome this problem we doubledthe number of subjects participating in No incentive Fixedincentive A incentive and B incentive which were playedby 54 subjects each organized in two sessions each oneinvolving 27 participants


B Instructions

Before the beginning of the game participants read the gameinstruction on their computer screen Subsequently theyfilled a short test designed to check their understanding ofthe game Participants could also ask the experimenters anyfurther questions or issues Here is the English translationof the original Italian instructions Sentences in italics aretreatment or experiment specific whereas a normal font isused for instructions common to all treatmentsexperiments

Screen 1 Overall Information on the Experiment

(i) All these instructions contain true information andare the same for all participants

(ii) Please read them very carefully At the end somequestions will be asked by the system to test yourunderstanding of the experiment

(iii) The experiment concerns economic problems(iv) During the experiment youwill be asked to take deci-

sions upon which your final earnings will dependEarnings will be paid in cash at the end of the experi-ment

(v) Each decision will take place through your computerscreen

(vi) During the experiment it is prohibited to talk withanyone If you do so you will be excluded from theexperiment and you will lose your earnings Pleaseturn your mobile phones off

(vii) For any information and question put your hands upand wait until an experimenter comes to your posi-tion

(viii) During the experiment virtual monetary units (MU)are used that have a fixed exchange rate with realEuros For each MU earned in the experiment youwill receive 25 Euro Cents For example if at the endof the experiment your earning is 500MU thismeansthat you will receive 1250 Euros plus a fixed show-upfee of 5 Euros

Screen 2 Interaction Rules

(i) The experiment consists of a sequence of interactionrounds between pairs of players (Baseline)

(ii) The experiment consists of a sequence of interactionrounds between groups of three players (all threeperson treatments)

(iii) Pairs are randomly matched and change each roundtherefore they are made up of different individualseach round (Baseline)

(iv) Groups are randomly matched and change eachround therefore they are made up of different indi-viduals each round (all three person treatments)

(v) There is no way to know who you are playing withnor is it possible to communicate with herhim

(vi) In each pair each participant will perform a differentrole roles are called ldquoPlayer Ardquo and ldquoPlayer Brdquo (Base-line)

(vii) In each group each participant will perform a differ-ent role roles are called ldquoPlayer Ardquo ldquoPlayer Brdquo andldquoPlayer Crdquo (all three person treatments)

(viii) Roles are randomly assigned at the beginning of theexperiment and then regularly changed for the rest ofit For example one participant in the pair will play asequence of rounds such as Player A Player B PlayerA Player B Player A and Player B and the other onewill play a sequence such as Player B Player A PlayerB Player A Player B and Player ATherefore over theexperiment all participants will play both roles thesame number of rounds (alternating roles experimentBaseline)

(ix) Roles are randomly assigned at the beginning of theexperiment and then regularly changed for the rest ofit For example one participant in the group will playa sequence of rounds such as PlayerA Player B PlayerC Player A Player B and Player C and another onewill play a sequence such as Player B Player C PlayerA Player B Player C and Player A Therefore overthe experiment all participants will play all roles thesame number of rounds (alternating roles experimentall three person treatments)

(x) Roles are randomly assigned at the beginning of theexperimentThismeans that each participantwill playin the same role throughout the whole experiment(fixed roles experiment all treatments)

(xi) Each participant should make one decision eachround

(xii) The experiment lasts 30 rounds

Screen 3 Task Structure(i) Player A plays first and receives an endowment of

10MU Shehe has to decide howmuchof it to keep forherhimself and howmuch to send to Player B PlayerA can send to Player B any amount from 0MU to thewhole endowment (=10MU)

(ii) The amount of MU kept by Player A is part of herhisearningThe amount of MU sent to Player B is tripledand assigned to Player B

(iii) Player B also receives an endowment of 10MU asPlayer A did

(iv) Example 1 if Player A sends 2MU Player B receives 2times 3 + 10 = 16MU

(v) Example 2 if Player A sends 5MU Player B receives5 times 3 + 10 = 25MU

(vi) Example 3 if Player A sends 8MU Player B receives8 times 3 + 10 = 34MU

(vii) Player B should decide how much of the wholeamount received to send to Player A Player B cansend any amount to Player A from 0MU to the entiresum


(viii) The amount ofMUkept by Player B represents herhisearning the amount of MU sent to Player A will addup to Player A earning

(ix) Before A and B players respectively take their deci-sions Player C is asked to rate the decision Player Btook the previous round (since no decision has beentaken by B in the first round yet C is not asked to ratein the first round) (all three person treatments)

(x) Player C can rate Player Brsquos decision as ldquonegativerdquoldquoneutralrdquo or ldquopositiverdquo and the rating is communi-cated to both A and B Players with whom heshe isgrouped (in the first rounds since there is no decisionundertaken by Player B yet the rating of Player B isassigned by the systemas ldquounknownrdquo) (all three persontreatments)

(xi) Player C does not receive any earnings for herhisaction (No incentive treatments)

(xii) Player C earning is fixed and is equal to 10MU (Fixedincentive treatments)

(xiii) Player C earning is equal to what Player A will earn atthe end of the round (A incentive treatments)

(xiv) Player C earning is equal to what Player B will earn atthe end of the round (B incentive treatments)

(xv) Earnings will be added up each round to give the finalreward for each participant which will be paid at theend of the experiment



Acknowledgments

The authors would like to gratefully acknowledge commentson a preliminary version of this paper by Karl-Dieter OppGianluca Carnabuci Michael Maes and the audience atthe INAS Conference on Analytical Sociology ColumbiaUniversity New York 8-9 June 2012 and at the ASA 2012Annual Meeting Denver 17ndash20 August 2012 They alsowould like to thankMarco Castellani and Niccolo Casnici forhelp during the experiment and Fabio Recinella for technicalassistance in the labThis work was supported by the CollegioCarlo Alberto and the University of Brescia The authorsalso gratefully acknowledge support from the Intra-EuropeanFellowship Program of the European Union (Grant no GA-2009-236953)

References

[1] J S Coleman Foundations of SocialTheory Harvard UniversityPress Harvard Mass USA 1990

[2] R S Burt and M Knez ldquoKinds of third-party effects on trustrdquoRationality and Society vol 7 no 3 pp 255ndash292 1995

[3] K S Cookm and R Hardin ldquoNorms of cooperativeness andnetworks of trustrdquo in Social Norms M Hechter and K-D Opp

Eds pp 327ndash347 Russel Sage Foundation New York NY USA2001

[4] M Bateson D Nettle and G Roberts ldquoCues of being watchedenhance cooperation in a real-world settingrdquo Biology Lettersvol 2 no 3 pp 412ndash414 2006

[5] I Bohnet and B S Frey ldquoThe sound of silence in prisonerrsquosdilemma and dictator gamesrdquo Journal of Economic Behavior andOrganization vol 38 no 1 pp 43ndash57 1999

[6] T N Cason and V-L Mui ldquoA laboratory study of group polar-isation in the team dictator gamerdquo Economic Journal vol 107no 444 pp 1465ndash1483 1997

[7] K JHaley andDMT Fessler ldquoNobodyrsquos watching Subtle cuesaffect generosity an anonymous economic gamerdquo Evolution andHuman Behavior vol 26 no 3 pp 245ndash256 2005

[8] M Rigdon K Ishii M Watabe and S Kitayama ldquoMinimalsocial cues in the dictator gamerdquo Journal of Economic Psychol-ogy vol 30 no 3 pp 358ndash367 2009

[9] R Boero G Bravo M Castellani and F Squazzoni ldquoReputa-tional cues in repeated trust gamesrdquo Journal of Socio-Economicsvol 38 no 6 pp 871ndash877 2009

[10] M Bacharach and D Gambetta ldquoTrust in signsrdquo in Trust andSociety K S Cook Ed vol 2 pp 148ndash184 Russell Sage NewYork NY USA 2001

[11] F SquazzoniG Bravo andKTakacs ldquoDoes incentive provisionincrease the quality of peer review An experimental studyrdquoResearch Policy vol 42 no 1 pp 287ndash294 2013

[12] K Cook R Hardin and M Levi Cooperation without TrustRussell Sage New York NY USA 2005

[13] R Hardin Trust and Trustworthiness Russell Sage New YorkNY USA 2004

[14] J P Bailey and Y Bakos ldquoAn exploratory study of the emergingrole of electronic intermediariesrdquo International Journal of Elec-tronic Commerce vol 1 no 3 pp 7ndash20 1997

[15] JW Palmer J P Bailey and S Faraj ldquoThe role of intermediariesin the development of trust on the WWW the use andprominence of trusted third parties and privacy statementsrdquoJournal of Computer-Mediated Communication vol 5 no 32000

[16] O Sorenson and T E Stuart ldquoSyndication networks and thespatial distribution of venture capital investmentsrdquo AmericanJournal of Sociology vol 106 no 6 pp 1546ndash1588 2001

[17] J Berg J Dickhaut and K McCabe ldquoTrust reciprocity andsocial historyrdquo Games and Economic Behavior vol 10 no 1 pp122ndash142 1995

[18] N D Johnson and A A Mislin ldquoTrust games a meta-analysisrdquoJournal of Economic Psychology vol 32 no 5 pp 865ndash889 2011

[19] V Buskens and W Raub ldquoRational choice research on socialdilemmas embeddedness effects on trustrdquo in Handbook ofRational Choice Social Research RWittek T A B Snijders andV Nee Eds Russell Sage New York NY USA 2008

[20] V Buskens W Raub and J van der Veer ldquoTrust in triads anexperimental studyrdquo Social Networks vol 32 no 4 pp 301ndash3122010

[21] E Fehr and U Fischbacher ldquoThe nature of human altruismrdquoNature vol 425 no 6960 pp 785ndash791 2003

[22] C Keser ldquoExperimental games for the design of reputationmanagement systemsrdquo IBM Systems Journal vol 42 no 3 pp498ndash506 2003

[23] I Seinen and A Schram ldquoSocial status and group normsindirect reciprocity in a repeated helping experimentrdquoEuropeanEconomic Review vol 50 no 3 pp 581ndash602 2006


[24] D Barrera and G G Van De Bunt ldquoLearning to trust networkseffects through timerdquo European Sociological Review vol 25 no6 pp 709ndash721 2009

[25] B S Frey and F Oberholzer-Gee ldquoThe cost of price incentivesan empirical analysis of motivation crowding- outrdquo AmericanEconomic Review vol 87 no 4 pp 746ndash755 1997

[26] RDAlexanderTheBiology ofMoral Systems Basic BooksNewYork NY USA 1987

[27] M A Nowak and K Sigmund ldquoThe dynamics of indirect reci-procityrdquo Journal of Theoretical Biology vol 194 no 4 pp 561ndash574 1998

[28] J-J Laffont and D Martimort The Theory of Incentives ThePrincipal-Agent Model Princeton University Press PrincetonNJ USA 2002

[29] D Ariely A Bracha and S Meier ldquoDoing good or doing wellImage motivation and monetary incentives in behaving proso-ciallyrdquo American Economic Review vol 99 no 1 pp 544ndash5552009

[30] J Heyman and D Ariely ldquoEffort for payment a tale of twomarketsrdquo Psychological Science vol 15 no 11 pp 787ndash793 2004

[31] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2012

[32] P Kollock ldquoSocial dilemmas the anatomy of cooperationrdquoAnnual Review of Sociology vol 24 pp 182ndash214 1998

[33] I Almas A W Cappelen E Oslash Soslashrensen and B TungoddenldquoFairness and the development of inequality acceptancerdquo Sci-ence vol 328 no 5982 pp 1176ndash1178 2010

[34] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo Quarterly Journal of Economics vol 114 no3 pp 817ndash868 1999

[35] M A Nowak K M Page and K Sigmund ldquoFairness versusreason in the ultimatum gamerdquo Science vol 289 no 5485 pp1773ndash1775 2000

[36] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquo American Economic Review vol 83 pp 1281ndash13021993

[37] F Cochard P Nguyen Van and M Willinger ldquoTrusting behav-ior in a repeated investment gamerdquo Journal of Economic Behav-ior and Organization vol 55 no 1 pp 31ndash44 2004

[38] S Bowles ldquoPolicies designed for self-interested citizens mayundermine ldquothe moral sentimentsrdquo evidence from economicexperimentsrdquo Science vol 320 no 5883 pp 1605ndash1609 2008

[39] R Boero G Bravo M Castellani and F Squazzoni ldquoWhybother with what others tell you An experimental data-drivenagent-based modelrdquo Journal of Artificial Societies and SocialSimulation vol 13 no 3 article 6 2010

[40] M A Nowak and R Highfield SuperCooperators AltruismEvolution and WhyWe Need Each Other to Succeed Free PressNew York NY USA 2011

[41] B S Frey and R Jegen ldquoMotivation crowding theoryrdquo Journal ofEconomic Surveys vol 15 no 5 pp 589ndash611 2001

[42] F Squazzoni ldquoA social science-inspired complexity policybeyond the mantra of incentivizationrdquo Complexity vol 19 pp5ndash13 2014

[43] E ShafirThe Behavioral Foundations of Public Policy PrincetonUniversity Press Princeton NJ USA 2013

[44] S Bowles and S Polanıa-Reyes ldquoEconomic incentives andsocial preferences substitutes or complementsrdquo Journal of Eco-nomic Literature vol 50 no 2 pp 368ndash425 2012

[45] B Greiner ldquoAn online recruitment system for economic exper-imentsrdquo in Forschung und Wissenschaftliches Rechnen 2003 KKremer and V Macho Eds pp 79ndash93 Gesellschaft fur wis-senschaftliche Datenverarbeitung Gottingen Germany 2004

[46] U Fischbacher ldquoZ-Tree zurich toolbox for ready-made eco-nomic experimentsrdquo Experimental Economics vol 10 no 2 pp171ndash178 2007

Research ArticleAltruism Noise and the Paradox of Voter TurnoutAn Experimental Study

Sarah A Tulman

United States Department of Agriculture Economic Research Service 1400 Independence Avenue SW Mail Stop 1800Washington DC 20250-0002 USA

Correspondence should be addressed to Sarah A Tulman sarahtulmanersusdagov


Academic Editor Arthur Schram

Copyright copy 2015 Sarah A Tulman This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper addresses the paradox of voter turnout wherein observed voting participation rates are far greater than what rationalchoice theory would predict Voters facemultiple voting choices stochastic voting costs and candidates offering different economicplatforms A combination of two approaches attempts to resolve this paradox quantal response equilibrium (QRE) analysis whichintroduces noise into the decision-making process and the possibility of ethical (altruism-motivated) voting A series of laboratoryexperiments empirically tests the predictions of the resultingmodel Participants in the experiments are also given opportunities forcommunicating online with their immediate neighbors in order to enhance the chances that subjects would realize the possibilityof ethical voting The results show that ethical voting occurs but gains momentum only in the presence of a vocal advocate andeven then it mostly dissipated by the second half of the session The QRE-based model was able to explain some but not all of theovervoting that was observed relative to the Nash equilibrium predictionThere is evidence to suggest that communication via thechat feature generated some of the voting and also some of the ethical voting

1 Introduction

Theparadox of voter turnout wherein observed turnout ratesin elections are far greater than what is predicted underrational choice theory has been a focus of researchers ineconomics and political science for over 50 years (see Geys[1] for a survey of the theoretical literature) and remainsunresolved Under rational choice theory the probability ofcasting a pivotal votemdashmaking or breaking a tiemdashis a keycomponent of the motivation to vote If the electorate islarge then the probability of casting the pivotal vote fallsto almost zero and rational choice theory predicts that noone will vote This probability declines quickly so that evenelections with relatively small electorates face this issue Andyet observed turnout rates in elections are far greater thanzero For example in the 2012 general election in the UnitedStates the estimated turnout rate among eligible voters was586 (All turnout rates are from the United States ElectionProject httpwwwelectprojectorg) Even in years without apresidential election when turnout is generally lower it is stillconsiderably higher than zero such as the 418 turnout ratein the 2010 House and Senate elections Therefore there has

to be some othermotivation for voting beyond the probabilityof onersquos vote being pivotal

Why does this paradox matter As the fields of electionforecasting and polling analysis have expanded in order totake advantage of the explosion in available data interest inthese forecasts has rapidly increased During the 2012 USpresidential campaign there was an endless flow of politicaland polling analysis covered in almost every media outletand the presidential campaigns devoted vast resources toldquoget out the voterdquo efforts while using the rapidly increasingquantities of available data to microtarget voters In spite ofthis increasing sophistication one thing that can and oftendoes cause election outcomes to diverge from forecasts is thematter of who actually votes on Election Day

The model presented in this paper and the laboratoryexperiments which test its predictions focus on the economicmotivations for voting The payoffs that are offered by thecandidates and the equitability (or lack thereof) of thesepayoffs across groups can be thought of as their economicplatforms Voters also face a cost associated with voting thatmodels the explicit and implicit costs of going to the pollssuch as transportation costs lost wages due to taking time



off from work or the opportunity cost of time I draw fromseveral threads of the literature which are at the intersectionof economics and political science and my model andexperiments augment previous work on voting behavior

The first thread from which my model and experimentsare drawn focuses on the use of quantal response equilibrium(QRE) a generalization of the Nash equilibrium that allowsfor decision errors to relax the assumption of perfect ratio-nality Building upon the framework presented in McKelveyand Palfrey [2] and also drawing from Goeree and Holt [3]Levine and Palfrey [4] used laboratory experiments to testvoter turnout predictions in the presence of heterogeneousand privately known participation costs However in theirpaper the subjects could only choose between voting andabstaining whereas in most elections potential voters canchoose between at least two candidates plus the option ofabstainingWould their results hold up if another choice wereadded so that potential voters could choose between threeoptions vote for one candidate vote for the other candidateand abstain The model and experimental design presentedhere extend those of Levine and Palfrey [4] in this manner

Another thread of the literature focuses on the role of eth-ical or altruism-motivated voting in generating voter turn-out Morton and Tyran [5] separated the motivations forvoting choices into selfish voting and ethical voting followingthe terminology used in the literaturemdashvoting for onersquoseconomic self-interest versus voting altruistically if one of thecandidates offers a lower payoff but also a more equitabledistribution of payoffs across groups of voters howevertheir equilibrium predictions were qualitative in nature Byincorporating a similar payoff structure into the QRE-basedframework from Levine and Palfrey [4] my model can gen-erate quantitative predictions of turnout rates against whichthe experimental results can be compared Additionally itcan be argued that incorporating bounded rationality andnoise into the decision-making process may lead to a morerealistic representation of a world in which individuals donot always (indeed almost never) behave perfectly rationallyThis combination was also used in Goeree et al [6]

Another aspect ofMorton andTyranrsquos [5] approach albeitone that was necessary given the nature of the experimentwas that their experimentwas a one-shot game Although thisis a more accurate representation of elections than a repeatedgame in an actual election voters usually have several weeksor months leading up to the election during which voterscan learn about and discuss the issues and the candidatesrsquoplatforms My experiments allowed for these learning effectsduring the experiment through the inclusion of 20 rounds(elections)

My experiments also incorporated an online chat featurewhich allowed subjects to communicate before each roundThe initial purpose of this was an attempt to increase theprobability that the subjects would grasp the possibility ofethical voting in treatments where that was relevantmdashthehope was that the subjects could clarify this (and otherpoints) for each other In the end although this effect wasindeed observed in certain cases it also provided insightinto subjectsrsquo motivations The chat feature was also found toincrease voting participation rates

Group identity and communication within those groupshave also been shown to increase voting participation as seenin papers byMorton [7] Groszliger and Schram [8] Schram andvan Winden [9] Schram and Sonnemans [10] and Charnesset al [11] Kittel et al [12] which was developed concurrentlywith this paper incorporated multiple-candidate electionscostly voting as in Levine and Palfrey [4] and prevotingcommunication Although the focus and context of theirpaper differ from mine they found that the distributionof earnings was more equitable when intergroup commu-nication was allowed (analogous in some respects to themixed-type chat groups in this paper)Their results also showthat voting participation increases when communication isallowed which matches the positive effect of communicationon voting turnout that I found

The main findings of this paper are summarized asfollows High rates of voter turnout were observed in theexperimental data especially when a subject voted for thecandidate offering that subjectrsquos voter type a higher payoffthan what the other candidate was offering Some ethical vot-ing was observed but not enough to explain the overvotingAlso most of the overvoting occurred when voters of eachtype voted in their own economic self-interest and by its verynature ethical voting is not a possible motive for that

One feature that could potentially explain overvotingunexpectedly is the chat feature (This feature was notincluded in the experimental design for the purpose ofincreasing voter participation but rather as a method tohopefully spread the word about the possibility of ethicalvoting and also to gain insight into the motivations behinddecisions This result was unexpected) Because the chat wasrepeated before every round and the groups remained fixedthroughout the session it set up a dynamic in which theremay have been some accountability between chat partnersFor example onemight ask the otherswhether they had votedin the previous round and if so then for which candidateThe transcripts also showed that subjects would often urgeothers to vote in a certain way or would agree to vote ina certain way and sometimes would ask about whetherforwhom others had voted in the previous round Additionallyin spite of the differences in design and context a few patternswere observed that had also appeared in the group identityliterature such as the emergence of subjects who persistentlyencouraged others to vote (analogous to Schram and vanWindenrsquos [9] producers of social pressure)

The rest of the paper proceeds as follows Section 2describes the theoretical model and results and the exper-imental design which is influenced by the predictionsgenerated by the theoretical model Section 3 discusses theempirical results Finally Section 4 concludes

2 Materials and Methods

21 Model This model in which voter turnout is a partici-pation game (Palfrey and Rosenthal [13]) is an extension ofthe logit quantal response equilibrium (QRE) model used inLevine andPalfrey [4] andGoeree andHolt [3] incorporatingthe possibility of ethical voting along the lines of Mortonand Tyran [5] Whereas Levine and Palfrey [4] only allowed


Table 1 Payoffs (symmetric)

Type 1 Type 2Candidate119883 119902 119902 minus 120579

Candidate 119884 119902 minus 120579 119902

potential voters (also referred to simply as ldquovotersrdquo unless adistinction is needed between actual and potential voters) tochoose between voting and abstaining here they have threechoices vote for Candidate 119883 vote for Candidate 119884 andabstain This better captures the reality of most electionsin which voters choose between two or more candidatesin addition to the option of abstaining from voting Thisextension of the model also allows for a range of payoffstructures including one which may induce ethical votingthrough the presence of a ldquoselfishrdquo candidate who offers a highpayoff to one group of voters and a low payoff to the other andan ldquoethicalrdquo candidate who offers an equitable distribution ofpayoffs across groups of voters

In this model the voting rule is a simple plurality Ties arebroken fairly with the winner being picked randomly (eg bya coin toss) Payoffs the distribution of which can be thoughtof as the candidatesrsquo economic platforms are paid accordingto which candidate wins regardless of whether or for whomthe participant voted The payoff structure for each type iscommon knowledge to everyone regardless of type

Potential voters are separated into two ldquotypesrdquo whichdetermine the payoff that each one will receive dependingon the winner of the election Each candidate offers differ-ent payoffs to the two different voter types (Table 1) Thistable shows payoffs that are symmetric in both the payoffdifferences and the actual payoffs This symmetry is nottheoretically necessarymdashthe main takeaway from this tableis how the candidates offer different payments to each typemdashbut matches the experimental design used in this researchWith this payoff structure voters of both types would faceidentical incentives to vote for the candidate offering theirtype the higher payoff

Voting is modeled as being costly This mirrors thecosts associated with voting in real-world elections such astransportation costs childcare lost wages due to time offfrom work or the opportunity cost of whatever someonewould otherwise do This cost is only incurred if someonevotes Costs are independent draws from the same uniformdistribution varying across both voters and elections andboth voter types face the same distribution of costs Ineach election each voter knows his or her own cost beforedeciding whether to vote but only knows the distributionfrom which othersrsquo costsmdashand his or her own costs duringfuture electionsmdashare drawn

Following Levine and Palfrey [4] a quasi-symmetricequilibrium in this model is a set of four turnout strategies(1205911119883 1205911119884 1205912119883 1205912119884) specifying the probabilities that a member

of type 119894 will vote for candidate 119895 as a function of thevoting cost For a quasi-symmetric equilibrium it is assumedthat within each type everyone will use the same strategymdashalthough it is possible for members of each type to vote foreither candidate in the Nash equilibrium any voting will be

for the candidate that offers the higher payoff to his or hertype

The aggregate voting probabilities for each type-candi-date combination are

119901lowast

119894119895= int

infin

minusinfin

120591119894119895(119888) 119891 (119888) 119889119888 = int

119888lowast

119894119895

minusinfin

119891 (119888) 119889119888 = 119865 (119888lowast

119894119895) (1)

for each combination of voter type 119894 and candidate 119895There is a cutpoint cost level 119888lowast

119894119895below which in the Nash

equilibrium the probability of voting equals one and abovewhich the probability of voting is zero At the cutpoint thevoter is indifferent These cutpoints are expressed as

119888lowast

119894119895=

(119881119894119895minus 119881119894119898)

2times (Pr (make tie)

119894119895+ Pr (break tie)

119894119895)

(2)

where 119881119894119895is the payoff received by a member of voter type

119894 if candidate 119895 wins the election and where 119898 indexes theother candidate This is the difference between the payoffreceived and the payoff that would have been received if theother candidate had won (the payoff difference) In the Nashequilibrium nobody would vote for the candidate offeringonersquos type the lower payoff because the payoff differencewould be negative implying a negative cost cutpoint in whichcase no cost would be low enough to induce voting Thesecutpoints are set to zero because costs cannot be negative

The term (12) times (Pr(make tie)119894119895+ Pr(break tie)

119894119895) is

the probability that a vote cast by a member of type 119894 forcandidate 119895 will be pivotal (henceforth referred to as theldquopivotal probabilityrdquo) The one-half is there because in thepresence of two candidates if someonersquos votemakes a tie thenthere is a 50 chance that this voterrsquos chosen candidate willwin (and that the vote will have been pivotal)mdashthe winner ofa tie is determined randomly If someonersquos vote breaks a tiethen there is a 100 chance that his or her chosen candidatewill win However because there was already a 50 chancethat this candidate would have won in the event of a tie therewould only be a 50 chance that the vote that broke the tiechanged that candidatersquos outcome from losing to winningFurther details about the calculation of the pivotal probabilitycan be found in Appendix A

The expected payoff differences of voting for candidate 119895as a member of type 119894 net of costs are expressed as

120587119896119894119895= (119881119894119895minus 119881119894119898) times Pr (pivotal)

119894119895minus 119888119896 (3)

where 119881119894119895is the payoff received by a member of type 119894 if

candidate 119895 wins the election and 119888119896is the voting cost faced

by a given person in a given election 120587119896119894119895

can be thought ofas a component of

119896119894119895= 120587119896119894119895+120576119896119894119895

where 120576119896119894119895

is independentand identically distributed extreme value This produces thelogit model that is shown below

Quantal response equilibrium (QRE) analysis intro-duced in McKelvey and Palfrey [2] allows for ldquonoiserdquo inthe decision-making process for voting This process is ofteninfluenced by factors such as emotions perception biasesvoter error unobserved individual heterogeneity and other


sources of noise The QRE introduces decision errors via alogit probabilistic choice rule according to which participantsmake their voting choices

Under QRE the best response functions are probabilisticrather than deterministic Although ldquobetterrdquo responses aremore likely to be observed than ldquoworserdquo responses theldquoworserdquo responses are still observed because choice has astochastic element Thus the assumption of perfect ratio-nality is relaxed in favor of bounded rationality As theamount of noise decreasesmdashand the QRE approaches theNash equilibriummdashvoters become increasingly likely tomakechoices that are consistent with the Nash equilibrium In itsmost general form the logit probability of choice ldquo119909rdquo takesthe form 119875(119909) = 119890120582120587119909sum

119894119890120582120587119894 where 120587

119894is the expected payoff

from making choice 119894 and 120582 is the logit precision parameterThe denominator sum

119894119890120582120587119894 sums across all possible choices

summing to 1As the precision parameter 120582 increases the level of noise

decreases and the level of precision in decision makingimproves As this happens the ratio of ldquobetterrdquo responsesto ldquoworserdquo responses improves When 120582 is very large (veryhigh precisionvery low noise) the solution approaches theNash equilibrium with all or almost all decisions match-ing the Nash equilibrium outcome (Strictly speaking thisconvergence only holds for the principal branch of theQRE when there are multiple equilibria However for themodel presented here the results were robust across a widerange of starting points) As 120582 approaches zero (very lowprecisionvery high noise) voting participation decisionsapproach randomness (50 if there are two choices 33if there are three choices etc) as the probabilities of ldquobet-terrdquo responses and ldquoworserdquo responses become increasinglysimilar

In the context of this model a Nash equilibrium wouldcorrespond to a sharp drop in the response function at thecost cutpoint where the probability of votingwould equal oneif facing a cost below 119888lowast and would equal zero if facing a costabove 119888lowast For an intermediate level of noise the cost cutpointwould affect the decision of whether or not to vote but theresponse function would not be as sharp because noise wouldlead to some instances of voting (or abstaining) evenwhen thecost is above (or below) the cutpoint A very low 120582 (very highnoise levelvery low level of precision) would correspond toa cutpoint that is effectively irrelevant where the size of thecost would have very minimal or no effect on the decision ofwhether or not to vote This can be seen in Figure 1

The logit quantal response function which can bethought of as the logit probability of a member of type 119894making voting choice 119895 (where the choices are vote for119883 votefor 119884 and abstain) and facing cost 119896 is shown by

119901119896vote119894119895 =

119890120582120587119896119894119895

119890120582120587119896119894119883 + 119890

120582120587119896119894119884 + 119890

120582120587119896119894119886

(4)

This can be simplified for a type 119894 voter choosing 119895 (where thechoices are only vote for119883 or vote for 119884) and facing cost 119896 as

119901119896119894119895=

119890120582120587119896119894119895

1 + 119890120582120587119896119894119883 + 119890

120582120587119896119894119884

(5)

0

01

02

03

04

05

06

07

08

09

1

Pr (v

ote)

clowast

Nash equilibriumSome noiseHigh noise

Figure 1 Logit response function relative to cost cutpoint 119888lowast

Table 2 Payoffs (asymmetric)

Type 1 Type 2Candidate119883 119902 + 120579 119902 minus 120579

Candidate 119884 119902 119902

In the presence of random costs these response functionshave to be averaged across all possible costs (for a giventype-candidate combination) in order to arrive at the averageresponse probability of type 119894 voting for candidate 119895 Afteraveraging across costs the probabilities are (119901

119894119883 119901119894119884 1 minus

119901119894119883minus 119901119894119884) where the third probability is the probability of

abstainingThe logit precision parameter 120582 is estimated using the

likelihood function

log 119871 = sum119897

(1198731119883119897times ln (119901

1119883119897) + 1198731119884119897

times ln (1199011119884119897) + (num obs

1119897minus 1198731119883119897minus 1198731119884119897)

times ln (1 minus 1199011119883119897minus 1199011119884119897) + 1198732119883119897

times ln (1199012119883119897) + 1198732119884119897times ln (119901

2119884119897)

+ (num obs2119897minus 1198732119883119897minus 1198732119884119897)

times ln (1 minus 1199012119883119897minus 1199012119884119897))

(6)

where 119873119894119895119897

is the total number of votes by members of type119894 for candidate 119895 that were observed during treatment 119897 (orsession) and num obs

119894119897is the total number of observations

for someone of type 119894 in treatment 119897 These observationsinclude decisions for any of the three possible choices votefor Candidate119883 vote for Candidate 119884 and abstain

One of the contributions of this paper is the incorporationof ethical (altruism-motivated) voting into aQRE frameworkTable 2 shows the payoffs for an ethical candidate and a selfishcandidate (following the terminology used in the literature)


The selfish candidate offers a high payoff to one voter typeand a low payoff to the other type The ethical candidateoffers payoffs that have a more equitable distribution acrosstypes In this respect the payoff structure followsMorton andTyran [5] In Table 2 the ethical candidate offers payoffs thatare equal across types Also the difference between the highand low payoffs is the same for both types Neither of thesefeaturesmdashthe ethical candidate offering exactly equal payoffsor setting payoffs so that the payoff difference is the same forboth typesmdashare necessary but it matches the experimentaldesign that was used in this research

In Table 2 119902 and 120579 stand for the baseline equitable payoffand the payoff difference respectively Candidate 119883 is theselfish candidate and Candidate 119884 is the ethical candidateCandidate 119884rsquos payoffs maximize the minimum payoff andalso minimize the difference between the payoffs received bythe two types Members of Type 1 receive a higher payoffif Candidate 119883 wins but may be motivated by altruism tovote for Candidate 119884 even though they would receive a lowerpayoff if this candidatewon the electionThis is ethical votingOn the other handmembers of Type 2 receive a higher payoffif Candidate 119884 wins and would have no rational motive forvoting for Candidate 119883 because this candidate offers them alower payoff and offers inequitable payoffs

Selfish voting happens when economic self-interest isacted upon by voting for the candidate who offers onersquostype a higher payoff This would happen when Type 1 votesfor Candidate 119883 or when Type 2 votes for Candidate 119884As discussed above ethical voting happens when altruismoutweighs economic self-interest and this would happenmdashat least potentiallymdashwhen Type 1 votes for Candidate 119884 It isalso possible that this would occur because of noise This isone example of how the chat feature proves useful for gaininginsight into the motives behind voting decisions There is norational motive for Type 2 to vote for Candidate119883 thereforethis voting choice would be due to noise

One difference between this design and Morton andTyranrsquos [5] is that in this design the aggregate payoffs are thesame across candidates I designed it this way so that in theabsence of ethical motives equilibrium predictions would bethe same across symmetric and asymmetric payoff structuresfor a given electorate size and a given noise level

Ethical voting can be divided into two categories ethicalexpressive and ethical instrumental For ethical expressivevoting utility (or in this model implicit payoff) is gainedfrom the act of voting for the ethical candidate regardless ofthe outcome of the election This is along the lines of a warmglow parameter as in Andreoni [14] It is captured by the 120572term in the net expected payoff difference equation belowfor someone of Type 1 voting for Candidate 119884 and facing aperson- and election-specific cost 119896


1119884minus 119888119896+ 120572 (7)

Because ethical expressive voting is an additive term it isindependent of the pivotal probability Therefore it wouldincrease the turnout rate even as the pivotal probabilitydecreases mitigating or even possibly outweighing the effectof the declining pivotal probability The other three net

expected payoff difference equations for the other type-candidate combinations remain the same as in (3)

For ethical instrumental voting utility is gained if theethical candidate wins Therefore the utilitypayoff gain istied to the probability of onersquos vote for that candidate beingpivotal This is captured in the 120575 term in the equation below

1205871119884= (1198811119884minus 1198811119883+ 120575) times Pr (pivotal)

1119884minus 119888119896 (8)

In the presence of either ethical instrumental or ethicalexpressive voting the estimation procedure for 120582 and 120572 or for120582 and 120575 is the same as before except that now two parametersare being estimated In each case either 120572 or 120575 respectively isidentified through an exclusion restriction because the ethicalvoting parameter only appears in one of the four net expectedpayoff difference equations Therefore 120582 is pinned down bythe other three net expected payoff difference equations andthen 120572 or 120575 is pinned down given 120582

In this model it is not possible to simultaneously modelboth ethical expressive and ethical instrumental preferencesIn that case the net expected payoff difference equationwould be


1119884minus 119888119896+ 120572 (9)

However because 120572 = 120575 times Pr(pivotal)1119884

there are an infinitenumber of possible combinations and 120572 and 120575 cannot beseparately identified

22 Experimental Design These theoretical predictions wereempirically tested through a series of laboratory experimentsAll sessions were conducted in the Veconlab ExperimentalEconomics Laboratory at the University of Virginia onundergraduate students using Veconlab softwareThere were180 subjectsmdash109 male and 71 femalemdashacross a total of 12sessions Each subject could only participate in one sessiontherefore all subjects were inexperienced at the start oftheir respective sessions Additionally each session utilizedonly one treatment so each subject only participated inone treatment The average payout per subject was $35including a $6 payment for showing up and individualpayouts depended upon individual decisions as well as thecollective voting outcome

Each session consisted of 20 rounds or elections Eachsubject was randomly assigned to one of two voting typesType 1 and Type 2 at the start of the session and remainedat that same type throughout the entire session in order toavoid reciprocity effects Within each session equal numbersof subjects were assigned to both types

There were three possible choices vote for Candidate 119883vote for Candidate 119884 and abstain Anyone from either votertype could vote for either candidate In the experiment elec-tions were referred to as rounds and candidates were referredto as options as in ldquoOption119883rdquo and ldquoOption 119884rdquo in an attemptto keep the language as neutral as possible Throughout thispaper ldquoroundrdquo and ldquoelectionrdquo will be used interchangeablyand the options will be referred to as ldquoCandidate 119883rdquo andldquoCandidate 119884rdquo except in instances in which the word choiceis relevant to the discussion at hand


As discussed in the previous section whichever candidategarnered the most votes within a round was declared thewinner with ties decided with a virtual coin flip Afterthe conclusion of each round all subjects were told whichcandidate had won and the total number of votes for eachcandidate (but not the breakdown of how many of thosewere cast by members of each type) Each candidate offereddifferent payoffs to each voter type and subjects were paid theamount that had been promised to their type by the candidatewho won in that round regardless of whether or for whichcandidate they had voted At the end of the session subjectswere paid their cumulative earnings

In each round each subject drew a voting cost fromthe same distribution Costs ranged from $0 to 042 inincrements of $002 This increment was chosen so that therewere 22 possible costs and 20 rounds meaning that onaverage there would be a good chance of a subject drawingalmost all of the possible costs over the course of the session

The upper bound of the range of costs was chosento generate at least some abstention Along the lines ofLevine and Palfrey [4] the payoff from making or breakinga tie would equal (119881

119894119895minus 119881119894119898)2 for a member of type 119894

voting for candidate 119895 where 119898 is the other candidate Forboth the asymmetric and symmetric treatments the payofffrom making or breaking a tie would equal $025 for bothtypes according to the payoffs detailed below Therefore forsubjects who drew a cost greater than this abstention wasthe strictly dominant strategy If there were a cost equal to$025 then abstaining would be the weakly dominant strategyof a subject who drew this cost but this was not a possibledrawbecause of the $002 increment sizeThehighest possiblecost was set above the payoff from making or breaking atie so that there would not exist an equilibrium in whichthe probability of voting was equal to one (ie in order togenerate at least some abstention in equilibrium)Note that119881

119894119895

is assumed to be the higher payoff As detailed earlier therewould be no cross-voting in the Nash equilibrium becausethe payoff differences would be negative leading to a negativecost cutpoint

At the end of the session subjects participated in anexercise to elicit the subjectsrsquo perceived probabilities that theirvote waswould have been pivotal in the final round to testthe relationship between voting turnout and the (perceived)probability of casting a pivotal vote An analysis of the elicitedprobabilities showed a significant and positive relationshipbetween the elicited probabilities and voting decisionsmdashsubjects who thought that they had a higher probability ofbeing pivotal weremore likely to have voted in the final roundand with lower elicited probabilities were less likely to havevotedThis section is available from the author upon request

The subjectsrsquo computer screens showed short ldquobacksto-riesrdquo next to each candidatersquos offered payoffs in the type-candidate payoff chart The purpose of these was to helpthe subjects in the treatments with asymmetric payoffsrealize the possibility of ethical voting However in orderto avoid inadvertently encouraging or leading some subjectsto vote in a certain waymdashas had happened in some of thepilot experimentsmdashneutral wording was used here The newbackstories merely repeated the payoffs in words

The backstories for asymmetric payoffs read as followsldquoOption 119883 will implement an investment that pays $2 toType 1 voters and $1 to Type 2 votersrdquo and ldquoOption 119884 willimplement an investment that pays $150 to both Type 1 andType 2 votersrdquo For symmetric payoffs the backstories were asfollows ldquoOption 119883 will implement an investment that pays$150 to Type 1 voters and $1 to Type 2 votersrdquo and ldquoOption119884 will implement an investment that pays $1 to Type 1 votersand $150 to Type 2 votersrdquo

Before the voting began in each round subjects were ableto chat online with their ldquoneighborsrdquo according to the IDnumber that had been assigned upon first logging into theprogram For example ID 3 could chat with ID 2 and ID 4Meanwhile ID 4 could chat with ID 3 and ID 5 The subjectswith ID numbers at either endmdashID 1 and ID 12 or ID 18depending on the session sizemdashwere in the same chat groupcreating a circular network Subjects could also see the votertype of each of the other two subjects in their chat groupsThechat period lasted exactly one minute in each round and itwas not possible for anyone to submit a voting decision untilafter the chat ended

This design models some features of real world discus-sions about politics For example most people only discussan upcoming election with people whom they know directly(friends family members possibly coworkers etc) Evenfor those who broadcast their views on social media thenumber of people who are on the receiving end of that isstill small relative to the total size of the electorate Given thesmall electorate size in this experiment limiting a subjectrsquoschat circle to only two other people roughly mirrors thisAdditionally the overlapping group structuremirrors the factthat social groups are not self-contained so that ideas orinformation can potentially be spread to a large number ofpeople

Furthermoremost of the chat groups containedmembersof both voting types This was done to reflect the factthat most people have family members friends colleaguesacquaintances and so forth who hold different political viewsthan they do (This differs from the focus ofmuch of the groupidentity literature on the effects of communication withingroups of voters) Some subjects were part of a majoritywithin their chat group and others were the minority and ina few cases the subject was in a group consisting entirely ofthat subjectrsquos type

Finally the chat transcripts are extremely useful forproviding insight into the subjectsrsquo motivations for voting acertain way or for abstaining altogether

The experiment was divided into four treatments whichwere designed to test the effects of electorate sizemdashandthereby the role of the probability of casting a pivotal votemdashand the possibility of ethical voting First the number ofsubjects per sessionmdashwhich can also be thought of as thesize of the electoratemdashwas varied Half (six) of the sessionshad 12 subjects per session and the remaining sessions had18 subjects According to the underlying theory the turnoutrate should fall as the electorate size increases because theprobability of onersquos vote being pivotal decreases Second thepayoffs were varied so that in half of the sessions the subjectsfaced symmetric payoffs and in the other half the payoffswere


Table 3 Symmetric and asymmetric payoffs

Asymmetric SymmetricType 1 Type 2 Type 1 Type 2

Candidate119883 $200 $100 $150 $100Candidate 119884 $150 $150 $100 $150

asymmetric These payoffs which were common knowledgewithin a session can be seen in Table 3

In the treatments with asymmetric payoffs Candidate 119883was the selfish candidate offering a high payoff ($200) toType 1 subjects and a lowpayoff ($100) to Type 2 subjectsTheethical candidate Candidate 119884 offered equal payoffs to bothvoter typesThese payoffsmaximize theminimumpayoff andminimize the difference between the payoffs received by thetwo groupsmdashindeed in this design members of both groupswould receive the same payoff The payoff of $150 for eachgroup that was received by subjects of both voter types ifCandidate 119884 won was lower than what Type 1 subjects wouldreceive and higher than what Type 2 subjects would receiveif Candidate119883 wonThis design was inspired by Morton andTyran [5]

To account for the possibility that Candidate 119883 beinglisted first on the table (or119883 coming before 119884 alphabetically)might have affected voting decisions the roles of ethicaland selfish candidates were reversed in roughly every otherasymmetric session so that Candidate 119883 offered the equalpayoffs and Candidate 119884 offered the inequitable payoffsEverything else remained the sameNomeaningful differencewas observed in the data In the final data 119883 and 119884 werereversed back for these sessions to make the data consistentwith the other sessions

Under this payoff design members of Type 1 may faceconflicting incentives in choosing a candidate Type 1 subjectswouldmaximize their earnings if Candidate119883won butmightalso decide to vote for Candidate 119884 for altruistic reasons

In the symmetric payoff design the two candidatesoffered payoffs that were mirror images of each other Thismeant that members of both voter types had the sameeconomic incentives to vote for the candidate who wasoffering onersquos type the higher payoff Each candidate wasequally inequitable so there was no possibility of ethicalvoting here These treatments served as a control to measurethe effect of noise in the decision-making process in theabsence of any potential for ethical voting

Each of the four treatments was a combination of sessionsize (12 or 18) and payoff structure (asymmetric or symmet-ric)

The payoffs shown in Table 3 were set such that theequilibriumpredictions of voting turnout rates were the sameacross asymmetric and symmetric payoff structures becausethe difference between the two candidatesrsquo offered payoffsis identical across the two payoff structures so that the netexpected payoff difference in functions from (3) remains thesame

Meanwhile the number of subjects per session wouldaffect the turnout predictions through the effect of the prob-ability of onersquos vote being pivotal on the net payoff function

Table 4 (a) Predicted turnout probabilities and cost cutpoints (12subjects) (b) Predicted turnout probabilities and cost cutpoints (18subjects)

(a)

120582 = 100 (Nash) 120582 = 7 (L and P 2007) 120582 = 02 (high noise)1199011119883

02996 03069 033331199011119884

0 00793 03241199012119883

0 00793 03241199012119884

02996 03069 033331198881119883

01221198881119884

01198882119883

01198882119884

0122

(b)


02608 03069 033331199011119884

0 00793 03241199012119883

0 00793 03241199012119884

02608 03069 033331198881119883

01221198881119884

01198882119883

01198882119884

0122

These both decrease (increase) as the electorate size increases(decreases)Therefore the predicted turnout probabilities fora sessionwith 18 subjectsmdashregardless of whether it is a sessionwith asymmetric or symmetric payoffsmdashare lower than thosefor sessions with 12 subjects all else equal

Tables 4(a) and 4(b) show the predicted turnout proba-bilities for the Nash equilibrium (extremely high 120582) for 120582 = 7(the value thatwas estimated in Levine andPalfrey 2007) andfor extremely high noise (very low 120582) The predictions for asession with 12 subjects are shown in (a) and the predictionsfor a session with 18 subjects are shown in (b)

In Tables 4(a) and 4(b) 119901119894119895is the predicted probability

that a member of type 119894 would vote for candidate 119895 Theprobability of a member of type 119894 abstaining is (1minus119901

119894119883minus119901119894119884)

As shown in Tables 4(a) and 4(b) the probability ofvoting is increasing in the amount of noise (decreasing inthe amount of precision) This is true for both own-votingand cross-voting where own-voting is defined as voting forthe candidate offering onersquos type the higher payoff and cross-voting is defined as voting for the candidate offering onersquostype the lower payoff However it is much more pronouncedfor cross-voting In the Nash equilibrium there is no cross-voting because in the presence of extremely high levels ofprecision (low noise levels) in the decision-making processand in the absence of altruism nobody would choose tovote for the candidate offering them a lower payoff becauseit would imply a negative expected payoff difference and anegative cost cutpoint As the level of noise increasesthe levelof precision falls it becomes increasingly likely that at leastsome of the subjects would cross-vote in spite of the negativeexpected payoff difference


Table 5 Predicted turnout probabilities for different levels of ethical preferences (120572)

For 120582 = 7 120572 = 0 120572 = 01 120572 = 025 120572 = 05 120572 = 075 120572 = 1

Type 1 for Candidate119883 03071 02708 01606 00295 00052 00009Type 1 for Candidate 119884 00793 01575 03988 08232 09637 09934Type 2 for Candidate119883 00793 00818 01059 01629 01702 01707Type 2 for Candidate 119884 03071 02921 02347 0175 0171 01708Candidate119883 ( of all voters) 01932 01763 013325 00962 00877 00858056Candidate 119884 ( of all voters) 01932 02248 031675 04991 056735 05821

For a session with 12 subjects (Table 4(a)) the costcutpoint is $0122 which means that there are seven possiblecostsmdashbetween $000 and $012mdashthat are low enough thatsomeone would vote (for the candidate offering onersquos type thehigher payoff) with probability of 1 Seven out of 22 possiblecosts equal approximately 30 which is in line with theturnout probabilities in the Nash equilibrium For a sessionof 18 subjects (Table 4(b)) the cost cutpoint is $01047 whichmeans that there are six possible costsmdashbetween $000 and$010mdashthat are low enough to induce someone to vote Sixout of 22 possible costs equal approximately 27 which is inline with the turnout probabilities in the Nash equilibrium

One other thing to note is that as the noise level increasesto the pointwhere subjects are deciding randomly the highestpossible predicted turnout rate will be 33 (33 of each typewould choose Candidate119883 33 would choose Candidate 119884and 33 would abstain) This is much lower than the 50probabilities that would be the case if they could only choosebetween voting and abstaining such as in Levine and Palfrey[4]Thiswill affect the ability ofQRE to explain large amountsof overvoting as will be discussed in the results section

Additionally the model predicts that increasing levelsof ethical preferences can increase the percentage of theelectorate voting for the ethical candidate and decrease thepercentage voting for the selfish candidate (Table 5) (As inTable 4 the first four rows of Table 5 measure the percentageof that votersrsquo type who vote for a given candidate The lasttwo rows capturing the voting decisions across voter typesmeasure the percentage of all voters) As the level of altruismincreases ethical voting would reduce the closeness of therace which in turn would decrease the probability of castinga pivotal vote This loss of pivotality would not matter for theethical voters whose increasingly strong altruisticmotivationwould outweigh this but it would lead those who would havevoted for the selfish candidate to abstain

3 Results and Discussion

There are several themes that appear throughout the resultsThe first is that voter turnout far exceeded even the highestpossible turnout predicted by QRE This was largely concen-trated in votes for the candidate offering the higher payoffto that subjectrsquos voter type and as such is not due to ethicalvoting This indicates a strong economic self-interest Inaddition contrary to the predictions of the modelmdashwhichwere that in the absence of altruism the number of subjects(size of the electorate) would affect turnout but the symmetryof the payoff structure would notmdashthe number of subjects

had no significant effect and turnout was significantly higherin sessions with symmetric payoffs relative to those withasymmetric payoffs (However the lack of a size effect is notentirely surprising considering that the theoretical differencesin economically self-interested voting rates were not verylarge as seen in Tables 4(a) and 4(b)) Third some ethicalvoting (where motives were confirmed through the chattranscripts) and potentially ethical voting were observedhowever it was mostly found only under certain conditionsand not enough was observed to explain the overvoting

31 Terminology Before getting into the analysis some ter-minology that is used throughout this section needs to bedefined ldquoTurnoutrdquo refers to all voting regardless of type orcandidate ldquoFavoredrdquo refers to votes for the candidate whooffers that subjectrsquos type a higher payoff than what the othercandidate offers Type 1 for Candidate 119883 and Type 2 forCandidate 119884 Similarly ldquounfavoredrdquo refers to votes againsttheir economic self-interestmdashType 1 for Candidate 119884 andType 2 for Candidate 119883 The unfavored category does notdistinguish between potentially ethical votingmdashType 1 votingfor119884 in a session with asymmetric payoffsmdashand cross-votingthat is done for other reasons

32 Voting Turnout Rates

321 Overview The overall turnout rates averaged across allrounds and broken down by type-candidate combinationscan be seen in Table 6(a)

The turnout rates for the candidates offering each typea higher payoffmdashCandidate 119883 if Type 1 and Candidate 119884 ifType 2mdashare much higher than even the highest turnout ratespredicted under QRE for any type-candidate combinationwhich topped out at 33 even in the presence of the highestpossible levels of noise Possible reasons for this overvotingeven relative to QRE predictions will be addressed laterThe second observation is that potentially ethical votingmdashType 1 voting for Candidate119884 in treatments with asymmetricpayoffsmdashis higher than nonaltruistic cross-voting in thesesame treatments which is Type 2 for Candidate 119883 Inthe treatments with symmetric payoffs where there is nopossibility of ethical voting this is not the case

There are two other notable relationships in the data Thefirst is that the turnout rate is higher for all type-candidatecombinations in the treatments with symmetric payoffsthan in those with asymmetric payoffs This contradictsthe modelrsquos prediction which was that there would be nodifference in turnout between the two payoff structures


Table 6 (a) Voting turnout rates (all rounds) (b) Voting turnout rates (rounds 11ndash20) (c) Voting turnout rates (rounds 1ndash10)

(a)

All treatments Symmetric Asymmetric 12 18Type 1 for Candidate119883 04911 05467 04356 04500 05185Type 1 for Candidate 119884 00639 00656 00622 00597 00667Type 2 for Candidate119883 00544 00733 00356 00653 00472Type 2 for Candidate 119884 05250 05533 04967 05417 05139

(b)


(c)


except for possibly higher turnout for Type 1 voting forCandidate in treatmentswith asymmetric payoffs (potentiallyethical voting) Also there is no systematic difference inturnout between the two session sizes This contradicts themodelrsquos predictions in which turnout would be higher insessions of 12 than in sessions of 18

The turnout rates in the second half of the data (rounds11ndash20) can be seen in Table 6(b)

This focus on the second half of the sessions (largely)sidesteps the effect of the learning curve that the subjectsexperienced in the early rounds Once again the effects ofsize and symmetry run counter to the modelrsquos predictionsTurnout is higher in treatments with symmetric payoffsrelative to those with asymmetric payoffs and session size hasno systematic effect

In rounds 11ndash20 the turnout rate for potentially ethicalvotingmdashType 1 voting for Candidate 119884 in treatments withasymmetric payoffsmdashis even lower than the nonaltruisticcross-voting done by Type 2 for Candidate 119883 again in theasymmetric treatments This points to altruism dissipatingand economic self-interest taking over fairly quickly Giventhat any possible altruism died out by the second half itwould be useful to look at the first half of the data (rounds1ndash10) This is shown in Table 6(c)

Here we see potential altruism with a turnout probabilityof 01089 for Type 1 subjects voting for Candidate 119884 inasymmetric treatments versus 00489 for Type 2 subjectsvoting for Candidate 119883 in those same treatments Also thesize and symmetry relationships are the same as before albeitnot as strongly

It is also helpful to see the trends over time not just theaverages Figure 2 shows the turnout ratemdashoverall turnoutrather than being broken down into the different type-candidate combinationsmdashover the course of the experiment

Within Figure 2 the top row shows the turnout rates for 12subjects (a) and for 18 subjects (b) and the bottom row showsthe turnout rates for asymmetric payoffs (c) and symmetricpayoffs (d)

A few trends appear in these figures First in all treat-ments the turnout rate decreases as the rounds go on Secondthere is no systematic difference between the 12- and 18-subject turnout rates This is contrary to the theoreticalresults which showed higher turnout for smaller electoratesizes And finally there is a clear difference between thetreatments with asymmetric versus symmetric payoffs Asbefore turnout in treatments with symmetric payoffs isalmost always higher This too is contrary to the theoreticalresults which were identical across payoff structures for agiven electorate size

322 Nonparametric Permutation Test A nonparametricpermutation test (also known as a randomization test) wasused to determine whether the turnout rates are differentacross treatments This nonparametric test was useful whenlooking at session averages because there are only 12 datapoints The results from the permutation tests showed thesame trends that were observed in the stylized facts thenumber of subjects per session does not significantly affectthe probability of voting (except for symmetric sessionsacross all rounds) and subjects in treatments with symmetricpayoffs are significantly more likely to vote than are subjectsin treatments with asymmetric payoffs

Details about the test and the results can be found in theAppendix

323 Econometric Analysis In the logit analysis the depen-dent variables are the different categories of voter turnoutmdash(ldquoturnoutrdquo) favored and unfavored In each of the following


1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1Pr

(vot

e)

12 asymmetric12 symmetric

(a)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)


(b)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)

12 asymmetric18 asymmetric

(c)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)

12 symmetric18 symmetric

(d)Figure 2 Voting turnout rates (top row symmetric versus asymmetric bottom row 12 versus 18)

regressions a random effects term was used to account forcorrelation within each subjectrsquos 20 decisions that cannot beattributed to any of the other independent variablesmdashsomesubjects were inherently more (or less) likely to vote or tovote in a certain way than other subjects

Table 7 shows the results for the following regressionsturnout

119894119905= 1205730+ 1205731(average vote difference)

119894119905

+ 1205732( of previous losses)

119894119905+ 1205733cost119894119905

+ 1205734type119894119905+ 1205735size119894119905+ 1205736asymmetric

119894119905

+ 1205737round

119894119905+ 1205738(type 1 times asymmetric)

119894119905

+ 1205739( same type)

119894119905

+ 12057310( group favored votes)

119894119905

+ 12057311( group unfavored votes)

119894119905

+ 119908turnout119894 + 119906turnout119894119905

favored119894119905= 1205740+ 1205741(average vote difference)

119894119905


119894119905+ 1205743cost119894119905


119894119905

+ 1205747round


119894119905

+ 1205749( same type)

119894119905


119894119905


119894119905

+ 119908favored119894 + 119906favored119894119905

unfavored119894119905= 1205790+ 1205791(average vote difference)

119894119905


119894119905+ 1205793cost119894119905


119894119905


Table 7 Logit with subject-level random effects and cluster-robust standard errors (marginal effects)

All rounds Rounds 11ndash20 Rounds 1ndash10Turnout Favored Unfavored Turnout Favored Unfavored Unfavored

History average |119883 minus 119884| minus0367 minus0508 0037 minus0415 minus0435 minus2863 0462(010) (010) (031) (023) (023) (136) (034)

Number consecutive losses minus0008 minus0012 0001 minus0004 minus0009 0000 minus0001(000) (000) (000) (000) (001) (000) (000)

Type 1 minus0013 minus0002 minus0004 0011 minus0017 0000 minus0011(006) (005) (001) (006) (006) (000) (001)

Cost minus1554 minus1620 minus0343 minus2248 minus2210 minus0735 minus0254(013) (012) (019) (022) (021) (036) (024)

Asymmetric minus0129 minus0094 minus0020 minus0144 minus0140 minus0001 minus0031

(006) (006) (001) (006) (006) (000) (002)

Size 12 minus0022 minus0031 000 minus004 minus0026 0000 0002(004) (003) (001) (004) (005) (000) (001)

Round minus0009 minus0001 minus0002 minus0005 minus0002 minus0000 minus0004

(000) (000) (000) (000) (000) (000) (000)

Type 1 asymmetric minus0023 minus0064 0016 minus0034 minus0005 0000 0035

(008) (008) (001) (009) (009) (000) (002)

Number same type minus0045 minus0032 minus0002 minus0001 0025 minus0001 minus0002(004) (004) (001) (004) (004) (000) (001)

Number favored 0118 0163 minus0013 0086 0095 minus0000 minus0025

(002) (002) (001) (003) (003) (000) (001)

Number unfavored 0025 minus0019 0005 0004 000 0000 0010

(002) (001) (000) (002) (002) (000) (001)LR test (119901 value) 0000 0000 0000 0000 0000 0000 0000119901 lt 010 119901 lt 005 119901 lt 001

Marginal effects elasticities used for continuous variables history average |119883 minus 119884| cost All others are marginal effects

+ 1205797round


119894119905

+ 1205799( same type)

119894119905


119894119905


119894119905

+ 119908unfavored119894 + 119906unfavored119894119905

(10)

The last line in Table 7 is the likelihood-ratio test of whetherthe residual intraclass correlation equals zero (not a marginaleffect even though it was consolidated into this table) Thistests whether the random effects term should be omittedTheresults show that the use of random effects was appropriate

Table 7 shows the marginal effects where marginal effectelasticities were used for continuous variables and the usualmarginal effects were used for binary or count variables Thisis because ldquoby what percentage does (dependent variable)increase if Asymmetric increases by 10rdquo is not meaningfulgiven that Asymmetric is binary Note that the scales aredifferent so that an elasticity of 034 and a marginal effectof 0034 (using an arbitrarily chosen example) would bothbe interpreted as a 034 increase in the dependent variableThis is why the results for ldquoHistory average |119883 minus 119884|rdquo andldquoCostrdquo look so much larger than the others

The independent variables are as follows ldquoHistory aver-age |119883 minus 119884|rdquo measures the average up until any given roundof the absolute value of the differences between the numberof votes received by each candidate in previous roundsThis captures how close the outcomes of the elections hadbeen and might have played a role in subjectsrsquo perceptionof the probability of their vote being pivotal in the currentround The second variable ldquonumber consecutive lossesrdquo isthe number of consecutive rounds immediately precedingthe round in question that a subjectrsquos typersquos favored candidate(offering that type the higher payoff) has lost This capturesdiscouragement ldquoType 1rdquo is the voter type and equals one ifType 1 and zero if Type 2 ldquoCostrdquo is the voting cost between$0 and $042 ldquoAsymmetricrdquo equals one if the observationis from a session with asymmetric payoffs and zero if it isfrom a session with symmetric payoffs ldquoSize 12rdquo equals one ifthe observation is from a session with 12 subjects and equalszero if it is from a session with 18 subjects ldquoRoundrdquo is theround number between 1 and 20 ldquoType 1 asymmetricrdquo isan interaction term for a Type 1 subject in a session withasymmetric payoffsmdashthe only combination for which ethicalvoting (Type 1 for Candidate 119884) is possiblemdashand equalsone if the subject fits that description and zero otherwiseldquoNumber same typerdquo is the number of members of a subjectrsquoschat group (excluding the subject) who are members ofthe same voter type as the subject ldquoNumber favoredrdquo and


ldquonumber unfavoredrdquo are the number ofmembers of a subjectrsquoschat group (again excluding the subject) who voted for thesubjectrsquos favored or unfavored candidate respectively in thecurrent round I also tested this using versions of ldquonumberfavoredrdquo and ldquonumber unfavoredrdquo that use a rolling averageof the previous 3 rounds and the results were qualitativelysimilar in both magnitude and significance Therefore Idecided to use the version above to focus on the moreimmediate effect of the most recent chat

This analysis was done for all rounds and for the secondhalf (rounds 11ndash20) for overall turnout favored and unfa-vored and for the first half (rounds 1ndash10) for unfavored onlyIt is uncommon to focus on the first half of the experimentaldata because subjects often experience a learning curveduring the first several rounds however in this case all ofthe ethical (motives confirmed through the chat transcripts)and potentially ethical voting occurred in the first half Thisoccurs in unfavored voting therefore this column is includedin Table 7

History average |119883 minus 119884| which measures the averagenumber of votes separating the two candidates in previousrounds (the average margin) is significant and negative forboth overall turnout and favored turnout both for all roundsand for rounds 11ndash20 This means that the farther apartthe vote counts had been in the previous rounds the lesslikely subjects were to vote or conversely the closer theprevious rounds had been the more likely subjects were tovote Looking at the results for all rounds if the vote countsin the previous elections were 10 farther apart the turnoutrate would fall by 367 overall (or conversely if previouselections were 10 closer the turnout rate would increaseby 367) and 508 for favored The results in the secondhalf were qualitatively similar albeit less strongly significantpossibly due to the average becoming less volatile as roundsgo on To make sure that this was not simply because theaverage incorporates more rounds as the session goes onso that each incremental round has a smaller impact on theaverage I also tried a rolling average of the previous threerounds and got qualitatively similar results

The negative relationship between the size of the marginof victory in previous elections and the probability of votingin the current election (or the positive relationship betweenthe closeness of previous elections and the probability ofvoting) may be because of the effect on subjectsrsquo perceptionsof the probability of their vote being pivotal This resultand interpretation qualitatively matches the results of thebelief elicitation from the extended version of this paper(available upon request) which found that subjectsrsquo ex antebeliefs about the probability of their vote being pivotal werepositively related to the probability of votingThis combinedwith the lack of size effect in the data points to a possibilitythat votersrsquo perceptions of the probability could be the channelthrough which the probability of casting a pivotal voteimpacts voting participation decisions rather than the actualprobability

The number of consecutive losses immediately precedinga round is significant and negative for both overall turnoutand favored turnout the results show that with each addi-tional consecutive loss the probability of voting falls by 09

and the probability of voting for favored falls by 12 Forunfavored the number of consecutive losses is significantbut positive It is not entirely clear why but may reflect thefact that as the number of previous losses adds up thesubjects who have a better understanding of the game maybe more likely to abstain leaving more voters who have aworse understanding and who would be more likely to makevoting choices that go against their economic self-interestThis relationship weakens when focusing on only the secondhalf because in most sessions losingwinning streaks wereshorter in the second half

The next significant variable is cost which is highly sig-nificant and negative for all three dependent variables For a10 increase in voting cost the probability of voting for eithercandidate falls by 1554 the probability of voting for favoredfalls by 1620 and the probability of voting for unfavoredfalls by 343 The effect of cost on unfavored voting isboth smaller in magnitude and in significance level possiblybecause noneconomic factors would play a larger role inunfavored voting decisions and therefore the voting costwould have less impact The cost effect is even stronger whenfocusing on only the second half as subjects have gained abetter sense of when it is or is not worth it to vote In thefirst half cost does not significantly affect unfavored votingpossibly because of the ethical (and potentially ethical) votingand the higher level of noise that occurred in the first halfboth of which would have made subjects less sensitive tocosts

The results for size and symmetry match those fromthe permutation test and Figure 2 in that the probability ofvoting is lower in sessions with asymmetric payoffs relative tosessions with symmetric payoffsmdashfor all rounds subjects insessionswith asymmetric payoffswere 129 less likely to voteat all 94 less likely to vote for favored and 20 less likely tovote for unfavoredmdashand the number of subjects per sessionis not significant What could explain the higher turnoutprobability in sessions with symmetric payoffs relative tothose with asymmetric payoffs There is no clear answer

The results in Table 5 show that the model predictsincreases in both unfavored voting and overall turnout as thelevel of altruism increases but the relationship is negative inthe empirical results However the drop in favored votingin Table 5 as altruism increases does match the directionof the empirical results It is possible that in the absenceof any possibility for ethical voting the subjects may haveperceived the elections as being closer which spurred greateramounts of voting In the data the elections were actuallycloser in sessions with asymmetric payoffs but work withbelief elicitation could potentially uncover a connection tothe subjectsrsquo perceptions of election closeness rather than theactual margins

The interaction term of Type 1 and asymmetric capturesthe voters who have the option of voting ethically Theresults here show that potentially ethical votingmdashType 1subjects voting for Candidate 119884 (unfavored) in sessionswith asymmetric payoffsmdashis 35 more likely to occur thannonethical unfavored voting during the first half whichis when almost all of the ethical voting (both potentiallyethical and transcript-confirmed ethically motivated) took


Table 8 Maximum likelihood estimates

Parameters Rounds Sessions or treatments Coefficient Standard errors 119905-statistics

120582

2nd half All treatments 54909 05164 1063321st half All treatments 10853 02841 382051st half Asymmetric treatments 1732 04339 397041st half Symmetric treatments 05687 03767 15098

120582 and 120572 1st halfSession 1 session 7120582 05448 08978 06068120572 0129 03992 0323


place Note that all unfavored voting in sessions with sym-metric payoffs and unfavored voting by Type 2 subjects insessions with asymmetric payoffs are not even potentiallyethical

It is possible that ethical voting could have had a largerimpact if the asymmetric payoff design featured higheraggregate payoffs Fowler and Kam [15] found that altruistsonly had a larger incentive to participate relative to thosewho are self-interested when outcomes are perceived asbenefitting everyone Outcomes that are perceived as beingmerely distributive without a larger aggregate benefit areseen as being distributive and altruists gain nothing frommerely shifting wealth from one group to another In theexperimental design used here the aggregate payoffs areidentical across candidates so the ethical candidate could beperceived as being distributive

Round which captures learning effects is significant andnegative for all three dependent variables when looking atall rounds and also for unfavored in rounds 1ndash10 As asession progresses the probability of voting falls by 09 (foroverall turnout) 01 (for favored) or 02 (for unfavored)with each additional round This points to the presence of alearning effect as the session proceeds possibly as subjectsfigure out that they are less likely to be pivotal than they hadanticipated or that it does not make sense to vote when facinga high cost Then the size and the strength of the significancedecline by rounds 11ndash20 indicating that the learning effecthad dissipated because the steepest part of the learning curvehappened early in the session

The next three variables analyze the effect of subjectsrsquochat groups on voting participation decisions The groupaffiliation of the members of a subjectrsquos chat group did notsignificantly affect voting decisions but whetherfor whomtheir chat-mates voted did influence voting decisions Look-ing at the results for all rounds ldquonumber favoredrdquo increasedfavored voting by 163 showing that economically self-interested voting participation was substantially increased bythe influence of group membersrsquo voting decisions to votefor that same candidate Unfavored voting decreased as thenumber of group members voting for a candidatersquos favoredcandidate increased as voters who might have been leaningtowards to vote against type were encouraged to vote for theirfavored candidate instead

Unfavored voting increased in the number of groupmemberswho voted for that candidatesrsquo unfavored candidatepossibly due to factors such as group members encouragingeach other to vote for the ethical candidate (where applica-ble) voting in solidarity with group members regardless ofwhether it is in the subjectrsquos best interests and manipulationInstances of these as observed in the chat transcripts canbe found in Section 34 However this 05 increase (or1 in the first half) is much smaller than the effect ofldquonumber favoredrdquo on favored voting possibly because it is aneasier ldquosellrdquo to convince people to vote for someone who ispromising them more money than the other candidate

33 Estimating the QRE and Ethical Voting Parameters Themodel that was presented in Section 2 can be used toestimate the QRE precision parameter 120582 and the ethicalvoting parameters 120572 and 120575 from the experimental dataTo review 120582 is a measure of the level of precision in thedecision-making process of whether and for whom to voteThe ethical expressive parameter 120572 measures the utility(or nonpecuniary expected payoff) received from the actof voting ethically This is along the lines of a warm glowparameter as in Andreoni [14] The ethical instrumentalparameter 120575 measures the utility (or nonpecuniary expectedpayoff) received from voting ethically if that vote is pivotaland leads to a victory for the ethical candidate

Table 8 shows the results of the estimation using theexperimental data

First 120582 was estimated for the second half (rounds 11ndash20) Ethical voting had dissipated by then so limiting it tothese rounds made it possible to estimate the QRE precisionparameter in the (effective) absence of ethical voting Theestimated value for 120582 was 549 which is lower than whatLevine and Palfrey [4] had estimated (120582 = 7) meaning thatthere was more noiseless precision in the decision-makingprocess here relative to Levine and Palfrey [4] In that papersubjects only had a choice between voting and abstaining sothe higher noiselower precision may be due to the additionof a third option (here the possibility of choosing eithercandidate)

On a related note it may also reflect the fact that votingturnout rates for Type 1 voting for Candidate 119883 and Type 2voting for Candidate 119884 are substantially above the highest


0

02

04

06

08

1

1 for X 1 for Y 2 for X 2 for Y

Pr (v

ote)

0

02

04

06

08

1

Pr (v

ote)

12 asymmetric (data)12 symmetric (data)Size 12 (QRE)



Figure 3 QRE versus data rounds 11ndash20


0

02

04

06

08

1

1 for X 1 for Y 2 for X 2 for Y 1 for X 1 for Y 2 for X 2 for Y

Pr (v

ote)

0

02

04

06

08

1

Pr (v

ote)



possible predicted turnout under QRE which was 033 in thepresence of extremely high noise as shown in Tables 4(a) and4(b) This is much lower than the 05 predicted under highnoise in Levine and Palfreyrsquos model (with only two optionswhen noise is so highprecision is so low that decisions aremade randomly the probability of each option approaches50) leading to a smaller range within which overvotingcould be explained by noise

Figure 3 illustrates this overvoting contrasting the QREturnout predictions for 12 subjects and 18 subjects given a 120582of 549 with the observed turnout rates For favored voting(Type 1 for 119883 and Type 2 for 119884) the observed turnout ratesare much higher than the predicted rates and the observed

turnout rates for unfavored voting (cross-voting) are lowerthan the predicted rates

Next as a comparison 120582was estimated for the first half ofthe data As expected the estimate of 120582 for the first half is con-siderably smallermdashindicating lower levels of precisionhigherlevels of noisemdashthan those in the second half This is becauseof both higher turnout rates and greater levels of cross-votingA comparison of observed turnout rates andQRE predictionsis illustrated in Figure 4 Surprisingly the estimated precisionparameter is lower (higher noise) for sessions with symmetricpayoffs in spite of lacking any possibility of ethical votingHowever looking at the data there are comparable levels ofcross-voting and the turnout rates of Type 1 for Candidate


119883 and Type 2 for Candidate 119884 are generally higher in thesymmetric treatments This higher turnout in favored votingis what is driving the differenceThis is because the estimated120582 is pulled lower by the high favored voting rates above andbeyond what the model can explain which is treated as extranoise

However the end goal is not only to estimate the QREprecision parameter 120582 but also to estimate the ethical votingparameters This is done in the final two sections of Table 8(As explained earlier the ethical voting parameters 120572 and 120575cannot be separately identified so these are the results forseparately estimated models one with 120572 and one with 120575)Unfortunately for either of these specifications neither 120582 northe respective ethical voting parameter is significant evenwhen only focusing on the two sessions with the highest ratesof overvoting

34 Chat The online chat feature provides insight into themotivations behind subjectsrsquo voting decisions (The chattranscript for all sessions is available upon request) Twonoteworthy themes appear in the chat the first is the powerof the overlapping chat circles to affect outcomes in bothuseful and harmful ways and the second is peer pressurepeerinfluence even to the point of outright manipulation

341 Transmission of Information Although at least a smallamount of potentially ethical voting was observed in mostsessions with asymmetric payoffs it appeared in a substantialfashion in Sessions 1 and 7 What differentiated these sessionsfrom the other sessions with asymmetric payoffs was that ineach case one of the subjects picked up on the possibility ofethical voting and voted accordingly and then broadcastedthis realization and acted as a cheerleader of sorts to try to getType 1 subjects to vote for Candidate 119884 The ldquocheerleadersrdquowere both Type 1 and Type 2 although their motivationsdiffered according to type These cheerleaders could bethought of as being thematically related to the producers ofsocial pressure in Schram and van Winden [9] In Session 1ID 7mdashwhowas Type 1mdashstarted this chain of events in the chatbefore round 1 ID 7 picked up on the idea of ethical votingldquoDO NOT BE GREEDY WE WILL ALL MAKE MOREMONEY IF WE ALL CHOOSE THE LOWER OPTIONrdquo(All quotes appear exactly as in the chat transcript) Strictlyspeaking this was not entirely true because the aggregatepayoff was the same regardless of which candidate won andindeed Type 1 made less money when Candidate 119883 wonHowever the comment still demonstrates a desire to spreadthe payoffs more equitably This subject then repeated this inthe chat before round 2 Also in round 2 ID 8mdashwho was aType 2 and therefore was acting in his or her own economicself-interestmdashtook up ID 7rsquos argument by saying ldquoType 1 isbeing greedyrdquo which spread the idea to ID 9mdashwho was also aType 2mdashwho said ldquoPlease choose option 119884 we receive equalamounts regardless of typerdquo Yet another Type 2 ID 11 saidldquovote for option 119910 Otherwise Type 2 gets shaftedrdquo ID 10 wasa Type 1 subject chatting with IDs 9 and 11 and started thechat period in round 2 by stating ldquoAlways vote for option 119883please Same deal We can get our moneyrdquo but then changedhis or her mind and ended up voting for Candidate 119884 after

being convinced by IDs 9 and 11 Of course the fact that ID10 said ldquopleaserdquo in the chat may indicate that this person wasinherently more considerate and therefore maybemore likelyto be open to the possibility of voting ethically as opposed tosomeone like ID 5 whose sentiment was ldquosucks to be a 2rdquo

Other rounds followed in a similar fashion The amountof ethical voting never exceeded 2 subjects per round and itwas always done by IDs 7 and 10mdashand then only ID 7 afterID 10 started abstaining but with only 12 subjects that wasenough to make or break a tie in a few rounds The chat alsoincluded some amusing attempts to generate ethical votingsuch as ldquoa vote for 119883 is a vote for the bigwig corporationstaking our hard-earned tax dollarsrdquo ldquoa vote for 119883 is un-Americanrdquo and ldquohere is a list of people who would vote for119909 in this scenario bane john lee hooker ivan the terriblejudasrdquo

In Session 7 a similar phenomenon occurred (In Session7 the selfish and ethical candidates had been switched sothat Candidate 119883 was the ethical candidate and Candidate119884 was the selfish candidate In this discussion the standardcandidate name will be used surrounded by brackets ratherthan what was written in the transcript) In round 3 ID5 (Type 2) joked about the inequity of Candidate [119883]rsquosoffered payoffs After this ID 6 (Type 1) suggested voting forCandidate [119884] and ID 7 (Type 1) agreed This block of votingcontinued for several more sessions Also in round 3 ID 15(Type 2) urged Type 1 subjects to vote for Candidate [119884] forethical reasons saying ldquobut [119910] makes money for everyonerdquo(As explained earlier this is not entirely true due to equal-sized aggregate payoffs but it demonstrates a desire for amoreequitable distribution of payoffs) However it did not spreadbeyond this subjectrsquos original chat group

One difference between Sessions 1 and 7 was the presencein Session 7 of voting that appeared to be ethical but wasactually due to subjects not understanding how the payoffsand costs worked This was the case for IDs 11 and 12 twoType 1 voters who banded together to vote for Candidate [119884]because they thought that it was to their advantage basedon how high or low their costs were Therefore some of theseemingly ethical voting was actually due to noise or votererror This is an example of how voting that appears ethicalmay not be and illustrates the usefulness of this chat featurefor providing insight into motives

It is also possible for incorrect information to spread andto affect the outcome of an election In Session 12 ID 3mdasha Type 2 candidatemdashstarted off in the chat before round 1by saying that everyone should vote for Candidate 119883 Thechat period ended before he could follow up and explain whybut in round 2 the subject said ldquocontinue with option 119883 andforward it to your adjacent ID groups If everyone votes 119883throughout we all winrdquoThis strategy worked and by round 3there were 11 votes for Candidate 119883 1 vote for Candidate 119884and no abstentions

The logic behind ID 3rsquos strategy is a mystery becausethis was a session with symmetric payoffs so neither of thecandidates was offering equitable payoffs Also as a memberof Type 2 this subject received a lower payoff if Candidate119883 won In round 5 this same subject wrote ldquocontinue with119909 regardless of Type 1 or Type 2 it will all work out to


a guaranteed payout fwd it to your grouprdquo This subjecthad very quickly understood the power of the overlappingchat groups and yet apparently had not understood theinstructions at the beginning of the session

At the same time word was spreading that subjects wouldnot receive a payoff if they abstained or voted for the losingcandidate However there were also individual conversationsin which subjects were testing out themechanics of the gameFor example in round 5 ID 8 had abstained and ID 9 hadvoted for Candidate 119884 so in the chat before round 6 theirdiscussion focused on whether payoffs were still receivedeven if a subject abstained from voting ID 8 wrote ldquoYes Idid (abstain and still receive the payoff) It sucks that Option119884 cannot win But the better option now is to not vote if yoursquoreType 2rdquo ID 1 showed similar insight into this and also intothe possibility of free riding in the presence of large marginsfor the winning candidate ldquoshould we not vote this roundIf the pattern stays the same ppl will vote 119909 and we wonrsquotget chargedrdquo Meanwhile ID 3 continued his or her push forCandidate119883

In the next round there was chatter demonstratingcontinued confusion on the part of some subjects andamong other subjects a growing understanding of the correctmechanics of the experiment In this round five subjectsendedup abstaining indicating that the ldquoalways voterdquo strategywas breaking up However everyone who did vote choseCandidate 119883 regardless of type Evidently some were stillconfused about the link between payoffs and voting decisionsbecause in the chat before round 8 ID 5 said ldquowhy would 5people not vote and miss out on this cash cowrdquo ID 5 was aType 1 so indeed it had been a cash cow for this subject exceptfor the fact that he or she had incurred the voting cost in everyround

It was at this point that a rare event took place theexperiment was actually paused and the rules andmechanicsof payoffs and costs were explained again such as the factthat the payoff received is a function of your type and thecandidate that wins not (directly) whether or for whom asubject voted (This had already been stated several timesduring the instructions which had been shown inwriting andalso read aloud at the start of the session The instructionshad also included a practice question that was specificallydesigned to make sure that subjects understood this andto correct any misunderstandings before the experimentbegan) After this was explained again voting decisionssettled into a more normal pattern

This incident provides an excellent example how thespread of information can affect outcomes regardless ofwhether or not the information is correct It is also an exampleof how relatively easy it is to exploit peoplersquos uncertainty aswould have been the case at the beginning of the sessionwhen there is always a learning curve regardless of howthorough the instructions are It is important to note thatthis does not necessarily have to be done maliciouslymdashit isunclear whether ID 3rsquos actions were due tomisunderstandingor wanting to cause trouble

342 Peer PressurePeer InfluenceManipulation A secondtheme thatwas seen in the chat transcriptswas that of subjects

urging each other to vote and sometimes agreeing to do soThismay have increased the turnout rates beyondwhatmighthave occurred without chat and may help to explain the highturnout rates that were observed Also because the chat isrepeated over many rounds there is accountability to someextentmdashyour chat neighbor might ask if you had voted andif so for whom you voted in the previous roundThat sense ofaccountability could also increase voter turnout And finallysome people feel a sense of solidarity in voting the same wayas their friendsfamilycolleaguesand so forth

In these experiments this feeling of solidarity led subjectsto vote against their own economic interests even in theabsence of altruistic motives Usually this only lasted oneor two rounds before self-interest took over However in afew cases this happened repeatedly These present a possibleparallel to Kittel et alrsquos [12] swing voters who were influencedto vote differently than they would have otherwise In thiscontext swing voter status was not due to experimentaldesign but rather to certain subjects being more easilyinfluenced than others

One extreme case occurred in Session 10 ID 3 who wasa Type 2 subject was paired in a chat group with ID 2 and ID4 both of whom belonged to Type 1 ID 3 wanted to vote withhis or her chat group and voted for Candidate 119883mdashagainstthe subjectrsquos economic self-interest but benefitting those ofthe other group membersmdashin every round throughout thesessionThe subject askedwhether theywere all the same typebefore round 6 in spite of the types having been listed nextto ID numbers in the chat interface in every round Howevereven after ID 3rsquos group members clarified that they were bothType 2 ID 3 continued to vote for Candidate119883 ID 3 voted forCandidate 119883 in all but one round Coincidentally this extravote for Candidate 119883 either made or broke a tie in each ofthe first five rounds and set up a streak in which Candidate119883 won every round of the session as subjects who wouldhave voted for 119884 (mostly other Type 2s) got discouraged andstopped voting I spoke with the subject after the end of thesession to find out if this voting behavior resulted from alack of understanding about the way in which the payoffsworked (being careful to ask about it in terms of wanting tofind out whether the instructions were clear enough so thatchanges could be made in an attempt to sidestep feelingsof defensiveness that could have affected the answers) Thesubject had indeed understood everything but simply likedthe feeling of solidarity from voting with the other chat groupmembers

In Session 11 there were two Type 1 subjects who oftenvoted for Candidate 119884 The payoffs in this session weresymmetric so altruism was not a possible motive Both ofthem chatted with ID 15 a Type 2 candidate ID 15 usuallyvoted for 119884 but sometimes voted for 119883 usually when IDs14 and 16 also voted 119883 and this subject spent much of thechat trying to convince his or her chat group to vote as agroup When asked about it after the session IDs 14 and16 each expressed a desire to vote with their chat groupssimilar to the previous example But then in a separateconversation ID 15 explained his strategy of occasionallyvoting for Candidate 119883mdashespecially in the earlier roundsmdashfor the purpose of generating a feeling of solidarity in IDs 14


and 16 This subject then would urge IDs 14 and 16 to votefor Candidate 119884 which maximized ID 15rsquos payoff The endresult was that ID 15 only voted for Candidate 119883 four timeswhereas ID 14 voted for Candidate 119884 eight times and ID 16voted for Candidate 119884 twelve times so on a net basis thisstrategy gained many more votes than it lost for Candidate119884

This was the only incident of strategic manipulation thatwas observed in any of the sessions at least as far as thetranscripts indicated (and a bit of luck in running into thesubject shortly after the session) However it does illustratethe ability of a savvy political strategist to manipulate otherswho may not be as savvy even those whose interests divergefrom those of the strategist It is also thematically related tothe producers of social pressure in Schram and van Winden[9]

4 Conclusions

The analysis presented above attempted to resolve the para-dox of voter turnout by incorporating ethical voting into aquantal response equilibrium- (QRE-) based model whichallows for noise in the decision-making process and thentesting the modelrsquos predictions using a series of laboratoryexperiments The benefit of this modeling approach is that itgenerates predicted voting turnout probabilities which wereused in fine-tuning the experimental design and againstwhich the results of the experiments could be compared

High rates of voter turnout were indeed observed In factthe turnout rates for some of the type-candidate combina-tions were so high that QRE-based analysis could not accountfor all of the overvoting relative to the Nash equilibriumpredictionsThis is due at least in part to the presence of threevoting choicesmdashvote for Candidate 119883 vote for Candidate119884 and abstainmdashas opposed to the two choices of vote andabstain that were present in Levine and Palfrey [4] Givena cost-payoff ratio that generates at least some abstentionin equilibrium the highest possible payoff occurs in thepresence of extremely high levels of noiseWith three choicesthe probability of choosing any one of them is 033 This isconsiderably lower than the 050 when there are only twochoices which would not have explained all of the overvotingobserved for some of the type-candidate combinations butwould have at least explained more of it

Ethical voting was not able to explain the overvotingeither Some ethical voting was observed but not enough toexplain the extent of overvoting Also most of the overvotingoccurred when subjects voted for the candidate offering theirtype a higher payoff Ethical voting which is defined as votingagainst onersquos own economic self-interest in order to votefor the candidate offering payoffs that are distributed moreequitable across voter types by its very nature does not fallinto this category

The online chat feature that subjects used for commu-nicating with their virtual neighbors before each round(election) was shown to have increased voter participationIn this chat feature subjects chatted online with the twosubjects with adjacent ID numbers within the experimentThis mirrored the way in which people discuss politics

and upcoming elections with their friends families andcoworkers

Additionally because the chat was repeated before everyround it also set up a dynamic in which it may have generatedsome accountability between chat partners for exampleby asking onersquos chat group whether they had voted in theprevious round and if so then for which candidate(s) It wasalso observed in the chat transcripts that subjects would oftenask others to vote in a certain way or agree to vote in acertain way and then sometimes follow up afterwards whichsupports that conjecture Through these channels subjectsrsquovoting decisions were (mostly positively) influenced by thevoting decisions of their group members

There are several possibilities for future research in thisarea such as comparing sessions with a chat feature versus nocommunication or the use of different chat group structuresfor example the current chat setup versus one in whichsubjects only communicated with members of their ownvoter type Another possibility would be to focus on onlyethical voting or only communication to clarify the individualimpacts of these features on turnout rates and patterns Itis possible that the inclusion of both communication and(the potential for) ethical voting in this paperrsquos experimentaldesign obscured the importance of ethical voting The chatfeature was originally included for the insight that it wouldprovide rather than being a tool to increase voting participa-tion and its impact on turnout was unexpected Because ofthis the experimental design did not carefully test the relativeimpacts of these two features It would be interesting to designtreatments to test for the individual effects which I leave forfuture research

Several other themes were also observed in the resultsin which the patterns in the observed turnout rates differedfrom those that had been predicted There was a strong costeffect on abstention rates as predicted However there wasno significant size effect in contrast to the predictions inwhich turnout declined as the session size increased and theprobability of casting a pivotal vote decreased Second theturnout rate was higher in sessions with symmetric payoffs(in which there was no possibility of ethical voting) thanin sessions with asymmetric payoffs (in which there was apossibility of ethical voting) This contradicts the predictedoutcome however the reason for this pattern is unclear Beliefelicitation might provide some insight into this but wouldrequire further research

Appendices

A Pivotal Probabilities


119894119895) is the

probability that a vote cast by amember of type 119894 for candidate119895 will be pivotal (henceforth referred to as the ldquopivotalprobabilityrdquo) The one-half is there because in the presenceof two candidates if someonersquos vote makes a tie then thereis a 50 chance that this voterrsquos chosen candidate will win(and that the vote will have been pivotal)mdashremember thatthe winner of a tie is determined randomly If someonersquos votebreaks a tie then there is a 100 chance that his or her chosen


candidate will win However because there was already a 50chance that this candidate would have won in the event ofa tie there would only be a 50 chance that the vote thatbroke the tie changed that candidatersquos outcome from losingto winningTherefore the sum of the probabilities of makinga tie and of breaking a tie is multiplied by one-half

The pivotal probabilities are analogous to those in Levineand Palfrey [4] and Goeree and Holt [3] but complicatedby the fact that it is possible for votes received by each ofthe candidates to vote for either candidate to come fromeither type of voters For example even in the relativelystraightforward example of 1 vote (total) for Candidate119883 and1 vote (total) for Candidate119884 there are numerous possibilitiesfor how this outcome arose both votes came from Type1 voters Type 1 voted for Candidate 119883 and Type 2 votedfor Candidate 119884 Type 2 voted for Candidate 119883 and Type1 voted for Candidate 119884 and both votes came from Type 2voters Additionally there is a separate pivotal probability foreach type-candidate combination because it is possible formembers of either type to vote for either candidate

One example is the probability that a member of Type1 will be pivotal by voting for Candidate 119883 modeled as amultinomial probability

Pr (pivotal)1119883

=1

2(

min(11987311198732)

sum

119899119884=1

119899119884

sum

1198992119884=0

119899119884minus1

sum

1198992119883=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198731minus 1

119899119884minus 1 minus 119899

2119883 119899119884minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901

2119883minus 1199012119884)1198732minus1198992119883minus1198992119884

sdot 119901119899119884minus1minus1198992119883

1119883119901119899119884minus1198992119884

1119884

sdot (1 minus 1199011119883minus 1199011119884)1198731minus(119899119884minus1198992119883)minus(119899119884minus1198992119884)

+

min(11987311198732)minus1

sum

119899119884=0

119899119884

sum

1198992119884=0

119899119884

sum

1198992119883=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198732minus 1

119899119884minus 1198992119883 119899119884minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119884minus1198992119883

1119883119901119899119884minus1198992119884

1119884

sdot (1 minus 1199011119883minus 1199011119884)1198731minus1minus(119899

119884minus1198992119883)minus(119899119884minus1198992119884)

)

(A1)

where 119873119894is the total number of potential voters of type 119894 119899

119895

is the total number of votes for candidate 119895 (cast by voters ofeither type) 119899

119894119895is the total number of votes by members of

type 119894 for candidate 119895 and 119901119894119895is the probability of a member

of type 119894 voting for candidate 119895

The first term within the brackets is the probability thata member of Type 1 will make a tie by voting for Candidate119883 This means that before this incremental voting decisionis made there is one fewer vote for Candidate 119883 than thereis for Candidate 119884 or 119899

119883= 119899119884minus 1 The second term is the

probability that a member of Type 1 will break a tie by votingfor Candidate 119883 The fact that one vote would break a tiemeans that there is currently a tie and 119899

119883= 119899119884

For each term within the brackets the triple sum allowsthe calculation to sum over every possible combination ofvotes for every possible number of votesTheouter sum loopsover every possible value of 119899

119884(the total number of votes for

Candidate 119884) and therefore also over every possible value of119899119883The inner two sums loop over every possible combination

of 1198992119884

and 1198992119883

(the number of votes cast by members of Type2 for Candidates 119884 and 119883 resp) Because 119899

1119884and 119899

1119883mdashthe

number of votes cast by members of Type 2 for Candidates119883 and 119884 respectivelymdashare simply (119899

119884minus 1198992119884) and (119899

119883minus 1198992119883)

these summations also loop over every possible combinationof 1198991119884

and 1198991119883

In the first term ( 119873211989921198831198992119884

) 1199011198992119883

21198831199011198992119884

2119884(1minus1199012119883minus1199012119884)1198732minus1198992119883minus1198992119884

is the probability that there will be exactly 1198992119883

and 1198992119884

votes ifall1198732members of Type 2 have already made their voting (or

abstention) decisions ( 1198731minus1

119899119884minus1minus1198992119883119899119884minus1198992119884

) 119901119899119884minus1minus1198992119883

1119883119901119899119884minus1198992119884

1119884(1 minus

1199011119883minus 1199011119884)1198731minus(119899119884minus1198992119883)minus(119899119884minus1198992119884) is the probability that there will

be exactly 1198991119883

and 1198991119884

votes if all 1198731minus 1 other members of

Type 1 have already votedThe second term follows the same formatThe other probabilities of amember of type 119894 being pivotal

by voting for candidate 119895 equalPr (pivotal)

1119884

=1

2(

min(1198731 1198732)

sum

119899119883=1

119899119883

sum

1198992119883=0

119899119909minus1

sum

1198992119884=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198731minus 1

119899119883minus 1198992119883 119899119883minus 1 minus 119899

2119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119883minus1198992119883

1119883119901119899119883minus1minus1198992119884

1119884


+

min(1198731 1198732)minus1

sum

119899119883=0

119899119883

sum

1198992119883=0

119899119909

sum

1198992119884=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198732minus 1

119899119883minus 1198992119883 119899119883minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119883minus1198992119883

1119883119901119899119883minus1198992119884

1119884



)


Pr (pivotal)2119883

=1

2(

min(1198731 1198732)

sum

119899119884=1

119899119884

sum

1198991119884=0

119899119884minus1

sum

1198991119883=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119884minus 1 minus 119899

1119883 119899119884minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119884minus1minus1198991119883

2119883119901119899119884minus1198991119884

2119884


+

min(1198731 1198732)minus1

sum

119899119884=0

119899119884

sum

1198991119884=0

119899119884

sum

1198991119883=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119884minus 1198991119883 119899119884minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119884minus1198991119883

2119883119901119899119884minus1198991119884

2119884



)

Pr (pivotal)2119884

=1

2(

min(1198731 1198732)

sum

119899119883=1

119899119883

sum

1198991119883=0

119899119909minus1

sum

1198991119884=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1


1119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119883minus1198991119883

2119883119901119899119883minus1minus1198991119884

2119884


+

min(1198731 1198732)minus1

sum

119899119883=0

119899119883

sum

1198991119883=0

119899119909

sum

1198991119884=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119883minus 1198991119883 119899119883minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119883minus1198991119883

2119883119901119899119883minus1198991119884

2119884



)

(A2)

Table 9 Voting turnout rates by gender

Male FemaleTurnout 054633 05993Favored 050275 05162Unfavored 004358 00831Ethical 00375 001833

0

02

04

06

08

1

1 3 5 7 9 11 13 15 17 19

Pr (v

ote)

Round

MaleFemale

Figure 5 Voting turnout rates by gender

These pivotal probabilities are then used to calculate theexpected payoff differences net of costs

B Gender Differences

Another area of interest is the effect of gender on votingdecisions In various nonparametric and parametric analysesgender was not found to have any significant effect on votingturnout either for overall for favored or for unfavoredThe turnout rates are summarized in Table 9 and Figure 5illustrates the closeness of these turnout rates over time

C Nonparametric Permutation Test

Nonparametric permutation tests were used to test thesignificance of the difference between the means of twoindependent samples when the sample sizes are small Thenull hypothesis is that all of the observations are from thesame population Therefore rejecting the null hypothesiswould mean that all of the observations could not be fromthe same population and the fact that the observationsoccurred specifically in their respective treatments cannotbe attributed to coincidence This was tested by reassigningthe observations between the treatments then comparing thedifferences in themeans of the permutationswith those in theoriginalThe null hypothesis is rejected if the observed differ-ence between treatments is large relative to the hypotheticaldifferences in the other possible permutations

First looking at session-level average turnout rates foroverall turnout shown in Table 10 we see the same rela-tionships as before in the averages for all rounds and for


Table 10 Voting turnout rates by session

Treatment Session 1st round Average Average (2nd half)

12 asymmetricSession 1 08333 05625 04417Session 3 09167 05000 03750Session 9 08333 05208 04750

12 symmetricSession 2 05833 06292 06500Session 4 08333 05292 04667Session 12 10000 06083 04750



Table 11 Permutation test results (119901 value)mdashoverall turnout rates

First round only Average Average (2nd half)1-tailed 2-tailed 1-tailed 2-tailed 1-tailed 2-tailed

12 asymmetric versus symmetric 030 075 01 0200 0150 030018 asymmetric versus symmetric 050 100 005 01 005 01

Asymmetric 12 versus 18 030 060 0250 0500 0200 0450Symmetric 12 versus 18 040 090 005 01 0150 035012 versus 18 (asymmetric and symmetric) 04142 08270 02767 05520 01357 02719Asymmetric versus symmetric (12 and 18) 03419 06848 0003 0006 00038 00078119901 lt 010 119901 lt 005 119901 lt 001

the second half (rounds 11ndash20) The first round of voting wasalso included because inmost real-world elections one wouldonly vote a single time

The nonparametric permutation test then analyzeswhether there really is a difference across treatments Theresults are shown in Table 11 The results are reported as 119901values

These results show the same trends that were observedin the stylized facts the number of subjects per sessiondoes not significantly affect the probability of voting (exceptfor symmetric sessions across all rounds) and subjects intreatments with symmetric payoffs are significantly morelikely to vote than are subjects in treatments with asymmetricpayoffs Results for the first round only are not significantwhich is not surprising As opposed to an election outsideof the lab where potential voters have weeks or months tolearn about the candidates and issues and to talk with theirfriendsfamilycolleagues in the experiment there is quitea bit of learning that occurs during the first few roundswhich may obscure any meaningful relationships in thedata

Next the data for favored turnout and unfavored turnoutwere analyzed in the same manner Given that most of thevoting was for the favored candidate the results found herewere very similar to those for overall turnout For unfavoredturnout no significant differences were found even afterusing the averages for rounds 1ndash10 (because that is whenalmost all of the ethical voting happened)

D Instructions

The subjects saw the following set of instructions at the startof the experiment These were also read aloud The instruc-tions below are from a 12-person session with asymmetricpayoffs so the instructions for the other treatments wouldvary with respect to the number of subjects andor the payoffcharts but would otherwise be the same At the top of eachpage the subject sees ldquoInstructions (ID = mdash) Page mdash of 6rdquo

(i) Matchings The experiment consists of a series ofrounds You will be matched with the same groupof 11 other people in each round The decisions thatyou and the 11 other people make will determine theamounts earned by each of you

(ii) Voting Decisions At the beginning of each round youwill be asked to consider a vote for one of 2 optionsOption119883 and Option 119884

(iii) Earnings The votes that are cast by the members ofyour group will determine a winning option whichwill determine your earnings in a manner to beexplained next

(iv) Please Note Your earnings will depend on the out-come that receives the most votes (with ties decidedat random) regardless of whether or not you votedfor the winning option


(v) CommunicationsTherewill be a chat room open priorto the voting process which allows you to send andreceive messages Voters will be identified by their IDnumbers and you are free to discuss any aspect of thevoting process during the 1-minute chat period

(vi) Communications Network You will be able to com-municate with only two other voters in your groupwho are your neighbors in the sense of having IDnumbers adjacent to your ID You should avoidinappropriate language or attempts to arrange sidepayments

Page 2

(i) Voting Sequence The option that receives the mostvotes will determine earnings for all voters (irrespec-tive of how each individual voted)

(ii) Ties In the event of a tie one of the tied options willbe selected at random with each tied option havingan equal chance of being selected

(iii) Your Earnings The option selected in the votingprocess will determine your earnings For exampleyour earnings for each outcome in the first round are

Option119883 $(individual payoff )Option 119884 $(individual payoff )

(iv) Voting Cost If you do vote you will incur a costThis cost will change randomly from round to roundand it will vary randomly from person to personEach personrsquos voting cost will be a number that israndomly selected from a range between $000 and$042 You will find out your voting cost before youdecide whether to vote You will not know anyoneelsersquos voting cost Your total earnings will be theamount determined by the voting outcome minusyour voting cost (if any)

Page 3

(i) Individual Differences There are 12 voters in yourgroup who are divided into 2 ldquotypesrdquo as shown inthe table below Your type and your earnings foreach possible outcome are indicated in the bright bluecolumn

(ii) Payoff Backstory

Option 119883 will implement an investment thatpays $2 to Type 1 voters and $1 to Type 2 votersOption 119884 will implement an investment thatpays $150 to both Type 1 and Type 2 voters

Payoffs for all voters

Outcome type 1(you) type 2

Option119883 $200 $100Option 119884 $150 $150NumberVoters

6(you and 5 others) 6

Page 4 Please select the best answer Your answer will bechecked for you on the page that follows

Question 1 Suppose that your voting cost is $021 and you areof Type 1 in this round (different people will have differenttypes) If you decide to vote and incur this cost your earningswill be

◻ $179 if Option 119883 receives the most votes and $129if Option 119884 receives the most votes◻ $179 if you voted for Option119883 and Option119883wins$129 if you voted for Option 119884 and Option 119884 winsand $000 if the option you voted for does not win

Submit answers





Page 5


⊡ $179 if Option119883 receives the most votes and $129 ifOption 119884 receives the most votes

◻ $179 if you voted for Option 119883 and Option 119883 wins$129 if you voted for Option 119884 and Option 119884 winsand $000 if the option you voted for does not win

Your answer is Correct You receive the payoff from therelevant (blue) column in the payoff table regardless ofwhether you voted or not In this case the cost of voting isdeducted since you voted in this example

Continue





Page 6 (instructions summary page)

(i) You will be matched with the same group of 11 otherpeople in each round

(ii) All participants will begin by finding out their ownvoting cost which will be a randomly determinednumber between $000 and $042 This cost will vary


from person to person After seeing your votingcost you will decide whether to incur this cost andvote The option that receives the most votes will beselected Your earnings will depend on the optionselected although your voting cost (if any) will besubtracted

(iii) Please Note Your earnings will be determined bythe outcome that receives the most votes (with tiesdecided at random) regardless of whether or not youvoted for the winning option

(iv) Each round will begin with a 1-minute chat periodYou will be able to communicate with only two othervoters in your group who are your ldquoneighborsrdquo inthe sense of having ID numbers adjacent to your IDnumber You will have to press the Update buttonmanually to retrieve otherrsquos messages as they arrive

(v) There will be a number of rounds in this part of theexperiment Your earnings for each round will becalculated for you and added to previous earnings

Disclaimer

The views expressed are those of the author and should notbe attributed to the US Department of Agriculture or theEconomic Research Service


The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was based on a chapter from a PhD dissertationcompleted at the University of Virginia under the helpfulsupervision of Charles Holt Federico Ciliberto and SimonAndersonThe author would also like to thank Craig VoldenWayne-Roy Gayle Beatrice Boulu-Reshef Sean SullivanAngela Smith Jura Liaukonyte Jonathan Williams Christo-pher Clapp Nathaniel Higgins Christopher Burns JamesWilliamson Michael Harris Charles Hallahan and the par-ticipants of the Experimental Economics Workshop and thePolitical Psychology Working Group both at the Universityof Virginia for their helpful commentsThe author is gratefulto Arthur Schram and two anonymous referees for theircomments and suggestions In addition the author thanksthe Political Psychology Working Group the Bankard Fundfor Political Economy and the National Science Foundation(Award no BCS-0904795) for their financial support and theDepartment of Economics and the Veconlab ExperimentalEconomics Laboratory at the University of Virginia for theexperimental infrastructure

References

[1] B Geys ldquolsquoRationalrsquo theories of voter turnout a reviewrdquo PoliticalStudies Review vol 4 no 1 pp 16ndash35 2006

[2] R D McKelvey and T R Palfrey ldquoQuantal response equilibriafor normal form gamesrdquoGames and Economic Behavior vol 10no 1 pp 6ndash38 1995

[3] J K Goeree and C A Holt ldquoAn explanation of anomalousbehavior inmodels of political participationrdquoAmerican PoliticalScience Review vol 99 no 2 pp 201ndash213 2005

[4] D K Levine and T R Palfrey ldquoThe paradox of voter participa-tionA laboratory studyrdquoTheAmerican Political Science Reviewvol 101 no 1 pp 143ndash158 2007

[5] R BMorton and J-R Tyran ldquoEthical versus selfishmotivationsand turnout in small and large electoratesrdquoWorking Paper NewYork University 2012

[6] J K Goeree C A Holt and S K Laury ldquoPrivate costs andpublic benefits unraveling the effects of altruism and noisybehaviorrdquo Journal of Public Economics vol 83 no 2 pp 255ndash276 2002

[7] R B Morton ldquoGroups in rational turnout modelsrdquo AmericanJournal of Political Science vol 35 no 3 pp 758ndash776 1991

[8] J Groszliger andA Schram ldquoNeighborhood information exchangeand voter participation an experimental studyrdquo The AmericanPolitical Science Review vol 100 no 2 pp 235ndash248 2006

[9] A Schram and F van Winden ldquoWhy People Vote free ridingand the production and consumption of social pressurerdquo Jour-nal of Economic Psychology vol 12 no 4 pp 575ndash620 1991

[10] A Schram and J Sonnemans ldquoWhy people vote experimentalevidencerdquo Journal of Economic Psychology vol 17 no 4 pp 417ndash442 1996

[11] G Charness L Rigotti and A Rustichini ldquoIndividual behaviorand group membershiprdquo American Economic Review vol 97no 4 pp 1340ndash1352 2007

[12] B Kittel W Luhan and R Morton ldquoCommunication andvoting in multi-party elections an experimental studyrdquo TheEconomic Journal vol 124 pp 196ndash225 2014

[13] T R Palfrey and H Rosenthal ldquoA strategic calculus of votingrdquoPublic Choice vol 41 no 1 pp 7ndash53 1983

[14] J Andreoni ldquoImpure altruism and donations to public goodsa theory of warm-glow givingrdquoThe Economic Journal vol 100no 401 pp 464ndash477 1990

[15] J H Fowler and C D Kam ldquoBeyond the self social identityaltruism and political participationrdquoThe Journal of Politics vol69 no 3 pp 813ndash827 2007

Research ArticlePreference for Efficiency or Confusion A Note ona Boundedly Rational Equilibrium Approach to IndividualContributions in a Public Good Game

Luca Corazzini12 and Marcelo Tyszler3

1Department of Law Science and History of Institutions University of Messina Piazza XX Settembre 498122 Messina Italy2ISLA Bocconi University Via Roentgen 1 20136 Milan Italy3Center for Research in Experimental Economics and Political Decision Making (CREED) University of AmsterdamRoetersstraat 11 1018 WB Amsterdam Netherlands

Correspondence should be addressed to Marcelo Tyszler marcelotyszlercombr


Academic Editor Dimitris Fotakis

Copyright copy 2015 L Corazzini and M Tyszler This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

By using data froma voluntary contributionmechanism experimentwith heterogeneous endowments and asymmetric informationwe estimate a quantal response equilibrium (QRE) model to assess the relative importance of efficiency concerns versusnoise in accounting for subjects overcontribution in public good games In the benchmark specification homogeneous agentsovercontribution is mainly explained by error and noise in behavior Results change when we consider a more general QREspecificationwith cross-subject heterogeneity in concerns for (group) efficiency In this case we find that themajority of the subjectsmake contributions that are compatible with the hypothesis of preference for (group) efficiency A likelihood-ratio test confirmsthe superiority of the more general specification of the QRE model over alternative specifications

1 Introduction

Overcontribution in linear public good games representsone of the best documented and replicated regularities inexperimental economics The explanation of this apparentlyirrational behaviour however is still a debate in the literatureThis paper is aimed at investigating the relative importance ofnoise versus preference for efficiency In this respect we buildand estimate a quantal response equilibrium (henceforthQRE [1]) extension of the model presented by Corazzini et al[2] This boundedly rational model formally incorporatesboth preference for efficiency and noise Moreover in con-trast to other studies that investigate the relative importanceof error and other-regarding preferences the QRE approachexplicitly applies an equilibrium analysis

To reconcile the experimental evidence with the standardeconomic framework social scientists developed explana-tions based on refinements of the hypothesis of ldquoother-regarding preferencesrdquo reciprocity [3ndash6] altruism and spite-fulness [7ndash9] commitment and Kantianism [10 11] normcompliance [12] and team-thinking [13ndash15]

Recently an additional psychological argument to explainagentsrsquo attitude to freely engage in prosocial behavior isgaining increasing interest the hypothesis of preference for(group) efficiency There is evidence showing that exper-imental subjects often make choices that increase groupefficiency even at the cost of sacrificing their own payoff[16 17] Corazzini et al [2] use this behavioral hypothesisto explain evidence from linear public good experimentsbased on prizes (a lottery a first price all pay auction and



a voluntary contribution mechanism used as a benchmark)characterized by endowment heterogeneity and incompleteinformation on the distribution of incomes In particularthey present a simple model in which subjects bear psy-chological costs from contributing less than what is efficientfor the group The main theoretical prediction of theirmodel when applied to linear public good experiments isthat the equilibrium contribution of a subject is increasingin both her endowment and the weight attached to thepsychological costs of (group-) inefficient contributions inthe utility function The authors show that this model iscapable of accounting for overcontribution as observed intheir experiment as well as evidence reported by relatedstudies

However as argued by several scholars rather than beingrelated to subjectsrsquo kindness overcontribution may reflecttheir natural propensity to make errors There are severalexperimental studies [18ndash23] that seek to disentangle other-regarding preferences frompure noise in behavior by runningad hoc variants of the linear public good game A generalfinding of these papers is that ldquowarm-glow effects and randomerror played both important and significant rolesrdquo [20 p 842]in explaining overcontribution

There are several alternative theoretical frameworks thatcan be used to model noise in behavior (bounded rationality)and explain experimental evidence in strategic games Twoexamples are the ldquolevel-119896rdquo model (eg [24ndash26]) and (rein-forcement) learningmodels (eg [27]) In the ldquolevel-119896rdquomodelof iterated dominance ldquolevel-0rdquo subjects choose an actionrandomly and with equal probability over the set of possiblepure strategies while ldquolevel-119896rdquo subjects choose the action thatrepresents the best response against level-(119896 minus 1) subjectsLevel-119896 models have been used to account for experimentalresults in games in which other-regarding preferences do notplay any role such as 119901-Beauty contests and other constantsum games Since in public good games there is a strictlydominant strategy of no contribution unless other-regardingpreferences are explicitly assumed ldquolevel-119896rdquo models do notapply Similar arguments apply to learning models In thebasic setting each subject takes her initial choice randomlyand with equal probability over the set of possible strategiesAs repetition takes place strategies that turn out to be moreprofitable are chosen with higher probability Thus unlessother-regarding preferences are explicitly incorporated intothe utility function repetition leads to the Nash equilibriumof no contribution

The QRE approach has the advantage that even in theabsence of other-regarding preferences it can account forovercontribution in equilibrium Moreover we can use themodel to assess the relative importance of noise versusefficiency concerns

We start from a benchmark model in which the popula-tion is homogeneous in both concerns for (group) efficiencyand the noise parameter We then allow for heterogeneityacross subjects by assuming the population to be partitionedinto subgroups with different degrees of efficieny concernsbut with the same value for the noise parameter

In the QRE model with a homogeneous population wefind that subjectsrsquo overcontribution is entirely explained by

noise in behavior with the estimated parameter of concernsfor (group) efficiency being zero A likelihood-ratio teststrongly rejects the specification not allowing for randomnessin contributions in favor of the more general QRE model Adifferent picture emerges when heterogeneity is introducedin the QRE model In the model with two subgroups theprobability of a subject being associated with a strictlypositive degree of preference for (group) efficiency is approx-imately one-third This probability increases to 59 whenwe add a third subgroup characterized by an even higherefficiency concern A formal likelihood-ratio test confirmsthe superiority of the QRE model with three subgroupsover the other specifications These results are robust tolearning processes over repetitions Indeed estimates remainqualitatively unchanged when we replicate our analysis onthe last 25 of the experimental rounds The rest of thispaper is structured as follows In Section 2 we describethe experimental setting In Section 3 we present the QREextension of the model based on the preference for (group)efficiency hypothesis Section 4 reports results from ourstatistical analysis Section 5 concludes the paper

2 The Experiment

We use data from three sessions of a voluntary contributionmechanism reported by Corazzini et al [2] Each sessionconsisted of 20 rounds and involved 16 subjects At thebeginning of each session each subject was randomly andanonymously assigned with equal chance an endowmentof either 120 160 200 or 240 tokens The endowment wasassigned at the beginning of the experiment and was keptconstant throughout the 20 rounds The experiment was runin a strangers condition [28] such that at the beginningof each round subjects were randomly and anonymouslyrematched in groups of four playersThis procedurewas com-mon knowledge Thus in each round subjects made theirchoices under incomplete information on the distributionof the endowments in their group In each round everysubject had to allocate her endowment between an individualand a group account The individual account implied aprivate benefit such that for each token a subject allocatedto the individual account she received two tokens On theother hand tokens in the group account generated monetaryreturns to each of the group members In particular eachsubject received one token for each token allocated by heror by any other member of her group to the group accountThus the marginal per capita return used in the experimentwas 05 At the beginning of each round the experimenterexogenously allocated 120 tokens to the group accountindependently of subjectsrsquo choices thus implying 120 extratokens for each group member At the end of each roundsubjects received information about their payoffs Tokenswere converted to euros using an exchange rate of 1000

points per euro Subjects mainly undergraduate students ofeconomics earned 1225 euros on average for sessions lastingabout 50 minutes The experiment took place in May 2006in the Experimental Economics Laboratory of the University


of Milan Bicocca and was computerized using the z-Treesoftware [29]

The features of anonymity and random rematching nar-row the relevance of some ldquotraditionalrdquo behavioral hypothe-ses used to explain subjectsrsquo overcontribution For instancethey preclude subjectsrsquo possibility to reciprocate (un)kindcontributions of group members [30] Moreover under theseconditions subjects with preferences for equality cannotmake compensating contributions to reduce (dis)advan-tageous inequality [31 32] Rather the hypothesis of prefer-ence for (group) efficiency as a particular form of warm-glow[8 9] appears as a more plausible justification

3 Theoretical Predictions andEstimation Procedure

Consider a finite set of subjects 119875 = 1 2 119901 In a genericround subject 119894 isin 119875 with endowment 119908

119894isin 119873+ contributes

119892119894to the group account with 119892

119894isin 119873+ and 0 le 119892

119894le 119908119894 The

monetary payoff of subject 119894 who contributes 119892119894in a round is

given by

120587119894(119908119894 119892119894) = 2 (119908

119894minus 119892119894) + 120 + 119892

119894+ 119866minus119894 (1)

where 119866minus119894is the sum of the contributions of group members

other than 119894 in that round Given (1) if subjectsrsquo utility onlydepends on the monetary payoff zero contributions are theunique Nash equilibrium of each round In order to explainthe positive contributions observed in their experimentCorazzini et al [2] assume that subjects suffer psychologicalcosts if they contribute less thanwhat is optimal for the groupIn particular psychological costs are introduced as a convexquadratic function of the difference between a subjectrsquosendowment (ie the social optimum) and her contributionIn the VCM player 119894rsquos (psychological) utility function is givenby

119906119894(119908119894 119892119894 120572119894) = 120587119894(119908119894 119892119894) minus 120572119894

(119908119894minus 119892119894)2

119908119894

(2)

where 120572119894is a nonnegative and finite parameter measuring

the weight attached to the psychological costs (119908119894minus 119892119894)2119908119894

in the utility function Notice that psychological costs areincreasing in the difference between a subjectrsquos endowmentand her contribution Under these assumptions in eachround there is a uniqueNash equilibrium inwhich individual119894 contributes

119892NE119894

=2120572119894minus 1

2120572119894

119908119894 (3)

The higher the value of 120572119894 the higher the equilibrium

contribution of subject 119894 The average relative contribution119892119894119908119894 observed in the VCM sessions is 22 which implies

120572 = 064Following McKelvey and Palfrey [1] we introduce noisy

decision-making and consider a Logit Quantal Responseextension of (2) In particular we assume subjects choosetheir contributions randomly according to a logistic quantal

response function Namely for a given endowment 119908119894

and contributions of the other group members 119866minus119894 the

probability that subject 119894 contributes 119892119894is given by

119902119894(119908119894 119892119894 120572119894 120583) =

exp 119906119894(119908119894 119892119894 120572119894) 120583

sum119908119894

119892119895=0exp 119906

119894(119908119894 119892119895 120572119894) 120583

(4)

where 120583 isin R+is a noise parameter reflecting a subjectrsquos

capacity of noticing differences in expected payoffsTherefore each subject 119894 is associated with a 119908

119894-

dimensional vector q119894(119908119894 g119894 120572119894 120583) containing a value of

119902119894(119908119894 119892119894 120572119894 120583) for each possible contribution level 119892

119894isin g119894equiv

0 119908119894 Let q

119894(119908119894 g119894 120572119894 120583)119894isin119875

be the system includingq119894(119908119894 g119894 120572119894 120583) forall119894 isin 119875 Notice that since othersrsquo contribu-

tion 119866minus119894 enters the rhs of the system othersrsquo 119902

119894will also

enter the rhs A fixed point of q119894(119908119894 g119894 120572119894 120583)119894isin119875

is hence aquantal response equilibrium (QRE) qQRE

119894(119908119894 g119894 120572119894 120583)119894isin119875

In equilibrium the noise parameter 120583 reflects the disper-

sion of subjectsrsquo contributions around the Nash predictionexpressed by (3) The higher the 120583 the higher the dispersionof contributions As 120583 tends to infinity contributions arerandomly drawn from a uniform distribution defined over[0 119908119894] On the other hand if 120583 is equal to 0 the equilibrium

contribution collapses to the Nash equilibrium (more specif-ically for each subject 119894 equilibrium contributions converge to119902119894(119908119894 119892

NE119894 120572119894 0) = 1 and 119902

119894(119908119894 119892119894 120572119894 0) = 0 forall119892

119894= 119892

NE119894)

In this framework we use data from Corazzini et al [2]to estimate 120572 and 120583 jointly We proceed as follows Ourinitial analysis is conducted by using all rounds (119899 = 20)

and assuming the population to be homogeneous in both120572 and 120583 This gives us a benchmark that can be directlycompared with the results reported by Corazzini et al [2]In our estimation procedure we use a likelihood functionthat assumes each subjectrsquos contributions to be drawn froma multinomial distribution That is

119871119894(119908119894 g119894 120572 120583) =

119899

prod119908119894

119892119895=0119899 (119892119895)

119908119894

prod

119892119896=0

119902QRE119894

(119908119894 119892119896 120572 120583)

119899(119892119896)

(5)

where 119899(119892119895) is the number of times that subject 119894 contributed

119892119895over the 119899 rounds of the experiment and similarly for

119899(119892119896) The contribution of each person to the log-likelihood

is the log of expression (5) The Maximum Likelihoodprocedure consists of finding the nonnegative values of 120583 and120572 (and corresponding QRE) that maximize the summationof the log-likelihood function evaluated at the experimentaldata In otherwords we calculate themultinomial probabilityof the observed data by restricting the theoretical probabili-ties to QRE probabilities only

We then extend our analysis to allow for cross-subjectheterogeneity In particular we generalize the QRE modelabove by assuming the population to be partitioned into 119878


Table 1 Homogeneous population (all rounds)

Data CFS (1) 120583 120572 (2) 120583 120572 (3) 120583 120572120583 mdash mdash 1 2183 [1969 2434] 4159 [3911 4434]120572 mdash 064 064 064 0 [0 001](Predicted) avg contributions

Overall endowments 3791 3938 3941 6024 3791119908119894= 120 3402 2625 2634 4484 3412

119908119894= 160 2453 3500 3503 5568 3767

119908119894= 200 4750 4375 4376 6557 3948

119908119894= 240 4557 5250 5250 7486 4036

log 119897119897 minus871395 minus348379 minus317069Obs 960 960 960 960 960This table reports average contributions as well as estimates and predictions from various specifications of the model based on the efficiency concernsassumption using all 20 rounds of the experiment CFS refers to the specification not accounting for noise in subjectsrsquo contributions while (1) (2) and (3)are Logit Quantal Response extensions of the model In (1) 120572 and 120583 are constrained to 064 and 1 respectively In (2) the value on 120572 is set to 064 while120583 is estimated through (5) Finally (3) refers to the unconstrained model in which both 120572 and 120583 are estimated through (5) The table also reports for eachspecification the corresponding log-likelihood Confidence intervals are computed using an inversion of the likelihood-ratio statistic at the 001 level subjectto parameter constraints

subgroups that are characterized by the same 120583 but different120572 In this case the likelihood function becomes

119871119894(119908119894 g119894 1205721 1205722 120572

119878 1205741 1205742 120574

119878 120583)

=

119878

sum

119904=1

120574119904

119899

prod119908119894

119892119895=0119899 (119892119895)

119908119894

prod

119892119896=0

119902QRE119894

(119908119894 119892119896 120572119904 120583)119899(119892119896)

(6)

where 1205741 1205742 120574

119878 with sum

119878

119904=1120574119904

= 1 are the probabili-ties for agent 119894 belonging to the subgroup associated with1205721 1205722 120572

119878 respectivelyThis allows us to estimate the value

of 120583 for the whole population the value of 1205721 1205722 120572

119878

for the 119878 subgroups and the corresponding probabilities1205741 1205742 120574

119878 For identification purposes we impose that 120572

119904le

120572119904+1

The introduction of one group at a time accompanied bya corresponding likelihood-ratio test allows us to determinethe number of 120572-groups that can be statistically identifiedfrom the original data In the following statistical analysisestimates account for potential dependency of subjectrsquos con-tributions across rounds Confidence intervals at the 001

level are provided using the inversion of the likelihood-ratiostatistic subject to parameter constraints in line with Cookand Weisberg [33] Cox and Hinkley [34] and Murphy [35]

4 Results

Using data from the 20 rounds of the experiment Table 1reports (i) average contributions (by both endowment typeand overall) observed in the experiment (ii) average con-tributions as predicted by the model not accounting fornoise in subjectsrsquo contributions and (iii) estimates as well asaverage contributions from different parameterizations of theLogit Quantal Response extension of themodel In particularspecification (1) refers to a version of the model in whichboth 120572 and 120583 are constrained to be equal to benchmark values

based onCorazzini et al [2]Under this parameterization120572 isfixed to the value computed by calibrating (3) on the originalexperimental data 064 while 120583 is constrained to 1 (Table 4shows the Maximum Likelihood estimation value of 120572 whenwe vary 120583 It is possible to see that for a large range of values of120583 this value is close to 064 We choose 120583 = 1 as a sufficientlylow value in which the estimated 120572 is close to 064 and thusprovide a noisy version of the base model which can be usedfor statistical tests)

As shown by the table specification (1) closely replicatespredictions of the original model presented by Corazziniet al [2] not accounting for noise in subjectsrsquo contributionsIn specification (2) 120572 is fixed to 064 while 120583 is estimatedby using (4) The value of 120583 increases substantially withrespect to the benchmark value used in specification (1)A likelihood-ratio test strongly rejects specification (1) thatimposes restrictions on the values of both 120572 and 120583 in favorof specification (2) in which 120583 can freely vary on R

+(LR =

1046033 Pr1205942(1) gt LR lt 001) However if we comparethe predicted average contributions of the two specificationswe find that specification (1) better approximates the originalexperimental data This is because a higher value of the noiseparameter spread the distributions of contributions aroundthe mean Therefore even with mean contributions furtherfrom the data (induced by the fixed value of 120572) the spreadinduced by the noise parameter in specification (2) producesa better fit This highlights the importance of taking intoaccount not only the average (point) predictions but also thespread around it It also suggests that allowing 120572 to vary canimprove fit

In specification (3) 120572 and 120583 are jointly estimated using(5) subject to 120572 ge 0 If both parameters can freely varyover R

+ 120572 reduces to zero and 120583 reaches a value that

is higher than what was obtained in specification (2) Asconfirmed by a likelihood-ratio test specification (3) fitsthe experimental data better than both specification (1)


Table 2 Homogeneous population (last 5 rounds)

Data CFS (1) 120583 (2) 120583 (3) 120583 120572120583 mdash 1 1163 [980 1395] 2691 [2414 3017]120572 059 059 059 0 [0 003](Predicted) avg contributions


119908119894= 160 1763 2346 2404 4168 2601

119908119894= 200 3528 2932 2971 4830 2630

119908119894= 240 2970 3519 3545 5460 2639

log 119897119897 minus167503 minus98754 minus88562Obs 240 240 240 240 240This table reports average contributions as well as estimates and predictions from various specifications of the model based on the efficiency concernsassumption using the last 5 rounds of the experiment only The same remarks as in Table 1 apply

Table 3 Heterogeneous subjects (all and last 5 rounds)

120583 1205721 and 120572

2(119899 = 20) 120583 120572

1 and 120572

2(119899 = 5) 120583 120572

1 1205722 and 120572

3(119899 = 20) 120583 120572

1 1205722 and 120572

3(119899 = 5)

120583 2850 [2588 3126] 1507 [1290 1764] 2214 [2056 2395] 1425 [1204 1685]1205721

0 [0 001] 0 [0 002] 0 [0 001] 0 [0 002]1205722

053 [046 060] 054 [047 061] 043 [039 046] 048 [040 056]1205723

104 [092 116] 076 [053 101]1205741

066 [053 078] 063 [050 075] 041 [033 046] 059 [049 064]1205742

050 [043 055] 034 [023 040](Predicted) avg contributions

Overall endowments 3715 2557 3851 2572119908119894= 120 3206 2217 3197 2207

119908119894= 160 3604 2463 3678 2468

119908119894= 200 3901 2677 4083 2699

119908119894= 240 4148 2873 4445 2913

log 119897119897 minus311206 minus86575 minus308335 minus86516Obs 960 240 960 240This table reports estimates and predictions from two specifications of the model with efficiency concerns accounting for cross subject heterogeneity in thevalue of 120572 The analysis is conducted both by including all experimental rounds and by focusing on the last five repetitions only Parameters are estimatedthrough (6) Given the linear restrictionsum119878

119904=1120574119904 = 1 we only report estimates of 1205741 1205742 120574119878minus1 Confidence intervals are computed using an inversion of the

likelihood-ratio statistic at the 001 level subject to parameter constraints

(LR = 1108654 Pr1205942(2) gt LR lt 001) and specification(2) (LR = 62621 Pr1205942(1) gt LR lt 001) Thus under themaintained assumption of homogeneity our estimates sug-gest that contributions are better explained by randomness insubjectsrsquo behavior rather than by concerns for efficiency

In order to control for learning effects we replicate ouranalysis using the last five rounds only

Consistent with a learning argument in both specifi-cations (2) and (3) the values of 120583 are substantially lowerthan the corresponding estimates in Table 1 Thus repetitionreduces randomness in subjectsrsquo contributions The mainresults presented above are confirmed by our analysis onthe last five periods Looking at specification (3) in themodel with no constraints on the parameters the estimatedvalue of 120572 again drops to 0 Also according to a likelihood-ratio test specification (3) explains the data better than both

specifications (1) (LR = 157883 Pr1205942(2) gt LR lt 001)

and (2) (LR = 20385 Pr1205942(1) gt LR lt 001)These results seem to reject the preference for (group)

efficiency hypothesis in favor of pure randomness in subjectsrsquocontributions However a different picture emerges whenwe allow for cross-subject heterogeneity In Table 3 we dropthe assumed homogeneity We consider two models withheterogeneous subjects the first assumes the population tobe partitioned into two subgroups (119878 = 2) and the secondinto three subgroups (119878 = 3) (We have also estimated amodel with 119878 = 4 However adding a fourth subgroup doesnot significantly improve the goodness of fit of the modelcompared to the specification with 119878 = 3 In particular with119878 = 4 the point estimates for the model with all periodsare 120583 = 2181 120572

1= 0 120572

2= 038 120572

3= 061 120572

4= 104

1205741= 039 120574

2= 042 and 120574

3= 009) As before we conduct


Table 4

120583 120572 Log-likelihood100000 0 minus36376450000 0 minus35917933333 0 minus3548725000 0 minus35083420000 0 minus34706716667 0 minus34356414286 0 minus34031912500 0 minus33732511111 0 minus33457610000 0 minus3320649091 0 minus329788333 0 minus3277177692 0 minus3258657143 0 minus3242166667 0 minus3227616250 0 minus3214935882 0 minus3204015556 0 minus3194795263 0 minus3187185000 0 minus318114000 0 minus3171223030 014 minus3192522000 033 minus3247221000 050 minus344442909 052 minus348826800 053 minus355557704 055 minus363445599 056 minus375367500 057 minus391689400 058 minus417335300 059 minus461592200 060 minus554735100 061 minus850613090 061 minus918167080 061 minus1003207070 061 minus1113338060 061 minus1261263050 061 minus1469913040 061 minus1785264This table reports Maximum Likelihood estimates of 120572 for selected values of120583 (see (5)) The last column reports the corresponding log-likelihood value

our analysis both by including all rounds of the experimentand by focusing on the last five repetitions only

We find strong evidence in favor of subjectsrsquo heterogene-ity Focusing on the analysis over all rounds according tothe model with two subgroups a subject is associated with1205721

= 0 with probability 066 and with 1205722

= 053 withprobability 034 Results are even sharper in the model withthree subgroups in this case 120572

1= 0 and the two other 120572-

parameters are strictly positive 1205722= 043 and 120572

3= 104

Subjects are associated with these values with probabilities

041 050 and 009 respectively Thus in the more parsi-monious model the majority of subjects contribute in a waythat is compatible with the preference for (group) efficiencyhypothesis These proportions are in line with findings ofprevious studies [18 21 22] in which aside from confusionsocial preferences explain the behavior of about half of theexperimental population

Allowing for heterogeneity across subjects reduces theestimated randomness in contributions the value of 120583

reduces from 4159 in specification (3) of the model withhomogeneous population to 2850 and 2214 in the modelwith two and three subgroups respectively According to alikelihood-ratio test both the models with 119878 = 2 and 119878 = 3 fitthe data better than the (unconstrained) specification of themodel with homogeneous subjects (for the model with 119878 = 2LR = 11725 Pr1205942(2) gt LR lt 001 whereas for the modelwith 119878 = 3 LR = 17466 Pr1205942(4) gt LR lt 001) Moreoveradding an additional subgroup to the model with 119878 = 2significantly increases the goodness of fit of the specification(LR = 5742 Pr1205942(2) gt LR lt 001) As before all theseresults remain qualitatively unchanged when we control forlearning processes and we focus on the last 5 experimentalrounds

In order to check for the robustness of our results inTable 3 we have also estimated additional specificationsaccounting for heterogeneity in both concerns for (group)efficiency and noise in subjectsrsquo behavior Although the log-likelihood of the model with both sources of heterogeneitysignificantly improves in statistical terms the estimatedvalues of the 120572-parameters remain qualitatively the same asthose reported in the third column of Table 3

5 Conclusions

Is overcontribution in linear public good experimentsexplained by subjectsrsquo preference for (group) efficiency orrather does it simply reflect their natural attitude to makeerrors In order to answer this fundamental question weestimate a quantal response equilibrium model in which inchoosing their contributions subjects are influenced by botha genuine concern for (group) efficiency and a random noisein their behavior

In line with other studies we find that both concernsfor (group) efficiency and noise in behavior play an impor-tant role in determining subjectsrsquo contributions Howeverassessing which of these two behavioral hypotheses is morerelevant in explaining contributions strongly depends onthe degree of cross-subject heterogeneity admitted by themodel Indeed by estimating a model with homogeneoussubjects the parameter capturing concerns for (group) effi-ciency vanishes while noise in behavior entirely accountsfor overcontribution A different picture emerges when weallow the subjects to be heterogeneous in their concerns forefficiency By estimating a model in which the populationis partitioned into three subgroups that differ in the degreeof concerns for efficiency we find that most of the subjectscontribute in a way that is compatible with the preferencefor (group) efficiency hypothesis A formal likelihood-ratio


test confirms the supremacy of the QRE model with threesubgroups over the other specifications

Previous studies [18ndash23] tried to disentangle the effects ofnoise fromother-regarding preferences bymainlymanipulat-ing the experimental design Our approach adds a theoreticalfoundation in the form of an equilibrium analysis In contrastto studies which focus mostly on (direct) altruism we followCorazzini et al [2] and allow for preference for efficiencyOur results are in line with the literature in the sense that wealso conclude that a combination of noise and social concernsplays a role Our results however are directly supported by asound theoretical framework proven valid in similar settings(eg [36])

Recent studies [37 38] have emphasized the importanceof admitting heterogeneity in social preferences in order tobetter explain experimental evidence In this paper we showthat neglecting heterogeneity in subjectsrsquo social preferencesmay lead to erroneous conclusions on the relative importanceof the love for (group) efficiency hypothesis with respect tothe confusion argument Indeed as revealed by our analysisthe coupling of cross-subject heterogeneity in concerns for(group) efficiency with noise in the decision process seemsto be the relevant connection to better explain subjectsrsquocontributions

Appendix

Table 4 shows the Maximum Likelihood value of 120572 and thelog-likelihood according to (5) as 120583 decreases from 1000 to04 As shown by the table for high values of 120583 the estimatedvalue of 120572 is 0 When 120583 is equal to 10 the estimated value ofalpha is 050 Moreover for 120583 lower than 200 the estimatedvalue of 120572 is 061 For the specification tests presented inSection 4 we set 120583 = 1 This is a sufficiently low value of 120583in order to generate a noisy version of the base model Twoarguments indicate why this choice is valid First for a rangeof values including 120583 = 1 the estimated 120572 is stable Moreoversince the log-likelihood of a model with 120572 = 061 and 120583 = 1

is higher than that corresponding to a model with 120583 = 04

(and similarly for 120572 = 064) the choice of any 120583 lower than1 for the benchmark value would only reinforce the resultsof Section 4 More specifically both likelihood-ratio statisticscomparing specifications (1)with specifications (2) and (3) ofTables 1 and 2 would increase



Acknowledgments

The authors thank Arthur Schram Jens Grosser and par-ticipants to the 2010 Credexea Meeting at the Universityof Amsterdam 2011 IMEBE in Barcelona and 2011 AnnualMeeting of the Royal Economic Society in London for usefulcomments and suggestions

References


[2] L Corazzini M Faravelli and L Stanca ldquoA prize to givefor an experiment on public good funding mechanismsrdquo TheEconomic Journal vol 120 no 547 pp 944ndash967 2010

[3] R Sugden ldquoReciprocity the supply of public goods throughvoluntary contributionsrdquoTheEconomic Journal vol 94 no 376pp 772ndash787 1984

[4] H Hollander ldquoA social exchange approach to voluntary con-tributionrdquo The American Economic Review vol 80 no 5 pp1157ndash1167 1990

[5] A Falk andU Fischbacher ldquoA theory of reciprocityrdquoGames andEconomic Behavior vol 54 no 2 pp 293ndash315 2006

[6] U Fischbacher S Gachter and E Fehr ldquoAre people condition-ally cooperative Evidence from a public goods experimentrdquoEconomics Letters vol 71 no 3 pp 397ndash404 2001

[7] D K Levine ldquoModeling altruism and spitefulness in experi-mentsrdquo Review of Economic Dynamics vol 1 no 3 pp 593ndash6221998

[8] J Andreoni ldquoGiving with impure altruism applications tocharity and ricardian equivalencerdquo Journal of Political Economyvol 97 no 6 pp 1447ndash1458 1989

[9] J Andreoni ldquoImpure altruism and donations to public goodsa theory of warm-glow givingrdquo Economic Journal vol 100 no401 pp 464ndash477 1990

[10] J J Laffont ldquoMacroeconomic constraints economic efficiencyand ethics an introduction to Kantian economicsrdquo Economicavol 42 no 168 pp 430ndash437 1975

[11] M Bordignon ldquoWas Kant right Voluntary provision of publicgoods under the principle of unconditional commitmentrdquoEconomic Notes vol 3 pp 342ndash372 1990

[12] M Bernasconi L Corazzini and A Marenzi ldquolsquoExpressiversquoobligations in public good games crowding-in and crowding-out effectsrdquo Working Papers University of Venice ldquoCarsquo FoscarirdquoDepartment of Economics 2010

[13] M Bacharach Beyond Individual Choice Teams and Framesin Game Theory Edited by N Gold and R Sugden PrincetonUniversity Press 2006

[14] R Sugden ldquoThe logic of team reasoningrdquo Philosophical Explo-rations vol 6 no 3 pp 165ndash181 2003

[15] R Cookson ldquoFraming effects in public goods experimentsrdquoExperimental Economics vol 3 no 1 pp 55ndash79 2000

[16] G Charness and M Rabin ldquoUnderstanding social preferenceswith simple testsrdquoQuarterly Journal of Economics vol 117 no 3pp 817ndash869 2002

[17] D Engelmann and M Strobel ldquoInequality aversion efficiencyand maximin preferences in simple distribution experimentsrdquoAmerican Economic Review vol 94 no 4 pp 857ndash869 2004

[18] J Andreoni ldquoCooperation in public-goods experiments kind-ness or confusionrdquoTheAmerican Economic Review vol 85 no4 pp 891ndash904 1995

[19] T R Palfrey and J E Prisbrey ldquoAltruism reputation andnoise in linear public goods experimentsrdquo Journal of PublicEconomics vol 61 no 3 pp 409ndash427 1996

[20] T R Palfrey and J E Prisbrey ldquoAnomalous behavior inpublic goods experiments how much and whyrdquoThe AmericanEconomic Review vol 87 no 5 pp 829ndash846 1997


[21] J Brandts and A Schram ldquoCooperation and noise in publicgoods experiments applying the contribution function ap-proachrdquo Journal of Public Economics vol 79 no 2 pp 399ndash4272001

[22] D Houser and R Kurzban ldquoRevisiting kindness and confusionin public goods experimentsrdquo American Economic Review vol92 no 4 pp 1062ndash1069 2002


[24] D O Stahl and PWWilson ldquoOn playersrsquo models of other play-ers theory and experimental evidencerdquo Games and EconomicBehavior vol 10 no 1 pp 218ndash254 1995

[25] T-H Ho C Cambrer and K Weigelt ldquoIterated dominanceand iterated best response in experimental lsquop-Beauty ContestsrsquordquoAmerican Economic Review vol 88 no 4 pp 947ndash969 1998

[26] D O Stahl and E Haruvy ldquoLevel-n bounded rationality intwo-player two-stage gamesrdquo Journal of Economic Behavior ampOrganization vol 65 no 1 pp 41ndash61 2008

[27] I Erev and A E Roth ldquoPredicting how people play gamesreinforcement learning in experimental games with uniquemixed strategy equilibriardquo American Economic Review vol 88no 4 pp 848ndash881 1998

[28] J Andreoni ldquoWhy free ride Strategies and learning in publicgoods experimentsrdquo Journal of Public Economics vol 37 no 3pp 291ndash304 1988


[30] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquoTheAmerican Economic Review vol 83 no 5 pp 1281ndash1302 1993


[32] G E Bolton and A Ockenfels ldquoERC a theory of equityreciprocity and competitionrdquo American Economic Review vol90 no 1 pp 166ndash193 2000

[33] R D Cook and S Weisberg ldquoConfidence curves in nonlinearregressionrdquo Journal of the American Statistical Association vol85 no 410 pp 544ndash551 1990

[34] D R Cox andDVHinkleyTheoretical Statistics Chapman andHall London UK 1974

[35] S A Murphy ldquoLikelihood ratio-based confidence intervals insurvival analysisrdquo Journal of the American Statistical Associa-tion vol 90 no 432 pp 1399ndash1405 1995


[37] U Fischbacher and S Gachter ldquoHeterogeneous social prefer-ences and the dynamics of free riding in public goodsrdquo CeDExDiscussion Paper 2006-01 2006

[38] M Erlei ldquoHeterogeneous social preferencesrdquo Journal of Eco-nomic Behavior and Organization vol 65 no 3-4 pp 436ndash4572008

Research ArticleCompetition Income Distribution and the Middle ClassAn Experimental Study

Bernhard Kittel1 Fabian Paetzel2 and Stefan Traub2

1Department of Economic Sociology and FOR 2104 University of Vienna 1090 Vienna Austria2Department of Economics and FOR 2104 Helmut-Schmidt-University 22043 Hamburg Germany

Correspondence should be addressed to Bernhard Kittel bernhardkittelunivieacat

Received 5 December 2014 Revised 28 January 2015 Accepted 17 February 2015

Academic Editor Klarita Gerxhani

Copyright copy 2015 Bernhard Kittel et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study the effect of competition on income distribution by means of a two-stage experiment Heterogeneous endowments areearned in a contest followed by a surplus-sharing task The experimental test confirms our initial hypothesis that the existence ofa middle class is as effective as institutional hurdles in limiting the power of the less able in order to protect the more able playersfrom being expropriated Furthermore majoritarian voting with a middle class involves fewer bargaining impasses than grantingveto rights to the more able players and therefore is more efficient

1 Introduction

In recent years public awareness of an increasingly lopsideddistribution of income and wealth in Western countries hasstrongly increased [1] Most OECD countries have witnessedgrowing inequality over the past 20 years [2] In particular thegap between the bottom and the top deciles of the householdincome distribution has risen dramatically The decile ratiocurrently amounts to about 1 15 in the US and 1 9 in theOECD-34 and even in a Nordic welfare state like Sweden itis close to 1 6 Given the fact that the typical OECD incomedistribution is right-skewed [3] it does not come as a surprisethat income tax progression is much higher in the US than inSweden [4]

Observers diagnose the formation as well as deliberateestablishment of a winner-take-all society in which themiddle class is gradually eroded [5ndash7] A few superstarswhich may (or may not) be the most able practitionersin a particular area receive all the profit while the othersrsquoefforts are in vain (eg [8ndash10]) Some commentators expectthis development to result in increasing societal divisiondistributive struggle and violent conflict [11]

On the other hand the fall of communist regimes in1989 has demonstrated the crippling effects of excessive

egalitarianism on economic efficiency and growth [12 13]Societies or groups thus need to strike a balance betweenboth excessive inequality and excessive equality In principleeither an institutional or a structural condition is sufficient toprotect a society against excesses from both sides On the onehand institutional rules of collective decision-making givingveto power to all stakeholders [14 15] enforce a consensualdecision and therebymake the distributional struggle an issueof explicit negotiation On the other hand the existence ofa neutral but opportunistic middle class serves as a bufferbetween the upper and the lower classes because they fearexpropriation from both sides and will thus side with theweaker group in case of conflict [16ndash18] Moreover not beingthemain target of redistributionmembers of themiddle classwill let their decisions be guided more by moral values suchas justice norms than the other two socioeconomic groups[19ndash21]

The two conditions however differ with respect to theirdecision efficiency Although the unanimity rule minimizesexternal costs due to the inclusion of all members of thesociety it also allows every member to hold up the decisionuntil the outcome satisfactorily represents its particular inter-ests For this reason this arrangement is almost inexistent at



the level of nation states and is notorious for its detrimentaleffects in the security council of the United Nations Becauseof its crippling implications the unanimity rule has beenreplaced by qualified majority for many policy areas in theEuropean Council by the Lisbon Treaty in 2007 In contrastthe position-based interests of the middle class limit the costsburdened upon the minority because its members will shiftsides as soon as demands on the minority become expropri-atoryThe existence of a middle class thus quasiautomaticallycorrects for both excessive inequality and equality withoutgenerating the hold-up effect of an institutional rule

In this paper we experimentally test the compositehypothesis following from this argument that in democraticand competitive organizations both the existence of amiddleclass and institutional hurdles can protect the most ableorganization members from the demand of the least able forexcessive income distribution policies (Protection Hypothe-sis) yet setting up institutional hurdles is inferior to relyingon a middle class as institutional hurdles involve efficiencylosses due to bargaining impasses (Efficiency Hypothesis)

In our experiment we directly test whether a neutralmiddle class is as effective as an institutional rule to avoidexcessive expropriation or extremely low redistribution Thesecond goal of the paper is to check if the existence of amiddleclass leads to higher efficiency than institutional hurdles

The experiment was composed of two consecutive tasksDuring the first task subjects took part in a multiple-prizerank-order contest that involved a simple cognitive abilitytask The contest rewarded effort by assigning the first prizeto the most able subject the second prize to the second mostable subject and so on Following Moldovanu and Sela [22]the contest ensured that more able subjects should expendmore effort in order to win it Hereby we focus on theimpact of the skewness of the prize scheme on effort andits interaction with ex-post income distribution During thesecond task players first deliberated on sharing a surplus(using a group chat) As soon as the deliberation time hadelapsed they simultaneously and anonymously made theirbinding proposals for a distribution parameter The proposalthat achieved the quorum was played out and the surplusshares were added to the prize money won in the contest Ifa group missed the quorum the players received only theirprizes

We find that the ProtectionHypothesis is clearly supportedby our data The existence of a neutral middle class is aseffective as institutional hurdles The Efficiency Hypothesisis also supported Institutional hurdles come at the costs ofsignificantly less efficiency in terms of higher default ratesThe corroboration of both hypotheses allows us to give a clearrecommendation for the rich The rich should brace againstthe erosion of the middle class because the middle class is theefficient protection of the rich

The rest of the paper is organized as follows Section 2links our paper to the literature Section 3 introducesthe theoretical framework of our experiment and derivestestable hypotheses Section 4 explains how the models weretransferred into the experimental lab Section 5 reports theresults of the experiment Section 6 discusses the results andconcludes the paper

2 Literature Review

If we assume that subjects solely follow self-interest and thatself-interest is directly expressed in a social contract witha corresponding level of redistribution we find ourself inthe world of the median voter theorem [23] Following thistheory an increasing skewness must result in larger pressurefor equalization because the median voterrsquos income declinesand hence her interest in redistribution increases [23 24]Disregarding the possibility of revolutionary redistribution(tax rates exceeding one) the most extreme form of equaliza-tion within a society is to implement by means of collectivedecision ameasure which results in equal payoffsThis resulthowever is likely to generate socially inefficient outcomesbecause it undermines the willingness of the more able toexpend effort [25]

Our paper contributes to the growing literature ana-lyzing why the rich do not get expropriated Besides theabove-mentioned protection through institutional hurdles orthe existence of a neutral middle class another stream inthe literature stresses different explanations why the poormedian voter does not necessarily follow the median voterrsquosprediction The ldquoprospects of upward mobilityrdquo hypothesis[26] brings forward the argument that some voters whohave an income below the mean expect that their futureincome will be above the mean The expectation of thesevoters works against votes for high levels of redistributionRoemer [27] andRoemer and Lee [28] show that voting aboutredistribution is also affected by religion and race The morefragmented the population is with regard to both dimensionsthe lower is the willingness to redistribute Alesina andAngeletos [13] and Fong [29] show that if the poor believethat the rich are rich because they had invested high effortsand have high abilities the pressure for redistribution is lowAccording to extensive empirical and experimental literaturepeople tend to accept some amount of inequality if it can beattributed to individual effort and ability (eg [30ndash35]) Klorand Shayo [36] find that if the poor identify themselves withthe nation (rich and poor) the poor do not want to harmmembers of the nation and therefore do not expropriate therichMoreover Glaeser et al [18] highlight that beyond beingprotected from expropriation through institutions which aremaintained to enforce low levels of redistribution the better-off are able to manipulate institutions to their own profitOur paper adds an additional explanation of why the poorcannot enforce high levels of redistributionWe argue that themiddle class acts as an uninvolved spectator not affected byredistribution that does not have an interest in excesses fromeither side

We contribute to a recent wave of research investigatingvoting on redistribution This literature sheds some lighton the voting decision by using experiments The mainmethodological advantage of using experiments is that itallows us to detect how preferences affect the voting-decisionA stylized fact generated by this literature is that voters arewilling to sacrifice their own payoffs in order to achieve amore equal payoff distribution which has been explained bysocial preferences or social identity Tyran and Sausgruber[37] reported supporting evidence for inequity theory [38]


but emphasized the importance of being pivotal for votingon redistribution Similarly Hochtl et al [39] showed thatfor empirically plausible cases inequality-averse voters maynot matter for redistribution outcomes Moreover they didnot observe efficiency-loving voters Bolton and Ockenfels[40] investigated the trade-off between equity and efficiencymotives in a voting game with three voters They foundthat twice as many voters were willing to give up theirown payoff in favor of an equal distribution as comparedto a more efficient but unequal distribution Messer et al[41] studied the impact of majority voting on the provisionof a public good They detected substantial concerns forefficiency in the subjectsrsquo behavior but found little supportfor inequality aversion and maximin preferences Paetzelet al [42] investigated whether voters would be willing tosacrifice their own payoff in order to implement not onlyan efficiency-increasing but also an inequality-increasingreformThis case can be thought of as the reverse of voting onefficiency-reducing but inequality-decreasing redistributionThey showed that efficiency preferences of potential reform-losers outweighed the inequality aversion of reform-gainersKlor and Shayo [36] reported a trade-off between social-identity concerns and payoffs maximization a subset oftheir sample systematically deviated from monetary payoffmaximization towards a tax rate benefiting their group Suchdeviations can neither be explained by efficiency concernsnor by inequality aversion

Recently Balafoutas et al [43] reported on an exper-iment similar to ours (heterogenous initial endowmentsquiz versus random assignment majority vote and neutralmiddle player) which generated a number of interestingresults For example the players with the highest and thelowest endowments were mainly driven by material self-interest Low-endowment players however signalled theirwillingness to cooperate by increasing their contributions ifthe redistribution rate was determined bymajority vote1 Ourexperiment differs in various important respects the size ofthe pie to be redistributed is exogenously given our gameinvolves a communication phase before voting on the tax andwe vary both the initial distribution of endowments and theinstitutional background in terms of the quorum

A bit against the trend Esarey et al [44] and Duranteet al [45] still found redistribution to be strongly related tothe self-interest of voters Durante et al [45] showed thatsupport for redistribution vanishes if taxation was associatedwith costs and deadweight losses and if participants earnedtheir incomes in a real effort task

3 Theoretical Framework

In this section we introduce the theoretical framework ofour experiment in order to derive testable hypotheses Theexperiment was composed of two consecutive tasks namelya contest and a demand game The contest is explained inthe first subsection It was used for allocating subjectsrsquo initialendowments The protection hypothesis presupposes thatinitial endowments adequately reflect subjectsrsquo abilities Wedid not induce abilities but subjects took part in a multiple-prize rank-order contest that involved a simple knowledge

test Assuming that subjects knew their own abilities to solvethe knowledge test and had a guess about the distribution ofabilities within the sample which wasmade up of their fellowstudents the contest ensured that more able subjects wouldexpend more effort in order to win it Effort was rewardedby assigning the first prize to the most able subject thesecond prize to the second most able subject and so on Thetheoretical setup for the contest was taken from Moldovanuand Sela [22] Sheremeta et al [46] showed the rank-ordercontest to generate higher effort than lottery and proportionalcontests2

The demand game is explained in the second subsectionIt was used for investigating the impact of different alloca-tions of initial endowments and institutional setups on theacknowledgment of abilities and effort respectively Subjectsfirst deliberated on sharing a surplus As soon as the delibera-tion time had elapsed they simultaneously and anonymouslymade their binding proposals for a distribution parameterThe proposal that achieved the quorum was played out andthe surplus shares were added to the prize money won in thecontest If a group missed the quorum the subjects receivedonly their prizes Such coordination and bargaining gameswith preplay communication (also known as cheap talk) haveintensively been studied in the literature (for surveys see eg[47ndash49]) The task determined the distribution parameter byvote and required a communication phase prior to finalizingthe proposals such that group members could coordinatetowards a distribution parameter capable of winning thenecessary majority In the context of our experiment theresults of Roth [50 51] are highly relevant who showed thatcheap talk focuses subjectsrsquo attention on a small numberof fairness norms in unstructured bargaining experiments(see also [52]) More specifically subjects identify initialbargaining positions that have some special reason for beingcredible and these serve as focal points

In the third subjection we derive testable hypothesesfrom the theoretical models We first identify three potentialfocal points of the demand game namely the egalitariansolution equal sharing and proportional sharingThen eachexperimental setup is assigned its most likely sharing ruleby theoretical consideration We also analyze the possibilityof disagreement Finally we argue by backward inductionwhether and how the sharing rule enacted in the demandgame could impact subjectsrsquo willingness to expend effort inthe contest

31 The Contest Following Moldovanu and Sela [22] weconsider a contest with three prizes 120587119895 where 1205871 gt 1205872 ge 1205873 ge

0 The total prize money Π = sum3119895=1 120587119895 has been exogenously

fixed by the contest organizer The set of contestants is givenby K = 119860 119861 119862 Individual effort is denoted by 119909119894 wherewe set 119909119860 gt 119909119861 gt 119909119862 The contest success function isperfectly discriminatory that is contestant 119860 is awarded thefirst prize1205871 contestant119861 is awarded the second prize1205872 andcontestant 119862 is awarded the third prize 1205873 Making an effortinvolves a cost 119888119894120574(119909119894)

3120574 is a strictly increasing function with

120574(0) = 0 119888119894 gt 0 is an ability parameter The lower is 119888119894 thehigher is the ability of contestant 119894 and the lower is the cost ofeffort


Individual abilities are private information They areindependently drawn from the closed interval [119898 1] where119898 gt 0 and have a continuous distribution function 119865(119888)

which is common knowledge To put it simply we assumethat ability follows a uniform distribution 119865(119888) = ((1(1 minus

119898))119888 minus 119898(1 minus 119898))119868[1198981](119888) + 119868(1infin)(119888) where 119868 is an indicatorfunction taking a value of one if 119888 is inside the stated intervaland otherwise taking a value of zero It can be shown (seeAppendix C in [22]) that in the Nash equilibrium individualeffort is given by

119909 (119888) = 120574minus1(

3

sum

119895=1

120587119895 int

1

119888

minus

1

119886

1198651015840119895 (119886) 119889119886) (1)

where

1198651015840119895 (119886) =

(3 minus 1)

(119895 minus 1) (3 minus 119895)

(1 minus 119865 (119886))3minus119895minus1

119865 (119886)119895minus2

sdot 1198651015840(119886) ((1 minus 3) 119865 (119886) + (119895 minus 1))

(2)

In order to interpret (1) one has to realize that a subjectrsquosprobability of wining prize 119895 1 le 119895 le 3 given her ability119886 isin [119898 1] can be represented by an order statistic

119865119895 (119886) =(3 minus 1)

(119895 minus 1) (3 minus 119895)

(1 minus 119865 (119886))3minus119895

119865 (119886)119895minus1

(3)

This equation takes into account that subject 119895 meets 2

competitors of whom 3 minus 119895 are less able than 119895 and 119895 minus 1

are more able Equation (2) is the first derivative of (3) withrespect to ability Hence the payoff-maximizing equilibriumeffort stated in (1) is a function of the weighted sum of thethree prizes and the weights differ from subject to subjectaccording to their abilities 119888 Shaked and Shanthikumar [53]have shown that if 120574(119888) is a strictly increasing function effortis a monotonically decreasing function of the cost of ability 119888and is zero at 119888 = 1 Hence the most able subject will expendthe highest effort and win the first price the medium subjectwill win the second prize and the least able subject willexpend the lowest effort (but not necessarily zero effort) andwin the third prize Since there is a unique relation betweenability effort and prize in the following we will call the mostable subject the 119860-player the medium able subject the 119861-player and the least able subject the 119862-player

The bidding function (1) is too complex to be solvedanalytically for the ldquostructural parametersrdquo of the prizescheme that are of interest for us namely its mean varianceand skewness Nevertheless clear predictions concerningsubjectsrsquo effort reactions can be derived for each parameterchange by numerical simulation In Appendix A we reportthe results of simulating a contest with 119888 = 04 06 08

and 119898 = 02 assuming linear cost functions The predictedoutcome for each prize scheme applied in the experimentis shown in Table 4 The columns list groups of threehypothetical subjects with their abilities expected winningprobabilities expected returns efforts ranks prizes costsand net payoffs (recall that subjects did not know the abilitiesof the other subjects but supposedly only the distribution

of ability) Table 5 also located in Appendix A summarizesthe information given in Table 4 by listing each prize schemewith its parametrization4 Additionally it states the mean andthe coefficient of variation of predicted effort As can be seenfrom the table predicted effort doubles if the prize moneyis doubled and if the coefficient of variation of the prizescheme is doubled and it is almost halved when the prizescheme becomes right-skewed Furthermore within-groupeffort variation increases with a right-skewed prize scheme

32 The Demand Game The second task is a variation ofthe Nash demand game [54] After awarding the prizes tothe contestants the contest organizer asks the contestants tomake an arbitrary number of proposals 119904 = 119904119860 119904119861 119904119862 forsharing a surplus 119878 = Π that is total prize money is doubledHowever the surplus is made available only if one of theproposals is supported by a qualified majority of the groupof contestants If none prevails the contestants receive onlytheir prizes 120587119863 = 120587119860 120587119861 120587119862 and exit the game (defaultoption) If one of the proposals becomes accepted it is madebinding and the contestants receive their gross payoffs 120587120572 =120587119860 + 119904119860 120587119861 + 119904119861 120587119862 + 119904119862 The following restriction appliesto the set of feasible distributions of 119878

119904120572119894 = 120572120587119894 + (1 minus 120572) (

2

3

Π minus 120587119894) forall119894 isin 119860 119861 119862 0 le 120572 le 1

(4)

It is straightforward to show that any feasible agreement 120572 isin

[0 1] would be weakly preferred by each contestant over thedefault option if 119904119894 ge 0 or 0 le 120587119894 le (23)Π Hence assumingthat the latter condition holds the default option can never bea Nash equilibrium Conversely any agreement is a Pareto-efficient Nash equilibrium We refer to 120572 as the distributionparameter in the following5

This unstructured bargaining protocol in which subjectscould first freely communicate and thereafter had to submittheir binding proposals for 120572 simultaneously and anony-mously is ideal for studying the distribution of the surplusand its interaction with the contest for two reasons (for acritical survey of bargaining theories and their experimentaltests see eg [55]) First existing experimental evidenceclearly suggests that subjects coordinate on a small number offocal points or fairness norms in such settings6 Distributionsthat could potentially serve as focal points in our experimentare addressed below Second the bargaining protocol allowsfor studying apart from conflict of interest the impact ofdifferent institutional settings in terms of voting rules onthe frequency of disagreement as required for testing theEfficiency Hypothesis7 Sequential bargaining protocols suchas Rubinsteinrsquos [56] and in its finite version Baron andFerejohnrsquos [57] are inappropriate for studying our researchquestions because they exhibit unique subgame perfect Nashequilibria and they assign proposer and responder roles tothe players

33 Focal Points and Hypotheses The demand game outlinedin the previous subsection has three potential focal pointswhich correspond to different distribution principles or


120572 = 0

120572 = 05

120572 = 1

AR

E

D

0

12

26 33 100

120587120572

B

120587120572

C

120587120572

A

Figure 1 Sharing a surplus The figure shows the gross payoffs 120587120572119894of players 119894 = 119860 (horizontal axis) and 119861 and 119862 (vertical axis) forsharing a surplus of 119878 = Π = 50 with a right-skewed distribution ofprizes120587 = 26 12 12ER represents the bargaining set as well as thePareto frontier D marks the default option (no agreement) Threefocal points E (egalitarian) A (equal sharing) and R (proportionalsharing) are depicted togetherwith their corresponding values of thedistribution parameter 120572

fairness norms Figure 1 gives a self-explaining example fora right-skewed distribution of prizes 120587

119863= (26 12 12)

Total prize money and surplus are given by Π = 119878 = 50The ldquobudget linerdquo ER represent the bargaining set as wellas the Pareto frontier With regard to 120572 we can distinguishthree interesting special cases First for 120572 = 1 we obtain asurplus distribution 119904

119877= 119904119877119860 119904119877119861 119904119877119862 where 119904

119877119894 = 120587119894 forall119894 isin

119860 119861 119862 that is proportional to the prize money earned inthe contest and thus has a strong positive correlation witheffort In comparison to the distribution of prizes 120587119863 =

120587119860 120587119861 120587119861 (point D) the resulting gross payoff distribution120587119877= 2120587119860 2120587119861 2120587119862 (point R in the figure) preserves the

degree of inequalitymeasured by any relative (scale invariant)inequalitymeasureThe equity standard underlying120587119877 is wellknown as proportional sharing [58] It targets the individualopportunity costs of the players [59]

Second for 120572 = 05 surplus is shared equally 119904119860119894 =

(13)Π proportional to the size of the surplus and uncorre-lated with effort Compared with 120587

119863 the distribution 120587119860=

120587119860 + (13)Π 120587119861 + (13)Π 120587119862 + (13)Π does not changeany absolute (translation invariant) inequality measure Therespective focal point is marked by point A in Figure 1Moulin [58 page 162] argued that ldquoany symmetric solutionconcept would divide equally the surplusrdquo The equal sharingrule rewards cooperation rather than effort

In the third case 120572 = 0 yields 119904119864119894 = (23)Π minus 120587119894which is proportional to the gap between the equal sharein total resources (prize money and surplus) and own prize119904119864119894 is negatively correlated with effort An equal distributionof gross payoffs 120587119864 = (23)Π (23)Π (23)Π (point E)occurs (egalitarian solution) Here the surplus is devotedto wiping out all inequality with regard to gross payoff byimposing a distribution principle which resembles Rawlsrsquo

[60] difference principle (though the demand game does notinvolve a veil of ignorance) However if erroneously appliedto gross income instead of net income this principle penalizeseffort by reversing the rank ordering produced by the contest

As regards different setups of the demand game we havethe following expectations The unanimity rule assigns everysubject the same veto power regardless of her performancein the contest Hence subjects deal separately with thedifferent tasks In the default option of the demand gamethe experimenter retains the surplus that is all subjectsare treated alike in being given nothing In line with theliterature (see eg [61]) we therefore conjecture the so-calledsymmetric solution equal sharing (120572 = 05) to be the onlydistribution standard compatible with the unanimity rule

If the demand game is played with simple majority votinginstead of the unanimity rule we have to carefully distinguishbetween symmetric and right-skewed prize schemes Let 120587 =

(13)Π denote the equal shareThe first derivative of (4) withrespect to 120572 is given by

120597119904120572119894

120597120572

= 2 (120587119894 minus 120587) (5)

A payoff-maximizing subject would therefore vote for 120572 = 1

if her prize money exceeds the equal share she would votefor 120572 = 0 if her prize money falls short of the equal shareand she would be indifferent if her prize money equals 120587 Inour experimental setup right-skewed prize schemes consistof two prizes falling short of 120587 and one prize exceedingit Hence the 119861- and 119862-players are expected to agree on120572 = 0 giving rise to the egalitarian solution In contrastto this if a symmetric prize scheme applies the 119861-player isindifferent because she cannot change her personal surplusshare by voting on 120572 The 119861-playerrsquos vote on the distributionparameter might be guided by social preferences Inequityaversionwould induce the119861-player to agreewith the119862-playeron an 120572 lt 05 On the other hand if she acknowledges thehigher ability of the 119860-player she might agree with the 119860-player on an 120572 gt 05 The 119861-player might not be interested in120572 at all and just randomizes between the proposals of 119860 and119862 in order to prevent the default option Since we have noa priori information about the distribution of 119861rsquos preferenceswe conjecture that any feasible agreement 120572 isin [0 1] is equallylikely Hence on average we should observe that 120572 = 05the equal sharing distribution standardThese considerationslead to a testable version of the Protection Hypothesis

(H1) Both unanimity rule and simple majority voting witha symmetric prize scheme result in a moderate levelof redistribution (120572 = 05 equal sharing) Simplemajority voting with a right-skewed prize schemeleads to the lowest possible value of the distributionparameter namely 120572 = 0 the egalitarian solution

Recall that any agreement 120572 isin [0 1] is a Pareto-efficientNash equilibrium Nevertheless default might occur if sub-jects decide to veto an agreement in order to prevent ldquounfairrdquooutcomes Then they would sacrifice potential payoff inexchange for keeping the respective prize scheme intactObviously subjectsrsquo veto power is highest if the decision


mode is the unanimity rule Furthermore it is more difficultfor three subjects to coordinate on a joint distribution stan-dard than just for two as in the case where 120572 is determined bysimple majority vote These considerations lead to a testableversion of the Efficiency Hypothesis

(H2) Fixing the distribution standard by the unanimity ruleinvolves more group disagreement in terms of defaultthan simple majority voting and therefore is lessefficient

Finally we have to address the issue whether subjectschange their behavior in the contest when anticipating theresults of the demand game Large distribution parameters1 ge 120572 ge 05 definitely preserve the rank ordering determinedby the contest in terms of net payoff A risk neutral contestantwould therefore plan her effort independent from the surplussharing task Small distribution parameters 05 gt 120572 ge 0

would punish effort and possibly reverse the rank ordering interms of net payoff if120572 is sufficiently close to zero In that casethe marginal return on effort becomes negative and effort isexpected to diminish stronglyHencewe can complement theProtection Hypothesis and the Efficiency Hypothesis by thefollowing hypothesis

(H3) Subjects on average expend less effort in contests withright-skewed prize schemes if the decision mode inthe demand game is simple majority voting

4 Experimental Design

The experiment was held in the experimental economicslaboratory at the University of Bremen and the Universityof Oldenburg using z-Tree [62]8 In total 216 subjects par-ticipated in 12 sessions in the 90-minute long experimentsSubjects were recruited using the recruitment system ORSEE[63] About half of the subjects were female (456) 37of the participants were students of economics 18 werestudents of social sciences and the rest came from all otherfields Participants were on average in the sixth semester Atthe outset subjects were drawn from the subject pool andrandomly assigned to a single session No subject was allowedto take part more than once We used anonymous randommatching of the subjects assigned to a certain treatment intotheir respective groups Hence individual decisions shouldbe independent from one another

41 Treatment and Round Structure The experiment in-volved four treatments that differed in

(i) the way subjects were allocated their initial endow-ments 120587119863 (contest random)

(ii) the quorum required for fixing the distributionparameter 120572 (majority unanimity)

Our Baseline treatment is contest times majority Subjects firsttook part in a contest and then determined the distributionparameter for sharing the surplus by majority vote9 Inthe Veto treatment unanimity was required that is everycontestant was given the same veto power We additionally

conducted Baseline and Veto without a contest in the firststep Here initial endowments were randomly allocated tothe subjects (Control-B and Control-V treatments) Eachtreatment involved eight rounds Rounds differed as to thefirst three moments (mean variance and skewness) of theprize schemes In each round the prize schemewas presentedto the subjects before the knowledge test started

Given the structure of our experiment treatment effectsregarding the impact of the quorum and endowment allo-cation procedure are investigated at the between-subjectslevel Treatment effects regarding the impact of the shapeof the prize scheme are investigated at the within-subjectslevel This procedure was chosen on the one hand in ordernot to confuse subjects with different voting and allocationmechanisms and on the other hand to study the impactof the variation of the moments of the prize scheme on thesubjectsrsquo effort and distribution preferences Apart from thisargument the number of subjects needed for the experimentshrunk by a factor of eight

42 First Task Knowledge Test Subjects participated undertime pressure in a contest More specifically they were pre-sented ten questions differing in complexity which had beentaken from an intelligence test Questions were displayedin sequential order Participants had a time restriction of15 seconds per question Going back and forth betweenquestions was not possible10 For each right (wrong) answerthey obtained (lost) one point Questions that remainedunanswered after the time limit of 150 seconds yieldedzero points After having completed the test subjects wererandomly matched with two other subjects into groupsof three A subjectrsquos success in terms of points collecteddetermined his or her rank 119894 isin 119860 119861 119862 in the group Arandom number drawn from the unit interval was addedto the number of collected points in order to avoid tiesParticipants were informed about the procedure but were notinformed whether they were actually affected

In each round each rank was endowed with a prize intokens exchangeable for money at the end of the experimentThe respective amount was taken from the left of Table 1Note that the number stated in the first column of the tableis given for reference purposes only The prize schemes werepresented in randomized order to each group11 After theend of the quiz subjects learned their own ranks and theranks of the other group members and received their prizemoney according to the relevant prize scheme12 They did notreceive information about the absolute number of correctlysolved tasks of any groupmemberThe group assignment wasrenewed in every round (stranger design) in order to avoidsupergame effects13

An important feature of the set of prize schemes isthat symmetric prize schemes render the 119861-player ldquoneutralrdquobecause her final payoff is independent of 120572 To see this weplug 120587119861 = (13)Π into (4) which yields 119904120572119861 = (13)Π forall 120572 isin [0 1] Hence it is only her ldquoimpartialrdquo distributionattitude which decides whether or not to agree later on inthe surplus-sharing task with player 119860 or 119862 on a specific 120572and not self-interest As opposed to this right-skewed prize


Table 1 Set of prize schemes

NumberRank Parameters

119860 119861 119862Coefficient of

Mean Variation Skewness1 33 17 0 l h s2 67 33 0 h h s3 25 17 8 l l s4 50 33 17 h l s5 36 7 7 l h r6 72 14 14 h h r7 26 12 12 l l r8 52 24 24 h l rNote L low h high s symmetric and r right-skewed

schemes discriminate against the 119861-player by allocating thesame prize money to her and 119862 Since 119861 tried harder than 119862it is likely that her net payoff from the contest will fall below119862rsquos Regardless of that 119861rsquos expected utility is higher than 119862rsquosbecause she is more able and therefore she will expendmoreeffort

If subjects were assigned to the Control treatments theirranks 119894 isin 119860 119861 119862 were determined by a random numbergenerator instead of a contest Everything else in particularthe parametrization of the rounds stayed the same

43 Second Task Surplus Sharing The second task involveda phase of preplay communication Subjects were allowed tochat with their group mates about the distribution parameterto be fixed For this purpose we arranged a separate chatroom for each group A calculator for the gross payoffdistributionwas available during the whole taskThe chat wasavailable for threeminutes at maximum Afterwards subjectswere prompted to type in simultaneously and anonymouslytheir preferred distribution parameter The subjects wereinformed that if there was no agreement on a distributionparameter according to the necessary quorum they wouldonly receive their prizemoney In Baseline at least two groupmembers had to type in exactly the same distribution param-eter In Veto all group members had to type in exactly thesame distribution parameter in order to find an agreementNote that the parametrization of the experiment secured thatthe distribution of the surplus for a given prize scheme wasequivalent to redistributing half of total income (the sumof prize money and surplus) Each distribution standard 120572

corresponded to a specific level of redistribution 120591 = 1 minus 12057214For example the egalitarian distributional standard 120572 = 0

corresponded to full redistribution 120591 = 100 percent Wefavored the design of the experiment as a distributionalproblem over its presentation as a redistribution task becausethe chosen design highlights that in case of group defaultsubjects are left alonewith their individual outcomes from thecontest and nobody can benefit from belonging to a group orsociety

At the end of the experiment after the eighth round hadbeen played one round was randomly selected for payoff

The distribution parameter agreed upon by the group wasapplied for calculating the gross payoff If subjects failed toreach an agreement in the surplus sharing task they receivedonly their prizes Finally tokens were converted at a rateof 4 1 into Euros All payments were made in cash and inprivate The minimum and the maximum possible payoffswere zero and 36 Euros respectively On average subjectsearned C1201 (asympUS$ 1367) plus a show-up fee of C5 (asympUS$569) for about 90 minutes of work

5 Results

We present the results of our experiment in three stepsFirst we investigate individual and group effort then wecompare the default rates between different treatments andprize schemes and finally we report the results of testing thehypotheses regarding the outcome of the vote and relate themto effort choice Furthermore the outcome of the surplussharing task after the contest is compared with a setup wherethe contest was replaced by a random generator

51 Effort Baseline and Veto were conducted each with72 subjects randomly forming 72 times 83 = 192 groupsWe start off with the analysis of individual effort expendedin the contest The results of running separate fixed effects(FE) regressions for both treatments are presented in Table 2We regressed individual effort on the parameters of theprize schemes mean (0 = low 1 = high) variance(0 = low 1 = high) and skewness (0 = symmetric1 = right skewed) The expected signs of their coefficientsare given in the last column of the table The fit of theseregressions is very low which is not too surprising giventhe usual noise in experimental data and given the fact thatwe did not induce abilities Varying mean and variance didnot have a significant influence on individual effort15 Forskewness we obtain a rather interesting pattern In Baselinesubjects behaved as expected by significantly reducing effortwith right-skewed prize schemes As opposed to this inVetosubjects increased effort as compared to symmetric prizeschemes In the following we will therefore concentrate onskewness quorum and their interaction

Mean group effort across all treatments is 825 pointsFigure 2 shows how group effort varied between treat-ments and prize schemes each bar representing 96 groupobservations As indicated by the FE regression the meandifference between Veto and Baseline is negative andinsignificant (minus013 SE 051 119875 = 0399)16 Due to theopposing effect of skewness between treatments the meaneffort difference between symmetric and right-skewed prizeschemes is insignificant too (027 SE 051 119875 = 0300)In fact mean group effort like individual effort decreases by166 points (SE 068) in Baseline which is highly significant(119875 le 001) while it increases by 113 points (SE 077) inVetowhich is significant at the 10 level (119875 = 0065)

We conclude that hypothesis H3 is clearly supportedby our data both at the individual and at the group levelSubjects expended less effort in contests with right-skewed


Table 2 Individual effort by treatment

Coefficient SE 119875Expected

signBaseline

Mean minus0281 0231 0228 +Variance 0038 0209 0855 +Skewness minus0552 0194 0006 minus

Constant 3170 0200 00001198772 00132

119865 376 0015Veto

Mean minus0313 0208 0137 +

Variance minus0063 0181 0731 +

Skewness 0375 0222 0095 minus

Constant 2729 0180 00001198772 0008

119865 196 0128Note Fixed effects regression (119899 = 72 in both regressions) Endogenousvariable effort SE robust standard errors Subject and period dummies 119875significance level of a two-tailed 119905-test

915

0

2

4

6

8

Mea

n gr

oup

effor

t

10

Symmetric SkewedSymmetric SkewedBaseline Veto

749 763

875

Figure 2 Mean effortThe figure showsmean group effort in pointsby treatment and prize scheme 119899 = 96 for each bar

prize schemes when the surplus sharing task involved thesimple majority rule Additionally we observed the oppositeeffect for the Veto treatment though the effect was lesspronounced In order to analyze our subjectsrsquo effort reactionsto the anticipation of surplus distribution in the second partof the experiment more closely we compiled the respectiveinformation onmean effort separated by rank and treatmentin Table 6 in Appendix A First focusing on simple majorityvoting it becomes apparent that the decline of group effortwas mainly due to the less able subjects 119861 and 119862 whoexpended significantly less effort when the contest involveda right-skewed prize scheme (119861-player 119875 = 0004 119862-player 119875 = 0043) while the most able subject 119860 alsoshowed a negative but insignificant reaction (119875 = 0326)The unanimity rule induced all subjects to expend a bit more

0

003001

021

016

005

01

015

02

Def

ault

rate


Figure 3 Default rates The figure shows the default rate bytreatment and prize scheme 119899 = 96 for each bar

effort when the contest involved a right-skewed prize schemebut all differences are insignificant (119860-player 119875 = 0167 119861-player 119875 = 0216 119862-player 119875 = 0114) Hence we refrainfrom speculating about the reasons for this effect

52 Default Before moving on to analyzing the distributionparameter we have to filter out those groups that failed tocome to an agreement The overall default rate is 102Figure 3 shows the default rates by treatment and prizescheme As expected by the Efficiency Hypothesis the una-nimity requirement for 120572 made default more likely in Vetothan in Baseline (mean difference 161 SE 30 119875 le

001) At first sight default rates also seem to be lower forright-skewed prize schemes but the difference turned out tobe insignificant at usual significance levels (36 SE 31119875 = 0119) Applying the test separately to each treatmentdoes not change the result (Baseline 21 SE 21 119875 =

0157 Veto 52 SE 56119875 = 0176)We conclude thatH2is clearly supported by the data Note that groups that failed atfinding an agreement on120572 exhibited significantly lower groupeffort (0769 points SE 0276 119875 = 0003) Default was notassociated with higher within-group effort variation (meandifference minus70 SE 512 119875 = 0554)

53 Distribution After excluding groups that defaulted345 group observations remained 188 (9395) in Baseline(symmetricright-skewed) and 157 (7681) in Veto Themean of the distribution parameter across all groups is0366 that is the average distribution attitude is a mixturebetween equal sharing (pointA in Figure 1) and the differenceprinciple (point E) However the black bars in Figure 4 show120572s of around 05 (equal sharing) for all situations exceptfor Baseline (majority) combined with right-skewed prizeschemes There 120572 is very close to zero that is the egalitariandistribution standard Comparingmeans between treatments(Veto versus Baseline 0142 SE 0029 119875 le 001) andbetween prize schemes (symmetric versus skewed 0236 SE0027 119875 le 001) supports the Protection Hypothesis that


Symmetric SkewedSymmetric

029 029

042

050

003

010

039047

SkewedMajority

RandomContest

Unanimity

0

01

02

03

04

05

Mea

n of120572

Figure 4 Distribution parameter The figure shows the mean ofthe distribution parameter 120572 by treatment and prize scheme Blackbars represent Baseline and Veto (contest) 119899 = 93 95 76 and81 group observations White bars represent Control (randomendowment) 119899 = 48 48 47 and 41 group observations

unanimity requirement and symmetric prize schemes makestrong (re)distribution less likely The effect of skewness ismore extreme in Baseline but significant in both treatments(Baseline 0400 SE 0038119875 le 001 Veto 0043 SE 0029119875 = 0069)

We conclude that H1 is clearly supported by the dataAdditionally we take a glance at the final proposals for 120572submitted by the different players (the respective figures canbe taken from the right panel of Table 6 in Appendix A)As implied by the Protection Hypothesis it is mainly the 119861-player who is responsible for this result When moving fromsymmetric to right-skewed prize schemes her 120572 drops from051 ndash punctuating her neutrality ndash to 010 (119875 le 001) It isalso interesting to see that both the119860-player and the 119861-playermade concessions when moving to a right-skewed prizescheme the 119860-player ldquoofferedrdquo 120572 = 04 instead of 120572 = 059

(119875 le 001) whenmoving to a symmetric prize scheme the 119861-player ldquoofferedrdquo 120572 = 041 instead of 120572 = 01 (119875 le 001) In theVeto treatment we observe significant differences betweensymmetric and right-skewed prize schemeswith regard to thesubmitted 120572 too (119860-player 119875 = 0052 119861-player 119875 = 0026119862-player 119875 = 0032) Although all results are relatively closeto equal sharing (120572 = 04) subjects seem to have perceivedthe right skewed prize scheme less fair than the symmetricand therefore were willing to compensate the losers with aslightly larger share of the surplus Finally we would like tonote that the dispersion of submitted distribution parameterswas highest in Baseline with right-skewed prize schemespointing to huge friction within the respective groups

One could argue that the 120572-values represented by theblack bars do not relate to subjectsrsquo approval of individualeffort or ability in the contest but to their acceptance ofthe prize-schemes specified by the experimenter In order tocheck for that objection we set up the Control treatmentwhere subjects were randomly endowed with the same set ofldquoprizesrdquo as in Baseline and Veto 72 subjects participated

in Control The white bars in Figure 4 represent themean 120572-values of the groups that did not default Thereare two striking observations First the average height ofthe white bars is significantly lower than the black ones(0247 versus 0366 mean difference minus0119 SE 0025 119875 le

001) This means that randomly endowing subjects gave riseto a more egalitarian distribution attitude Groups strovetowards wiping out inequality caused by chance rather thaneffort Second the white bars seem to replicate the black-barpattern produced by the contest The situation with majorityvote and right-skewed initial endowment exhibits the lowestdistribution parameter of only 003

Table 3 gives the difference of the group mean of 120572between contest and random endowment by quorum andprize scheme All mean differences exhibit the expectedsign and are significantly greater than zero at least at the 5percent level Figure 4 and Table 3 indicate that the differencebetween contest and random endowment is largest for situ-ations involving symmetric endowmentsprize schemes andmajority vote This conjecture is supported by the data thedifference of the mean differences of 120572 between symmetrictimes majority and symmetric times unanimity is 0147 (SE 0069119875 = 0017) right-skewed times majority is 0087 (SE 0068119875 = 0100) and right-skewed times unanimity is 0139 (SE0064 119875 = 0015) All other pairwise difference tests areinsignificant In other words if at all individual effort interms of success in a contest is acknowledged by a groupmajority only if the prize scheme is symmetric and thedistribution of the surplus is determined by themajority voteDespite the general acceptance of the outcome of the contestsgroups on average agreed on a distribution parameter close to05 making equal sharing the dominant distribution motiveThis setup was also associated with highest group effort (915see Figure 2) The acceptance of the outcome of the contestshrunk dramaticallywith a right-skewed prize scheme In thissituation the majority agreed willingly on counteracting ex-ante inequality by imposing the difference principle Hencedefault rates shrunk but at the cost of lower effort Theunanimity requirement obviously shifted the focus away fromeffort towards cooperation in order to obtain the surplusIn that situation groups geared themselves to equal sharingcombined with a small correction for ex-ante inequality Therespective 120572-values fall below 05 and are almost independentfrom the shape of the prize schemeendowment distribution

6 Conclusion

In this paper we experimentally test if both the existenceof a middle class and institutional hurdles can protect themost able members of the society from the demand ofthe least able for excessive income distribution policies(Protection Hypothesis) We operationalized the institutionalrule by the quorum needed for a group decision (majorityversus unanimity) and the social structure by the existenceof a middle class in contrast to a winner-take-all societyMoreover we have argued that setting up institutional hurdlesis inferior to relying on a middle class as institutional


Table 3 Contest versus random endowment

Prize scheme QuorumMajority Unanimity0218 0071

Symmetric (0057) (0035)119875 le 001 119875 = 0022

0079 0131Right-skewed (0029) (0031)

119875 le 001 119875 le 001

Note First row mean difference of 120572 between contest and random Secondrow standard errors Third row significance level of a one-tailed 119905-test

hurdles involve efficiency losses due to bargaining impasses(Efficiency Hypothesis)

The experiment directly tests our main hypotheses Wefind that the Protection Hypothesis is clearly supported byour data The existence of a neutral middle class is aseffective as institutional hurdles to protect the rich frombeingexpropriated Hereby the middle class balances the interestsof the poor and the rich The middle class does not havean interest in excesses from one side because redistributionshifts money from the rich to the poor We argue that themiddle class can be interpreted as an uninvolved spectatorwho has no interest to favor one group over another TheEfficiency Hypothesis is also supported Institutional hurdlescome at the cost of significantly less efficiency in terms ofhigher default rates Hereby a higher default rate means thatthe payoff for both the rich and the poor is lower Even atthe individual level we could observe that subjects expendmore effort if the prize scheme involved amoderatingmiddleplayer The corroboration of both hypotheses allows us todraw the conclusion that the middle class is the efficientprotection of the rich

We contribute to the literature about why the rich donot get expropriated Leaving aside the branch of the lit-erature which summarizes the reasons of why the poor donot expropriate the rich (eg beliefs identity and socialpreferences of the poor) we study a constellation inwhich therich can actively protect themselves from being expropriatedAlthough any extrapolation of our laboratory results has tobe done carefully we note that our finding that collectivedecisions have been less equalizing and more efficient underthe majority rule with a neutral middle player than in groupswith a right-skewed prize structure resonates well with theempirical data supporting the warnings about the impact ofrising inequality on societal stability and economic growthin contemporary societies Coming back to the empiricalfindings in the introduction not only the middle class itselfbut also the rich should have an interest in bracing against theerosion of the middle class

Appendices

A Further Tables

See Tables 4 5 and 6

B Instructions

B1 Welcome to the Experiment and Thank You for Partic-ipating Please do not talk to other participants during theentire experiment During the experiment you have to makeseveral decisions Your individual payoff depends on yourown decision and the decisions of your group members dueto the following rules You will be paid individually privatelyand in cash after the experiment During the experiment wewill talk about Tokens as the experimental currency After theexperiment Tokens will be transferred into Euros with thefollowing exchange rate

10 Tokens = 250 Euros (B1)

Please take your time reading the instructions and makingyour decisions You are not able to influence the duration ofthe experiment by rushing through your decisions becauseyou always have to wait until the remaining participantshave reached their decisions The experiment is completelyanonymous At no time during the experiment or afterwardswill the other participants knowwhich role youwere assignedto and how much you have earned

If you have any questions please raise your hand Oneof the experimenters will come to you and answer yourquestions privately Following this rule is very importantOtherwise the results of this experiment will be worthless forscientific purposes You will receive a show-up fee of 5 Eurosfor your participation Depending on your decisions and thedecisions of the other participants you can additionally earnup to 144 Tokens (36 Euros) The expected duration of theexperiment is 90minutesThe exact course of the experimentwill be described in the following

B2 Detailed Information The experiment consists of 8rounds which all follow the same course In each roundparticipants will be randomly and repeatedly assigned togroups of three members Your payoff will be determinedonly by your own decisions and decisions of the other groupmembers Decisions of the other groups do not affect yourpayment No participant receives information about othersubjectsrsquo final choices although group members have thepossibility to communicate via chat to find an agreementThefinal decisions are anonymous within each group

After the group assignment a knowledge test follows(in the control treatments ranks were assigned by a randomgenerator instead of the knowledge test) The more questionsyou answer correctly in the knowledge test the higher yourpotential payoff at the end of the experiment will be Theknowledge test consists of various tasks For each correctlysolved task you receive one point each question you answeredincorrectly a point is subtracted and for not answeringquestions you do not get any points Based on the collectedpoints you will be assigned a rank within the group Theplayer with the highest score in the knowledge test gets thehighest rank (rank 1) and also the highest token endowmentThe player with the lowest number of collected points will beassigned the lowest rank (rank 3) and also the lowest tokenendowment


Table 4 Predictions for the contest

Prizescheme

Player Ability Expected winningprobabilities ()

Expectedreturn Effort Rank Prize Cost Net

payoff119894 119888119894 1st 2nd 3rd 119864(120587119894)

119909119894 119895 120587119895

1 04 562 375 62 10 38 A 33 15 181 2 06 250 500 250 4 21 B 17 13 4

3 08 62 375 562 1 9 C 0 8 minus81 04 562 375 62 19 77 A 67 31 36

2 2 06 250 500 250 8 42 B 33 25 83 08 62 375 562 2 18 C 0 15 minus151 04 562 375 62 13 20 A 25 8 17

3 2 06 250 500 250 10 11 B 17 7 103 08 62 375 562 8 5 C 8 4 41 04 562 375 62 27 38 A 50 15 35

4 2 06 250 500 250 21 21 B 33 12 213 08 62 375 562 18 9 C 17 7 101 04 562 375 62 12 29 A 36 11 25

5 2 06 250 500 250 8 10 B 7 6 13 08 62 375 562 7 2 C 7 2 51 04 562 375 62 24 57 A 72 23 49

6 2 06 250 500 250 16 20 B 14 12 23 08 62 375 562 14 4 C 14 3 111 04 562 375 62 14 14 A 26 6 20

7 2 06 250 500 250 13 5 B 12 3 93 08 62 375 562 12 1 C 12 1 111 04 562 375 62 29 28 A 52 11 41

8 2 06 250 500 250 25 10 B 24 6 183 08 62 375 562 24 2 C 24 2 22

Note Predictions for a 3-player contest with linear cost functions and maximum ability119898 = 02

Table 5 Predicted effort by prize scheme

Prizescheme Mean Coefficient

of variation Skewness EffortMean Variation

1 0 1 0 229 62 1 1 0 459 63 0 0 0 119 64 1 0 0 225 65 0 1 1 136 106 1 1 1 272 107 0 0 1 66 108 1 0 1 131 10Note 0 = lowsymmetric 1 = highright-skewed Effort predictions accordingto Table 4

The knowledge test is designed with a time restrictionof 150 seconds for 10 screens with different tasks The taskscome from different fields Each question has only onecorrect answer You have 15 seconds per question Please payattention to the time restriction in the upper right corner

Table 6 Effort and distribution proposal by treatment

Prize schemeSubjectrsquos rank

Effort 120572

119860 119861 119862 119860 119861 119862

Simple majority voting

Symmetric 439 314 163 059 051 041(022) (020) (032) (003) (003) (003)

Right-skewed 426 238 085 040 010 010(017) (020) (030) (004) (002) (002)

Unanimity rule

Symmetric 392 255 116 048 049 047(021) (021) (031) (002) (002) (002)

Right-skewed 424 280 172 043 041 043(026) (022) (035) (002) (003) (002)

Note The figures give the subjectsrsquo mean scores in the intelligence test (perround) and the mean of the distribution parameter 120572 typed in after the chatphase Standard errors are in parentheses

of the screen By clicking on the OK button you can get tothe next screen After a total of 150 seconds the knowledge


test stops and the collected points are summed up The ranksand the associated endowments will be provided after theknowledge test explanations and further instructions

B3 After the Knowledge Test You have collected points inthe knowledge test According to the collected points of allgroup members each group member will be assigned a rankand a certain endowment of tokens The ranking reflectsdirectly the performance in the knowledge test of the grouprsquosmembers (in the control treatment the ranks and endowmentwere assigned randomly) The token endowment varies fromround to round A total of 8 rounds will be played Thesequential arrangement of token distributions will be chosenrandomly

B4 Decision about Payoff Distribution by Choosing a Dis-tribution Parameter In the experiment you will see on thenext screen your own rank and token endowment and theranks and token endowments of your group members In thefollowing you and your groupmembers have to determine thedistribution parameter You should set a distribution param-eter which distributes tokens between the group members insuch a way that your preferences are met The payoff of thegroup member 119883 = 1 2 3 at the end of the experiment isthen calculated as follows

Payoff 119883 = Tokens 119883 sdot (1 minus 119905)

+

(Tokens 1 + Tokens 2 + Tokens 3) sdot 1199053

(B2)

Distribution only takes place when at least 2 (3 underunanimity rule) groupmembers choose the same distributionparameterThe chosen distribution parameter determines thefinal distribution within the group in that period Samplecalculation player 1 has 3 tokens player 2 has 5 tokensand player 3 has 10 tokens Given that order players chosethe following distribution parameters 81 (50 underunanimity rule) 50 and 50 The distribution parameterwhich determines the level of redistribution is thus 50Thepayoff for player 1 is

Payoff 1 = 3 sdot (1 minus 05) +

(3 + 5 + 10) sdot 05

3

= 15 + 3 = 45

(B3)

The computer calculates the payoffs under the assumptionthat the distribution parameter you make is the relevantdistribution parameter which will be implemented

If not at least 2 (3 under unanimity rule) players fromthe group typed in the same distribution parameter nodistribution takes place and players receive only their tokenendowment determined by their rank in the knowledge testIn the example above this would mean that if the playerschose distribution parameters of 96 51 and 7 andtherefore no agreement is reached player 1 receives only 15tokens The same would apply to the endowments of theremaining players

B5 Possibility to Communicate (Open or Restricted) Youhave the possibility to chat with the other group members ina joint chat room You can chat freely (restricted in the waythat you can only type in numbers between 0 and 100 underrestricted communication rule) and discuss which distributionparameter is to be implemented but you are not allowed toreveal your identity

Chat time is restricted to 3 minutes If not at least 2(3 under unanimity rule) players have entered the samedistribution parameters in time and have confirmed theirchosen distribution parameters by clicking the OK buttonthen no distribution will take place

The calculator is available to you for trying out differentdistribution parameters To get an impression about the chatstructure the endowments and the tokens and the calculatora sample screen is given below In the upper part of thescreen you see the group chat room The rank number ofeach player is displayed directly above and within the chatwindow In the lower part of the screen the ranks andtokens of all group members before and after distributionare displayed In the right field you can enter as often asyou wish different distribution parameters to calculate thefinal token distribution In this example the final distributionis calculated as follows the distribution parameter is 45Hence 45 of 6 = 27 45 of 10 = 45 and 45 of 4 = 18 aresubtracted from the playersrsquo accounts resulting in a sum of 9TokensDividing the sumby 3 gives 3Tokens to be distributedto each groupmemberThe final distribution then is given by6minus27+3 = 63 10minus45+3 = 85 and 4minus18+3 = 52TokensFor example using the equation given above the payoff ofplayer 3 would be given by

Payoff = 4 sdot (1 minus 045) +

(4 + 6 + 10) sdot 045

3

= 22 + 3 = 52

(B4)

Pressing the OK button confirms and completes the decision

B6 Calculation of the Payoffs One of the 8 rounds israndomly selected for payoff Each round could thereforebe payoff relevant Individual payoffs will be calculatedaccording to the chosen distribution parameter and the rulesstated above The experiment will begin shortly If you havequestions please raise your hand and wait quietly until anexperimenter comes to you Speaking with other participantsis strictly prohibited throughout the experiment Thank youand have fun in the experiment



Acknowledgments

The authors thank two anonymous referees and the editorsin particular Klarita Gerxhani for their comments andsupportThis research was partly financed by the NOWETASFoundation under the project title ldquoDecision making among


collectives and the individual the interdisciplinary analysisof perceptions of justice in collective decision processes andtheir feedback to the individual levelrdquo and by the GermanResearch Foundation (DFG) research group FOR 2104ldquoNeeds-based justice and distribution proceduresrdquo (TraubTR 4586-1)They would like to thank Nicola Maaser and theparticipants of several research seminars and the workshopldquoPositive and Normative Aspects of Distributive JusticeAn Interdisciplinary Perspectiverdquo held at the University ofBremen in particular for their most helpful comments andsuggestions

Endnotes

1 A similar effect of voting was reported by Frohlich andOppenheimer [64]

2 For more general information on contest theory werefer to the surveys by Corchon [65] and Konrad 42]Experimental research on contestswas recently surveyedby Dechenaux et al [66] Multiple-prize contests haveintensively been investigated with respect to the opti-mality of the prize scheme in terms of maximum effortgeneration (see eg [67])

3 In general the cost of effort include all monetary andnonmonetary costs such as cognitive effort and time Inour experiment the only costs of solving questions in theintelligence test were cognitive Time consumption wasequal for all subjects and covered by a show-up fee

4 The numbers listed in both tables are only point esti-mates for a single distribution of ability One couldextend the simulation by randomly drawing 119888 from 119865(119888)

a sufficiently large number of times in order to obtainconfidence intervals for mean effort However since thestylized facts would stay unchanged we save the effort

5 Note that the restriction 119904119894 ge 0 implies that thesubjectsrsquo rank ordering according to their prizes cannotbe reversed by the surplus sharing task This parallelsthe condition stated by Meltzer and Richard [23] thatthe median voterrsquos tax rate is utility maximizing onlyif it does not exceed one and therefore preserves therank ordering between productive and less productiveindividuals

6 A loosely related study by Selten and Ockenfels [68]found a relative majority of subjects to comply with afairness norm they called ldquofixed total sacrificerdquo Subjectswere willing to donate a fixed amount of money to thelosers of a solidarity game irrespective of the numberof losers In a game studied by Okada and Riedl [69]proposers could form different coalitions of up to threeplayers and then distribute the coalition surplus amongthe coalitionmembers Inmost cases the proposer chosean inefficient two-person coalition and the responderreciprocated the choice of the proposer by not rejectingthe unfair proposal that excluded a third of the popula-tion from payoff

7 Goeree and Yariv [70] found that deliberation blursdifferences between institutions (eg in terms of vot-ing rules) and uniformly improves efficiency of groupoutcomes We borrow from the work of these authorsby considering two different voting rules under whichthe surplus-sharing game is conducted namely simplemajority voting andunanimity rule In so far our study isalso related to the literature on deliberation in collectivedecision making (see [70ndash72]) For an overview of theexperimental literature on deliberation see Karpowitzand Mendelberg [73]

8 For a transcript of the instructions see Appendix B

9 We have also varied the chat possibilities in such a waythat in half of the sessions participants could exchangefull text messages and in the remaining sessions partici-pants could only exchange numbers Due to the fact thatwe do not find any differences between these sessionswe decided to pool these sessions

10 The questions compassed mathematical linguisticaland combinatorial tasks

11 Note that the stranger matching protocol requires keep-ing the randomization of the ordering of the prizeschemes constant in each session Since we conductedfour sessions per treatment we used four predefinedrandomizations that stayed the same for all treatmentsin order to control for eventual path-dependencies

12 Prize schemes 5 and 6 slightly violate the nonnegativityconstraint for 119904120572119894 for large 120572-values This is due to theconstruction of the set of prize schemes where half ofthe eight different prize schemes exhibit the same meanvariation coefficient and skewness respectively

13 Note that the theoretical model of the contest outlinedin Section 3 has to assume that subjects know their ownabilities and the distribution of abilities in order to derivethe result that those who are more able also expendmore effort We did not collect subjectsrsquo expectationsabout their abilities and winning probabilities in theexperiment but it could be interesting to check whethertheir beliefs were correct and to compare their goodnessof fit with the outcome of the contest

14 As can be taken from the instructions we let subjectsenter 120591 instead of 120572 which turned out to be moreintelligible in pilot experiments

15 We also expected right-skewed prize schemes to increasewithin-group effort variation among contestants Ana-lyzing effort variation again speaks a slightly differentlanguage Indeed it seems to increase inBaselineHow-ever the mean difference turns out to be insignificant(034 SE 053 119875 = 0259) In Veto the mean differencehas the wrong sign but it is significant (065 SE 028119875 = 0012)

16 Significance level of a one-tailed 119905-test Tests regardingdirectional hypothesis are one-tailed unless otherwisespecified SE standard error


References

[1] T Piketty Capital in the Twenty-First Century Belknap PressCambridge Mass USA 2014

[2] OECD Growing Unequal Income Distribution and Poverty inOECD Countries OECD Paris France 2008

[3] S K Singh and G S Maddala ldquoA function for size distributionof incomesrdquo Econometrica vol 44 no 5 pp 963ndash970 1976

[4] C Seidl K Pogorelskiy and S Traub Tax Progression in OECDCountries An Integrative Analysis of Tax Schedules and IncomeDistributions Springer Berlin Germany 2013

[5] R H Frank and P Cook The Winner-Take-All Society FreePress New York NY USA 1995

[6] R H Frank How Rising Inequality Harms the Middle ClassUniversity of California Press Berkely Calif USA 2007

[7] J S Hacker and P Pierson Winner-Take-All Politics HowWashington Made the Rich Richermdashand Turned Its Back on theMiddle Class Simon amp Schuster New York NY USA 2010

[8] S Rosen ldquoThe economics of superstarsrdquo The American Eco-nomic Review vol 71 no 5 pp 845ndash858 1981

[9] T A DiPrete and G M Eirich ldquoCumulative advantage as amechanism for inequality a review of theoretical and empiricaldevelopmentsrdquoAnnual Review of Sociology vol 32 pp 271ndash2972006

[10] E Franck and S Nuesch ldquoTalent andor popularity what doesit take to be a superstarrdquo Economic Inquiry vol 50 no 1 pp202ndash216 2012

[11] J E StiglitzThe Price of Inequality How Todayrsquos Divided SocietyEndangers Our Future W W Norton amp Company New YorkNY USA 2012

[12] F Fukuyama The End of History and the Last Man Free PressNew York NY USA 1992

[13] A Alesina and G-M Angeletos ldquoFairness and redistributionrdquoAmerican Economic Review vol 95 no 4 pp 960ndash980 2005

[14] J M Buchanan and G Tullock The Calculus of ConsentLogical Foundations of Constitutional Democracy University ofMichigan Press Ann Arbor Mich USA 1962

[15] L Miller and C Vanberg ldquoDecision costs in legislative bargain-ing an experimental analysisrdquo Public Choice vol 155 no 3-4pp 373ndash394 2013

[16] F W Scharpf ldquoA game-theoretical interpretation of inflationand unemployment inWestern Europerdquo Journal of Public Policyvol 7 no 3 pp 227ndash257 1987

[17] W Easterly ldquoThe middle class consensus and economic devel-opmentrdquo Journal of Economic Growth vol 6 no 4 pp 317ndash3352001

[18] E Glaeser J Scheinkman and A Shleifer ldquoThe injustice ofinequalityrdquo Journal of Monetary Economics vol 50 no 1 pp199ndash222 2003

[19] W Arts and J Gelissen ldquoWelfare states solidarity and justiceprinciples does the type really matterrdquo Acta Sociologica vol44 no 4 pp 283ndash299 2001

[20] L drsquoAnjou A Steijn and D van Aarsen ldquoSocial positionideology and distributive justicerdquo Social Justice Research vol 8no 4 pp 351ndash384 1995

[21] MM Jaeligger ldquoWhat makes people support public responsibilityfor welfare provision self-interest or political ideology Alongitudinal approachrdquo Acta Sociologica vol 49 no 3 pp 321ndash338 2006

[22] B Moldovanu and A Sela ldquoThe optimal allocation of prizes incontestsrdquoAmerican Economic Review vol 91 no 3 pp 542ndash5582001

[23] A H Meltzer and S F Richard ldquoA rational theory of the sizeof governmentrdquo Journal of Political Economy vol 89 no 5 pp914ndash927 1981

[24] R Perotti ldquoGrowth income distribution and democracy whatthe data sayrdquo Journal of Economic Growth vol 1 no 2 pp 149ndash187 1996

[25] A Alesina and R Perotti ldquoIncome distribution political insta-bility and investmentrdquo European Economic Review vol 40 no6 pp 1203ndash1228 1996

[26] R Benabou and E A Ok ldquoSocial mobility and the demandfor redistribution the POUM hypothesisrdquo Quarterly Journal ofEconomics vol 116 no 2 pp 447ndash487 2001

[27] J E Roemer ldquoWhy the poor do not expropriate the rich an oldargument in new garbrdquo Journal of Public Economics vol 70 no3 pp 399ndash424 1998

[28] J E Roemer and W Lee ldquoRacism and redistribution in theUnited States a solution to the problem of American exception-alismrdquo Journal of Public Economics vol 90 no 6-7 pp 1027ndash1052 2006

[29] C Fong ldquoSocial preferences self-interest and the demand forredistributionrdquo Journal of Public Economics vol 82 no 2 pp225ndash246 2001

[30] N Frohlich and J A Oppenheimer Choosing Justice An Exper-imental Approach to Ethical Theory University of CaliforniaPress Ewing NJ USA 1993

[31] D Miller Principles of Social Justice Harvard University PressCambridge Mass USA 1999

[32] J Konow ldquoWhich is the fairest one of all A positive analysis ofjustice theoriesrdquo Journal of Economic Literature vol 41 no 4pp 1188ndash1239 2003

[33] S Traub C Seidl U Schmidt and M V Levati ldquoFriedmanHarsanyi Rawls Bouldingmdashor somebody else An experi-mental investigation of distributive justicerdquo Social Choice andWelfare vol 24 no 2 pp 283ndash309 2005

[34] A Sen The Idea of Justice Belknap Press Cambridge MassUSA 2009

[35] W Gaertner and E Schokkaert Empirical Social ChoiceQuestionnaire-Experimental Studies on Distributive JusticeCambridge University Press Cambridge UK 2012

[36] E F Klor and M Shayo ldquoSocial identity and preferences overredistributionrdquo Journal of Public Economics vol 94 no 3-4 pp269ndash278 2010

[37] J-R Tyran and R Sausgruber ldquoA little fairness may induce alot of redistribution in democracyrdquo European Economic Reviewvol 50 no 2 pp 469ndash485 2006


[39] W Hochtl R Sausgruber and J-R Tyran ldquoInequality aversionand voting on redistributionrdquo European Economic Review vol56 no 7 pp 1406ndash1421 2012

[40] G E Bolton and A Ockenfels ldquoInequality aversion efficiencyand maximin preferences in simple distribution experimentscommentrdquo American Economic Review vol 96 no 5 pp 1906ndash1911 2006

[41] K D Messer G L Poe D Rondeau W D Schulze and CVossler ldquoExploring voting anomalies using a demand revealing


random price voting mechanismrdquo Journal of Public Economicsvol 94 pp 308ndash317 2010

[42] F Paetzel R Sausgruber and S Traub ldquoSocial preferences andvoting on reform an experimental studyrdquo European EconomicReview vol 70 pp 36ndash55 2014

[43] L Balafoutas M G Kocher L Putterman and M SutterldquoEquality equity and incentives an experimentrdquo EuropeanEconomic Review vol 60 pp 32ndash51 2013

[44] J Esarey T C Salmon and C Barrilleaux ldquoWhat motivatespolitical preferences Self-interest ideology and fairness in alaboratory democracyrdquo Economic Inquiry vol 50 no 3 pp604ndash624 2012

[45] R Durante L Putterman and J van der Weele ldquoPreferencesfor redistribution and perception of fairness an experimentalstudyrdquo Journal of the European Economic Association vol 12 no4 pp 1059ndash1086 2014

[46] R M Sheremeta W A Masters and T N Cason ldquoWinner-take-all and proportional-prize contests theory and experi-mental resultsrdquo Working Paper 12-04 Economic Science Insti-tute Chapman University 2012


[48] V Crawford ldquoA survey of experiments on communication viacheap talkrdquo Journal of Economic Theory vol 78 no 2 pp 286ndash298 1998

[49] R Croson T Boles and J K Murnighan ldquoCheap talk inbargaining experiments lying and threats in ultimatum gamesrdquoJournal of Economic Behavior and Organization vol 51 no 2pp 143ndash159 2003

[50] A Roth ldquoToward a focal-point theory of bargainingrdquo in Game-Theoretic Models of Bargaining A Roth Ed pp 259ndash268Cambridge University Press Cambridge UK 1985

[51] A Roth ldquoBargaining phenomena and bargaining theoryrdquo inLaboratory Experimentation in Economics Six Points of View ARoth Ed pp 14ndash41 Cambridge University Press CambridgeUK 1987

[52] R Forsythe J Kennan and B Sopher ldquoAn experimentalanalysis of strikes in bargaining games with one-sided privateinformationrdquoThe American Economic Review vol 81 no 1 pp253ndash278 1991

[53] M Shaked and G J Shanthikumar Stochastic Orders and TheirApplications Academic Press San Diego Calif USA 1994

[54] J Nash ldquoTwo-person cooperative gamesrdquo Econometrica vol 21pp 128ndash140 1953

[55] A E Roth ldquoBargaining experimentrdquo in The Handbook ofExperimental Economics J H Kagel and A E Roth EdsPrinceton University Press Princeton NJ USA 1995


[57] D P Baron and J A Ferejohn ldquoBargaining in legislaturesrdquoTheAmerican Political Science Review vol 83 no 4 pp 1181ndash12061989

[58] H Moulin ldquoEqual or proportional division of a surplus andother methodsrdquo International Journal of Game Theory vol 16no 3 pp 161ndash186 1987

[59] L J Mirman and Y Tauman ldquoDemand compatible equitablecost sharing pricesrdquoMathematics of Operations Research vol 7no 1 pp 40ndash56 1982

[60] J Rawls A Theory of Justice Harvard University Press Cam-bridge Mass USA 1971

[61] N Anbarci and I Boyd ldquoNash demand game and the Kalai-Smorodinsky solutionrdquo Games and Economic Behavior vol 71no 1 pp 14ndash22 2011


[63] BGreiner ldquoTheonline recruitment systemORSEE 20mdasha guidefor the organization of experiments in economicsrdquo Tech RepUniversity of Cologne Cologne Germany 2004

[64] N Frohlich and J A Oppenheimer ldquoChoosing justice inexperimental democracies with productionrdquo The AmericanPolitical Science Review vol 84 no 2 pp 461ndash477 1990

[65] L C Corchon ldquoThe theory of contests a surveyrdquo Review ofEconomic Design vol 11 no 2 pp 69ndash100 2007

[66] E Dechenaux D Kovenock and R M Sheremeta ldquoA survey ofexperimental research on contests all-pay auctions and tourna-mentsrdquo Discussion Paper No SP II 2012-109Wissenschaftszen-trum fur Sozialforschung Berlin Germany 2012

[67] C Harbring and B Irlenbusch ldquoAn experimental study ontournament designrdquo Labour Economics vol 10 no 4 pp 443ndash464 2003

[68] R Selten and A Ockenfels ldquoAn experimental solidarity gamerdquoJournal of Economic Behavior and Organization vol 34 no 4pp 517ndash539 1998

[69] A Okada and A Riedl ldquoInefficiency and social exclusion ina coalition formation game experiment evidencerdquo Games andEconomic Behavior vol 50 no 2 pp 278ndash311 2005

[70] J K Goeree and L Yariv ldquoAn experimental study of collectivedeliberationrdquo Econometrica vol 79 no 3 pp 893ndash921 2011

[71] D Austen-Smith and T J Feddersen ldquoDeliberation prefer-ence uncertainty and voting rulesrdquo American Political ScienceReview vol 100 no 2 pp 209ndash217 2006

[72] D Gerardi and L Yariv ldquoDeliberative votingrdquo Journal ofEconomic Theory vol 134 no 1 pp 317ndash338 2007

[73] C F Karpowitz and TMendelberg ldquoAn experimental approachto citizen deliberationrdquo in Cambridge Handbook of Experimen-tal Political Science J N Druckman D P Green J H Kuklinskiand A Lupia Eds pp 258ndash272 Cambridge University PressCambridge Mass USA 2011









Editorial Board













Contents





















Acknowledgment



Jens Groszliger

References














1 Introduction















Player 2119889 ℎ

Player 1 119889 1 1 119909 119910

ℎ 119910 119909 025 025












(119904lowast

1 119904lowast

2) (119880

lowast

1 119880lowast

2) Pareto optimal

C1 (119889 119889) (1 1) Yes

C2 (119889 119889) (1 1) Yes(ℎ ℎ) (025 025) No

C3 (119889 ℎ) (119909 119910) Yes(ℎ 119889) (119910 119909) Yes

C4 (ℎ ℎ) (025 025) No










(a) C1 119910 minus 119909 = 0

2119889 ℎ

1 119889 1 1 05 05ℎ 05 05 025 025

(b) C2 119910 minus 119909 = 05

2119889 ℎ

1 119889 1 1 0 05ℎ 05 0 025 025

(c) C3 119910 minus 119909 = 1

2119889 ℎ

1 119889 1 1 05 15ℎ 15 05 025 025

(d) C4 119910 minus 119909 = 15

2119889 ℎ

1 119889 1 1 0 15ℎ 15 0 025 025







d h

d h d h

1

1

x

y

y

x

025

025

P1

P2P2

u1

u2







2= 11990421198891199042ℎrepresents







119894(1199041 1199042)

is defined by 119906119894(119889 11990421198891199042ℎ) = 119906

119894(119889 1199042119889) and 119906

119894(ℎ 11990421198891199042ℎ) =




2) is a SPNE


2) ge 1199062(1199041 1199042) for every 119904


1isin 1198781

and 1199061(119904lowast

1 119904lowast

2) ge 1199061(1199041 119904lowast

2) for every 119904

1isin 1198781


2119889119904lowast






2119889119904lowast




(119904lowast

1 119904lowast

2) (119906

lowast

1 119906lowast

2) Pareto optimal




2) is a SPNE in


1 119904lowast

2) = 119909

and 1199062(119904lowast

1 119904lowast

2) = 119910









1119898 119898lowast

2) ge 119906

2(119904lowast

1 119904lowast

2) where

1199061(119904lowast

1119898 119898lowast

2) ge 119906

1(1199041 119898lowast

2) for every 119904


1119898is






1 119906lowast

2) Threat (119906

1 1199062) Promise (119906

1 1199062)




2= 119904lowast

2


only if 1199062(119889119898lowast

2) = 1199062(119889 119904lowast

2) and 119906

2(ℎ119898lowast

2) lt 1199062(ℎ 119904lowast

2)


only if 1199062(119889119898lowast

2) lt 1199062(119889 119904lowast

2)



119894(119904119894 119889) gt





1= 119889




2such that 119906

2(119904lowast

1119898 119898lowast

2) gt 119906

2(119904lowast

1 119904lowast

2) if and

only if 119910 gt 1
















1isin 119889 ℎ





1205792isinΘ2

119901(1205792) = 1 In the second





2= (119898119889119889 119898119889ℎ 119898ℎ119889 119898ℎℎ) is a


2isin






1205792

= 11990421198891199042ℎ


2 and119898

2= (1198981205792

119898minus1205792


2




2isin 1205792 where 119904

1119889119889is




1199041(minus1198981205792) 1198981205792

119898minus1205792

) = 119906119894(1199041 1199042) for 119904

11198981205792

= 1199041and

1205792= 1199042


1205792|1198982




1205792isinΘ2

1205721205792|1198982



2= 119889119889 In the






2= (119898

lowast

119889119889 119898lowast

119889ℎ 119898lowast

ℎ119889 119898lowast

ℎℎ) a


1119889119889 119904lowast

1119889ℎ 119904lowast

1ℎ119889 119904lowast

1ℎℎ) and a





1198981205792isinΘ2

1199061205792

(119904lowast

1119898 1198981205792

119898minus1205792

)

(2) 119904lowast1119898


1isin1198781

sum1205792isinΘ2

1205721205792|1198982

sdot 1199061(11990411198981205792

1199041(minus1198981205792)

119898lowast

2)

(3) 120572lowast1205792|1198982


2 For all 120579

2who play the


1205792|119898lowast

2

=

1199011205792


2


1205792




1205792|1198982



u1

u2

d

h

d

h

d

h

d

h

x

y

x

y x

y

x

y

N

1 1

1

1

11

1

1

1

1

1 1

x y

x y

025 025

025

025

025

025

025 025

120572hh120572dh

120572dh

120572dh

120572dh

120572hd

120572hd

120572hd

120572hd

120572dd

120572dd

120572dd

120572dd

120579hh

120579dh

120579hd

120579dd

mhh

mdh

mhd

mdd

d

h

d

h

d

h

d

h

d h d h d h d h

d hd hd hd h

120572hh

120572hh120572hh

u1 u2

y x

y x

1 1

1 1

x y

x y

025 025

025 025

u1 u2

y x

y x

u1

u2 x

y

x

y x

y

x

y

1

1

1

1

025

025

025

025



2



2= 1205792 then 120572

1205792|2



1205792|1198982

= 1





2 then 119904lowast

11198981205792

= 119889


1205792


1205792






119889119889 119898lowast

119889ℎ 119898lowast

ℎ119889 119898lowast

ℎℎ) (119904

lowast

1119889119889 119904lowast

1119889ℎ 119904lowast

1ℎ119889 119904lowast

1ℎℎ) (120572

lowast

119889119889|119889119889 120572lowast

119889ℎ|119889ℎ 120572lowast

ℎ119889|ℎ119889 120572lowast

ℎℎ|ℎℎ)

C1 (119889119889 119889ℎ 119889119889 ℎℎ) (119889 119889 ℎ 119889) (119901119889119889

(119901119889119889+ 119901ℎ119889) 1 1 1)

C2 (119889119889 119889ℎ 119889119889 119889ℎ) (119889 119889 ℎ ℎ) (119901119889119889

(119901119889119889+ 119901ℎ119889)119901119889ℎ

(119901119889ℎ+ 119901ℎℎ) 1 1)

C3 (119889ℎ 119889ℎ ℎℎ ℎℎ) (ℎ 119889 ℎ 119889) (1119901119889ℎ

(119901119889119889+ 119901119889ℎ) 1119901ℎℎ

(119901ℎ119889+ 119901ℎℎ))

C4 (119889ℎ 119889ℎ 119889ℎ 119889ℎ) (ℎ 119889 ℎ ℎ) (1119901119889ℎ

(119901119889119889+ 119901119889ℎ+ 119901ℎ119889+ 119901ℎℎ) 1 1)









1205792|lowast

2(1205792)= 1 lowast

1205792


1205792

(119904lowast

1119898 lowast

1205792

lowast

minus1205792

) ge 1199061205792

(119904lowast

1119898

1198981205792

lowast

minus1205792

) for every1198981205792

isin Θ2




lowast

1205792



1205792




= 2is a








C1 120572lowast

119889119889|119889119889ge

120572lowast

ℎ119889|119889119889(119910 minus 119909)


C2 120572lowast

119889ℎ|119889ℎge

120572lowast

ℎℎ|119889ℎ(025 minus 119909)

075120572lowast

119889119889|119889119889ge

120572lowast

ℎ119889|119889119889(119910 minus 119909)

(1 minus 119910)

C3 Lie 120572lowast

ℎℎ|ℎℎge

120572lowast

ℎ119889|ℎℎ(119910 minus 119909)

(119909 minus 025)120572lowast

119889ℎ|119889ℎge

120572lowast

119889119889|119889ℎ(119910 minus 1)

075

C4 Lie 120572lowast

119889ℎ|119889ℎge

120572lowast

119889119889|119889ℎ(119910 minus 1) + 120572

lowast

ℎ119889|119889ℎ(119910 minus 119909) + 120572

lowast

ℎℎ|119889ℎ(025 minus 119909)

075






only if (1minus119910)120572119889119889|119898(120579

2)+(075)120572

119889ℎ|119898(1205792)+(119909minus119910)120572

ℎ119889|119898(1205792)+(119909minus

025)120572ℎℎ|119898(120579

2)ge 0





5 Conclusions










Appendix


1 119904lowast

2) = 119909 and

1199062(119904lowast

1 119904lowast

2) = 119910 then 119910 ge 1 If 119904lowast

2= ℎ119889 then 119906

1(119889 ℎ119889) ge


2= ℎℎ

then 1199061(119889 ℎℎ) ge 119906


1199062(ℎ ℎℎ) ge 119906



1 119904lowast

2) = 119909 and

1199062(119904lowast

1 119904lowast

2) = 119910



1 lowast

2) ge 119906

2(119904lowast

1 119904lowast

2)


2= 119904lowast

2


2) ge 1199062(119904lowast

1 119904lowast

2) If lowast

2= 119904lowast


2is


2) ge 1199062(119904lowast

1 119904lowast

2) then

119904lowast

1= 119889 or 119904lowast



2(ℎ lowast

2) lt

1199062(119889 119904lowast

2) If 119904lowast

1= ℎ and


2(ℎ lowast

2) lt

1199062(119904lowast

1 119904lowast

2) If 119904lowast

1= ℎ and


1199062(ℎ lowast

2) ge 119906

2(ℎ 119904lowast

2) and 119906

2(119889 lowast

2) lt 119906

2(119889 119904lowast

2) The C1



1= 119889



1 119904lowast

2) = 119910 and


1(ℎ 119889ℎ) then 119898

2= 119889ℎ is a


2(119904lowast

1 119904lowast

2)

If 1199062(119889 119889ℎ) gt 119906

2(119904lowast

1 119904lowast

2) then 1 gt 119906

2(119904lowast

1 119904lowast

2) to satisfy this


1 119904lowast

2) = (ℎ ℎ119904

lowast


2= ℎ119904lowast


then 1199062(119889 ℎ119904lowast

2ℎ) gt 1199062(119889 119889119904lowast



2= ℎ119904lowast


1(119904lowast

1 119904lowast

2) =


2(119904lowast

1 119904lowast






1119898 119898lowast

2) gt 1199062(119904lowast

1 119904lowast

2)


119889ℎ and 120579ℎ119889= ℎ119889 If 119898lowast




1199061(119889 1199041(minus119889ℎ) 119889ℎ119898

minus119889ℎ) = 1 and 119906

1(ℎ 1199041(minus119889ℎ) 119889ℎ119898

minus119889ℎ) =


1(119889 1199041(minusℎ119889) ℎ119889119898

minusℎ119889) =

119909 and 1199061(ℎ 1199041(minusℎ119889) ℎ119889119898




minus119889ℎ) = 1

and 119906ℎ119889(119889 1199041(minusℎ119889) ℎ119889119898






1205792


1119898lowast(1205792)= 119889 If lowast

1205792




(119889 1199041(minus119898lowast(120579

2)) lowast

1205792

119898lowast

minus1205792

) lt 1199061205792

(119904lowast

1119898 1198981205792

119898lowast

minus1205792


1205792

(119889 1199041(minus119898lowast(120579

2)) lowast

1205792

119898lowast

minus1205792

) ge 1199061205792

(119904lowast

1119898

119898lowast

1205792

119898lowast

minus1205792



1205792

= 2is a self-

committing message



1199061(1198891198981205792

) = 1120572119889119889|119898(120579

2)+ 1120572119889ℎ|119898(120579

2)+ 119909120572ℎ119889|119898(120579

2)+

119909120572ℎℎ|119898(120579

2)

1199061(ℎ1198981205792

) = 119910120572119889119889|119898(120579

2)+ 025120572

119889ℎ|119898(1205792)+ 119910120572ℎ119889|119898(120579

2)+

025120572ℎℎ|119898(120579

2)

therefore 1199061(1198891198981205792

) ge 1199061(ℎ1198981205792


2)+ (075)120572

119889ℎ|119898(1205792)+ (119909 minus 119910)120572

ℎ119889|119898(1205792)+ (119909 minus

025)120572ℎℎ|119898(120579

2)ge 0



Acknowledgments


References
























1 Introduction











2 Hypotheses



119875A = 119889A minus 119868 + 119877

119875B = 119889B + 119898119868 minus 119877(1)








max [119901 (119891) (119887 + 119890) + 119889 (119890 119891) minus 119896] (2)


119891gt 0)








119891(119887 + 119890) + 119889

1015840

119891= 0 (3)
























3 Methods





4 Results




A investments

2

3

4

5

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(a)

B returns

1

2

3

4

5

6

7

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(b)

Cooperation index

045

05

055

06

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(c)









A investments

1

2

3

4

5

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(a)

B returns

1

2

3

4

5

6

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(b)

Cooperation index

045

05

055

06

MU

Base

line

No

ince

ntiv

e

Fixe

d in

cent

ive

A in

cent

ive

B in

cent

ive

(c)







5 Discussion





Baseline


B ret (prop) 009 001 016 001 056 1370 0059CI 050 000 051 000 098 3190 0118

No incentive


B ret (prop) 024 001 014 001 171 1890 0006CI 063 001 054 000 117 3960 0000

Fixed inc


B ret (prop) 017 001 010 001 170 1500 0153CI 054 001 052 000 104 2640 0319

A incentive


B ret (prop) 023 001 021 001 110 1340 0333CI 060 001 056 001 107 3650 0002

B incentive


B ret (prop) 019 001 018 001 106 1300 0387CI 058 001 055 000 105 3070 0071







1509lowastlowast







1012dagger



0024lowast



lowastlowast



lowast





lowastlowastlowast

























Appendices










B Instructions











































Acknowledgments


References


















































Sarah A Tulman







1 Introduction





















Candidate 119884 119902 minus 120579 119902









119901lowast

119894119895= int

infin

minusinfin

120591119894119895(119888) 119891 (119888) 119889119888 = int

119888lowast

119894119895

minusinfin

119891 (119888) 119889119888 = 119865 (119888lowast

119894119895) (1)




119888lowast

119894119895=

(119881119894119895minus 119881119894119898)


119894119895+ Pr (break tie)

119894119895)

(2)




119894119895) is




119894119895minus 119888119896 (3)





119896119894119895= 120587119896119894119895+120576119896119894119895

where 120576119896119894119895






119894119890120582120587119894 where 120587








119901119896vote119894119895 =

119890120582120587119896119894119895

119890120582120587119896119894119883 + 119890

120582120587119896119894119884 + 119890

120582120587119896119894119886

(4)


119901119896119894119895=

119890120582120587119896119894119895

1 + 119890120582120587119896119894119883 + 119890

120582120587119896119894119884

(5)

0

01

02

03

04

05

06

07

08

09

1

Pr (v

ote)

clowast





Candidate 119884 119902 119902


119894119883 119901119894119884 1 minus



likelihood function

log 119871 = sum119897

(1198731119883119897times ln (119901

1119883119897) + 1198731119884119897

times ln (1199011119884119897) + (num obs

1119897minus 1198731119883119897minus 1198731119884119897)


times ln (1199012119883119897) + 1198732119884119897times ln (119901

2119884119897)



(6)

where 119873119894119895119897












1119884minus 119888119896+ 120572 (7)





1119884minus 119888119896 (8)




1119884minus 119888119896+ 120572 (9)












119894119895























(a)


02996 03069 033331199011119884

0 00793 03241199012119883

0 00793 03241199012119884

02996 03069 033331198881119883

01221198881119884

01198882119883

01198882119884

0122

(b)


02608 03069 033331199011119884

0 00793 03241199012119883

0 00793 03241199012119884

02608 03069 033331198881119883

01221198881119884

01198882119883

01198882119884

0122





119894119883minus119901119894119884)




For 120582 = 7 120572 = 0 120572 = 01 120572 = 025 120572 = 05 120572 = 075 120572 = 1















(a)


(b)


(c)














1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1Pr

(vot

e)


(a)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)


(b)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)


(c)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)






119894119905


119894119905+ 1205733cost119894119905


119894119905

+ 1205737round


119894119905

+ 1205739( same type)

119894119905


119894119905


119894119905

+ 119908turnout119894 + 119906turnout119894119905


119894119905


119894119905+ 1205743cost119894119905


119894119905

+ 1205747round


119894119905

+ 1205749( same type)

119894119905


119894119905


119894119905

+ 119908favored119894 + 119906favored119894119905


119894119905


119894119905+ 1205793cost119894119905


119894119905









(006) (006) (001) (006) (006) (000) (002)



(000) (000) (000) (000) (000) (000) (000)


(008) (008) (001) (009) (009) (000) (002)



(002) (002) (001) (003) (003) (000) (001)




+ 1205797round


119894119905

+ 1205799( same type)

119894119905


119894119905


119894119905


(10)


















120582














0

02

04

06

08

1


Pr (v

ote)

0

02

04

06

08

1

Pr (v

ote)






0

02

04

06

08

1


Pr (v

ote)

0

02

04

06

08

1

Pr (v

ote)
































4 Conclusions









Appendices



119894119895) is the






Pr (pivotal)1119883

=1

2(

min(11987311198732)

sum

119899119884=1

119899119884

sum

1198992119884=0

119899119884minus1

sum

1198992119883=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198731minus 1

119899119884minus 1 minus 119899

2119883 119899119884minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119884minus1minus1198992119883

1119883119901119899119884minus1198992119884

1119884


+

min(11987311198732)minus1

sum

119899119884=0

119899119884

sum

1198992119884=0

119899119884

sum

1198992119883=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198732minus 1

119899119884minus 1198992119883 119899119884minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119884minus1198992119883

1119883119901119899119884minus1198992119884

1119884



)

(A1)


119895








119883= 119899119884




of 1198992119884

and 1198992119883


1119884and 119899

1119883mdashthe


119884minus 1198992119884) and (119899

119883minus 1198992119883)


and 1198991119883

In the first term ( 119873211989921198831198992119884

) 1199011198992119883

21198831199011198992119884



and 1198992119884




) 119901119899119884minus1minus1198992119883

1119883119901119899119884minus1198992119884

1119884(1 minus


be exactly 1198991119883

and 1198991119884




1119884

=1

2(

min(1198731 1198732)

sum

119899119883=1

119899119883

sum

1198992119883=0

119899119909minus1

sum

1198992119884=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198731minus 1


2119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119883minus1198992119883

1119883119901119899119883minus1minus1198992119884

1119884


+

min(1198731 1198732)minus1

sum

119899119883=0

119899119883

sum

1198992119883=0

119899119909

sum

1198992119884=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198732minus 1

119899119883minus 1198992119883 119899119883minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119883minus1198992119883

1119883119901119899119883minus1198992119884

1119884



)


Pr (pivotal)2119883

=1

2(

min(1198731 1198732)

sum

119899119884=1

119899119884

sum

1198991119884=0

119899119884minus1

sum

1198991119883=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119884minus 1 minus 119899

1119883 119899119884minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119884minus1minus1198991119883

2119883119901119899119884minus1198991119884

2119884


+

min(1198731 1198732)minus1

sum

119899119884=0

119899119884

sum

1198991119884=0

119899119884

sum

1198991119883=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119884minus 1198991119883 119899119884minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119884minus1198991119883

2119883119901119899119884minus1198991119884

2119884



)

Pr (pivotal)2119884

=1

2(

min(1198731 1198732)

sum

119899119883=1

119899119883

sum

1198991119883=0

119899119909minus1

sum

1198991119884=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1


1119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119883minus1198991119883

2119883119901119899119883minus1minus1198991119884

2119884


+

min(1198731 1198732)minus1

sum

119899119883=0

119899119883

sum

1198991119883=0

119899119909

sum

1198991119884=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119883minus 1198991119883 119899119883minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119883minus1198991119883

2119883119901119899119883minus1198991119884

2119884



)

(A2)



0

02

04

06

08

1

1 3 5 7 9 11 13 15 17 19

Pr (v

ote)

Round

MaleFemale























D Instructions


Page 1













Page 3











Submit answers





Page 5





Continue













Disclaimer




Acknowledgments


References
























1 Introduction













2 The Experiment










119894isin 119873+ and 0 le 119892

119894le 119908119894 The


given by

120587119894(119908119894 119892119894) = 2 (119908

119894minus 119892119894) + 120 + 119892

119894+ 119866minus119894 (1)



119906119894(119908119894 119892119894 120572119894) = 120587119894(119908119894 119892119894) minus 120572119894

(119908119894minus 119892119894)2

119908119894

(2)




119892NE119894

=2120572119894minus 1

2120572119894

119908119894 (3)








119902119894(119908119894 119892119894 120572119894 120583) =

exp 119906119894(119908119894 119892119894 120572119894) 120583

sum119908119894

119892119895=0exp 119906

119894(119908119894 119892119895 120572119894) 120583

(4)



119894-




0 119908119894 Let q

119894(119908119894 g119894 120572119894 120583)119894isin119875



119894will also



119894(119908119894 g119894 120572119894 120583)119894isin119875




NE119894 120572119894 0) = 1 and 119902

119894(119908119894 119892119894 120572119894 0) = 0 forall119892

119894= 119892

NE119894)



119871119894(119908119894 g119894 120572 120583) =

119899

prod119908119894

119892119895=0119899 (119892119895)

119908119894

prod

119892119896=0

119902QRE119894

(119908119894 119892119896 120572 120583)

119899(119892119896)

(5)










119908119894= 160 2453 3500 3503 5568 3767

119908119894= 200 4750 4375 4376 6557 3948

119908119894= 240 4557 5250 5250 7486 4036



119871119894(119908119894 g119894 1205721 1205722 120572

119878 1205741 1205742 120574

119878 120583)

=

119878

sum

119904=1

120574119904

119899

prod119908119894

119892119895=0119899 (119892119895)

119908119894

prod

119892119896=0

119902QRE119894

(119908119894 119892119896 120572119904 120583)119899(119892119896)

(6)

where 1205741 1205742 120574

119878 with sum

119878

119904=1120574119904




119878



119904le

120572119904+1



4 Results




+(LR =









119908119894= 160 1763 2346 2404 4168 2601

119908119894= 200 3528 2932 2971 4830 2630

119908119894= 240 2970 3519 3545 5460 2639



120583 1205721 and 120572

2(119899 = 20) 120583 120572

1 and 120572

2(119899 = 5) 120583 120572

1 1205722 and 120572

3(119899 = 20) 120583 120572

1 1205722 and 120572

3(119899 = 5)

120583 2850 [2588 3126] 1507 [1290 1764] 2214 [2056 2395] 1425 [1204 1685]1205721

0 [0 001] 0 [0 002] 0 [0 001] 0 [0 002]1205722

053 [046 060] 054 [047 061] 043 [039 046] 048 [040 056]1205723

104 [092 116] 076 [053 101]1205741

066 [053 078] 063 [050 075] 041 [033 046] 059 [049 064]1205742



119908119894= 160 3604 2463 3678 2468

119908119894= 200 3901 2677 4083 2699

119908119894= 240 4148 2873 4445 2913










1= 0 120572

2= 038 120572

3= 061 120572

4= 104

1205741= 039 120574

2= 042 and 120574



Table 4








3= 104






5 Conclusions







Appendix






Acknowledgments


References
















































1 Introduction














2 Literature Review






















119909 (119888) = 120574minus1(

3

sum

119895=1

120587119895 int

1

119888

minus

1

119886

1198651015840119895 (119886) 119889119886) (1)

where

1198651015840119895 (119886) =

(3 minus 1)

(119895 minus 1) (3 minus 119895)


119865 (119886)119895minus2

sdot 1198651015840(119886) ((1 minus 3) 119865 (119886) + (119895 minus 1))

(2)


119865119895 (119886) =(3 minus 1)

(119895 minus 1) (3 minus 119895)

(1 minus 119865 (119886))3minus119895

119865 (119886)119895minus1

(3)








119904120572119894 = 120572120587119894 + (1 minus 120572) (

2

3


(4)






120572 = 0

120572 = 05

120572 = 1

AR

E

D

0

12

26 33 100

120587120572

B

120587120572

C

120587120572

A



119863= (26 12 12)


119877= 119904119877119860 119904119877119861 119904119877119862 where 119904

119877119894 = 120587119894 forall119894 isin













120597119904120572119894

120597120572

= 2 (120587119894 minus 120587) (5)

























119860 119861 119862Coefficient of









5 Results








signBaseline


Constant 3170 0200 00001198772 00132

119865 376 0015Veto

Mean minus0313 0208 0137 +



Constant 2729 0180 00001198772 0008


915

0

2

4

6

8

Mea

n gr

oup

effor

t

10


749 763

875



0

003001

021

016

005

01

015

02

Def

ault

rate










029 029

042

050

003

010

039047

SkewedMajority

RandomContest

Unanimity

0

01

02

03

04

05

Mea

n of120572









6 Conclusion





Symmetric (0057) (0035)119875 le 001 119875 = 0022

0079 0131Right-skewed (0029) (0031)

119875 le 001 119875 le 001





Appendices

A Further Tables


B Instructions









Prizescheme



payoff119894 119888119894 1st 2nd 3rd 119864(120587119894)

119909119894 119895 120587119895

1 04 562 375 62 10 38 A 33 15 181 2 06 250 500 250 4 21 B 17 13 4

3 08 62 375 562 1 9 C 0 8 minus81 04 562 375 62 19 77 A 67 31 36

2 2 06 250 500 250 8 42 B 33 25 83 08 62 375 562 2 18 C 0 15 minus151 04 562 375 62 13 20 A 25 8 17

3 2 06 250 500 250 10 11 B 17 7 103 08 62 375 562 8 5 C 8 4 41 04 562 375 62 27 38 A 50 15 35

4 2 06 250 500 250 21 21 B 33 12 213 08 62 375 562 18 9 C 17 7 101 04 562 375 62 12 29 A 36 11 25

5 2 06 250 500 250 8 10 B 7 6 13 08 62 375 562 7 2 C 7 2 51 04 562 375 62 24 57 A 72 23 49

6 2 06 250 500 250 16 20 B 14 12 23 08 62 375 562 14 4 C 14 3 111 04 562 375 62 14 14 A 26 6 20

7 2 06 250 500 250 13 5 B 12 3 93 08 62 375 562 12 1 C 12 1 111 04 562 375 62 29 28 A 52 11 41

8 2 06 250 500 250 25 10 B 24 6 183 08 62 375 562 24 2 C 24 2 22









Effort 120572

119860 119861 119862 119860 119861 119862


Symmetric 439 314 163 059 051 041(022) (020) (032) (003) (003) (003)

Right-skewed 426 238 085 040 010 010(017) (020) (030) (004) (002) (002)

Unanimity rule

Symmetric 392 255 116 048 049 047(021) (021) (031) (002) (002) (002)

Right-skewed 424 280 172 043 041 043(026) (022) (035) (002) (003) (002)








+


(B2)



(3 + 5 + 10) sdot 05

3

= 15 + 3 = 45

(B3)







(4 + 6 + 10) sdot 045

3

= 22 + 3 = 52

(B4)





Acknowledgments




Endnotes



















References


















































































Editorial Board













Contents





















Acknowledgment



Jens Groszliger

References














1 Introduction















Player 2119889 ℎ

Player 1 119889 1 1 119909 119910

ℎ 119910 119909 025 025












(119904lowast

1 119904lowast

2) (119880

lowast

1 119880lowast

2) Pareto optimal

C1 (119889 119889) (1 1) Yes

C2 (119889 119889) (1 1) Yes(ℎ ℎ) (025 025) No

C3 (119889 ℎ) (119909 119910) Yes(ℎ 119889) (119910 119909) Yes

C4 (ℎ ℎ) (025 025) No










(a) C1 119910 minus 119909 = 0

2119889 ℎ

1 119889 1 1 05 05ℎ 05 05 025 025

(b) C2 119910 minus 119909 = 05

2119889 ℎ

1 119889 1 1 0 05ℎ 05 0 025 025

(c) C3 119910 minus 119909 = 1

2119889 ℎ

1 119889 1 1 05 15ℎ 15 05 025 025

(d) C4 119910 minus 119909 = 15

2119889 ℎ

1 119889 1 1 0 15ℎ 15 0 025 025







d h

d h d h

1

1

x

y

y

x

025

025

P1

P2P2

u1

u2







2= 11990421198891199042ℎrepresents







119894(1199041 1199042)

is defined by 119906119894(119889 11990421198891199042ℎ) = 119906

119894(119889 1199042119889) and 119906

119894(ℎ 11990421198891199042ℎ) =




2) is a SPNE


2) ge 1199062(1199041 1199042) for every 119904


1isin 1198781

and 1199061(119904lowast

1 119904lowast

2) ge 1199061(1199041 119904lowast

2) for every 119904

1isin 1198781


2119889119904lowast






2119889119904lowast




(119904lowast

1 119904lowast

2) (119906

lowast

1 119906lowast

2) Pareto optimal




2) is a SPNE in


1 119904lowast

2) = 119909

and 1199062(119904lowast

1 119904lowast

2) = 119910









1119898 119898lowast

2) ge 119906

2(119904lowast

1 119904lowast

2) where

1199061(119904lowast

1119898 119898lowast

2) ge 119906

1(1199041 119898lowast

2) for every 119904


1119898is






1 119906lowast

2) Threat (119906

1 1199062) Promise (119906

1 1199062)




2= 119904lowast

2


only if 1199062(119889119898lowast

2) = 1199062(119889 119904lowast

2) and 119906

2(ℎ119898lowast

2) lt 1199062(ℎ 119904lowast

2)


only if 1199062(119889119898lowast

2) lt 1199062(119889 119904lowast

2)



119894(119904119894 119889) gt





1= 119889




2such that 119906

2(119904lowast

1119898 119898lowast

2) gt 119906

2(119904lowast

1 119904lowast

2) if and

only if 119910 gt 1
















1isin 119889 ℎ





1205792isinΘ2

119901(1205792) = 1 In the second





2= (119898119889119889 119898119889ℎ 119898ℎ119889 119898ℎℎ) is a


2isin






1205792

= 11990421198891199042ℎ


2 and119898

2= (1198981205792

119898minus1205792


2




2isin 1205792 where 119904

1119889119889is




1199041(minus1198981205792) 1198981205792

119898minus1205792

) = 119906119894(1199041 1199042) for 119904

11198981205792

= 1199041and

1205792= 1199042


1205792|1198982




1205792isinΘ2

1205721205792|1198982



2= 119889119889 In the






2= (119898

lowast

119889119889 119898lowast

119889ℎ 119898lowast

ℎ119889 119898lowast

ℎℎ) a


1119889119889 119904lowast

1119889ℎ 119904lowast

1ℎ119889 119904lowast

1ℎℎ) and a





1198981205792isinΘ2

1199061205792

(119904lowast

1119898 1198981205792

119898minus1205792

)

(2) 119904lowast1119898


1isin1198781

sum1205792isinΘ2

1205721205792|1198982

sdot 1199061(11990411198981205792

1199041(minus1198981205792)

119898lowast

2)

(3) 120572lowast1205792|1198982


2 For all 120579

2who play the


1205792|119898lowast

2

=

1199011205792


2


1205792




1205792|1198982



u1

u2

d

h

d

h

d

h

d

h

x

y

x

y x

y

x

y

N

1 1

1

1

11

1

1

1

1

1 1

x y

x y

025 025

025

025

025

025

025 025

120572hh120572dh

120572dh

120572dh

120572dh

120572hd

120572hd

120572hd

120572hd

120572dd

120572dd

120572dd

120572dd

120579hh

120579dh

120579hd

120579dd

mhh

mdh

mhd

mdd

d

h

d

h

d

h

d

h

d h d h d h d h

d hd hd hd h

120572hh

120572hh120572hh

u1 u2

y x

y x

1 1

1 1

x y

x y

025 025

025 025

u1 u2

y x

y x

u1

u2 x

y

x

y x

y

x

y

1

1

1

1

025

025

025

025



2



2= 1205792 then 120572

1205792|2



1205792|1198982

= 1





2 then 119904lowast

11198981205792

= 119889


1205792


1205792






119889119889 119898lowast

119889ℎ 119898lowast

ℎ119889 119898lowast

ℎℎ) (119904

lowast

1119889119889 119904lowast

1119889ℎ 119904lowast

1ℎ119889 119904lowast

1ℎℎ) (120572

lowast

119889119889|119889119889 120572lowast

119889ℎ|119889ℎ 120572lowast

ℎ119889|ℎ119889 120572lowast

ℎℎ|ℎℎ)

C1 (119889119889 119889ℎ 119889119889 ℎℎ) (119889 119889 ℎ 119889) (119901119889119889

(119901119889119889+ 119901ℎ119889) 1 1 1)

C2 (119889119889 119889ℎ 119889119889 119889ℎ) (119889 119889 ℎ ℎ) (119901119889119889

(119901119889119889+ 119901ℎ119889)119901119889ℎ

(119901119889ℎ+ 119901ℎℎ) 1 1)

C3 (119889ℎ 119889ℎ ℎℎ ℎℎ) (ℎ 119889 ℎ 119889) (1119901119889ℎ

(119901119889119889+ 119901119889ℎ) 1119901ℎℎ

(119901ℎ119889+ 119901ℎℎ))

C4 (119889ℎ 119889ℎ 119889ℎ 119889ℎ) (ℎ 119889 ℎ ℎ) (1119901119889ℎ

(119901119889119889+ 119901119889ℎ+ 119901ℎ119889+ 119901ℎℎ) 1 1)









1205792|lowast

2(1205792)= 1 lowast

1205792


1205792

(119904lowast

1119898 lowast

1205792

lowast

minus1205792

) ge 1199061205792

(119904lowast

1119898

1198981205792

lowast

minus1205792

) for every1198981205792

isin Θ2




lowast

1205792



1205792




= 2is a








C1 120572lowast

119889119889|119889119889ge

120572lowast

ℎ119889|119889119889(119910 minus 119909)


C2 120572lowast

119889ℎ|119889ℎge

120572lowast

ℎℎ|119889ℎ(025 minus 119909)

075120572lowast

119889119889|119889119889ge

120572lowast

ℎ119889|119889119889(119910 minus 119909)

(1 minus 119910)

C3 Lie 120572lowast

ℎℎ|ℎℎge

120572lowast

ℎ119889|ℎℎ(119910 minus 119909)

(119909 minus 025)120572lowast

119889ℎ|119889ℎge

120572lowast

119889119889|119889ℎ(119910 minus 1)

075

C4 Lie 120572lowast

119889ℎ|119889ℎge

120572lowast

119889119889|119889ℎ(119910 minus 1) + 120572

lowast

ℎ119889|119889ℎ(119910 minus 119909) + 120572

lowast

ℎℎ|119889ℎ(025 minus 119909)

075






only if (1minus119910)120572119889119889|119898(120579

2)+(075)120572

119889ℎ|119898(1205792)+(119909minus119910)120572

ℎ119889|119898(1205792)+(119909minus

025)120572ℎℎ|119898(120579

2)ge 0





5 Conclusions










Appendix


1 119904lowast

2) = 119909 and

1199062(119904lowast

1 119904lowast

2) = 119910 then 119910 ge 1 If 119904lowast

2= ℎ119889 then 119906

1(119889 ℎ119889) ge


2= ℎℎ

then 1199061(119889 ℎℎ) ge 119906


1199062(ℎ ℎℎ) ge 119906



1 119904lowast

2) = 119909 and

1199062(119904lowast

1 119904lowast

2) = 119910



1 lowast

2) ge 119906

2(119904lowast

1 119904lowast

2)


2= 119904lowast

2


2) ge 1199062(119904lowast

1 119904lowast

2) If lowast

2= 119904lowast


2is


2) ge 1199062(119904lowast

1 119904lowast

2) then

119904lowast

1= 119889 or 119904lowast



2(ℎ lowast

2) lt

1199062(119889 119904lowast

2) If 119904lowast

1= ℎ and


2(ℎ lowast

2) lt

1199062(119904lowast

1 119904lowast

2) If 119904lowast

1= ℎ and


1199062(ℎ lowast

2) ge 119906

2(ℎ 119904lowast

2) and 119906

2(119889 lowast

2) lt 119906

2(119889 119904lowast

2) The C1



1= 119889



1 119904lowast

2) = 119910 and


1(ℎ 119889ℎ) then 119898

2= 119889ℎ is a


2(119904lowast

1 119904lowast

2)

If 1199062(119889 119889ℎ) gt 119906

2(119904lowast

1 119904lowast

2) then 1 gt 119906

2(119904lowast

1 119904lowast

2) to satisfy this


1 119904lowast

2) = (ℎ ℎ119904

lowast


2= ℎ119904lowast


then 1199062(119889 ℎ119904lowast

2ℎ) gt 1199062(119889 119889119904lowast



2= ℎ119904lowast


1(119904lowast

1 119904lowast

2) =


2(119904lowast

1 119904lowast






1119898 119898lowast

2) gt 1199062(119904lowast

1 119904lowast

2)


119889ℎ and 120579ℎ119889= ℎ119889 If 119898lowast




1199061(119889 1199041(minus119889ℎ) 119889ℎ119898

minus119889ℎ) = 1 and 119906

1(ℎ 1199041(minus119889ℎ) 119889ℎ119898

minus119889ℎ) =


1(119889 1199041(minusℎ119889) ℎ119889119898

minusℎ119889) =

119909 and 1199061(ℎ 1199041(minusℎ119889) ℎ119889119898




minus119889ℎ) = 1

and 119906ℎ119889(119889 1199041(minusℎ119889) ℎ119889119898






1205792


1119898lowast(1205792)= 119889 If lowast

1205792




(119889 1199041(minus119898lowast(120579

2)) lowast

1205792

119898lowast

minus1205792

) lt 1199061205792

(119904lowast

1119898 1198981205792

119898lowast

minus1205792


1205792

(119889 1199041(minus119898lowast(120579

2)) lowast

1205792

119898lowast

minus1205792

) ge 1199061205792

(119904lowast

1119898

119898lowast

1205792

119898lowast

minus1205792



1205792

= 2is a self-

committing message



1199061(1198891198981205792

) = 1120572119889119889|119898(120579

2)+ 1120572119889ℎ|119898(120579

2)+ 119909120572ℎ119889|119898(120579

2)+

119909120572ℎℎ|119898(120579

2)

1199061(ℎ1198981205792

) = 119910120572119889119889|119898(120579

2)+ 025120572

119889ℎ|119898(1205792)+ 119910120572ℎ119889|119898(120579

2)+

025120572ℎℎ|119898(120579

2)

therefore 1199061(1198891198981205792

) ge 1199061(ℎ1198981205792


2)+ (075)120572

119889ℎ|119898(1205792)+ (119909 minus 119910)120572

ℎ119889|119898(1205792)+ (119909 minus

025)120572ℎℎ|119898(120579

2)ge 0



Acknowledgments


References
























1 Introduction











2 Hypotheses



119875A = 119889A minus 119868 + 119877

119875B = 119889B + 119898119868 minus 119877(1)








max [119901 (119891) (119887 + 119890) + 119889 (119890 119891) minus 119896] (2)


119891gt 0)








119891(119887 + 119890) + 119889

1015840

119891= 0 (3)
























3 Methods





4 Results




A investments

2

3

4

5

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(a)

B returns

1

2

3

4

5

6

7

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(b)

Cooperation index

045

05

055

06

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(c)









A investments

1

2

3

4

5

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(a)

B returns

1

2

3

4

5

6

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(b)

Cooperation index

045

05

055

06

MU

Base

line

No

ince

ntiv

e

Fixe

d in

cent

ive

A in

cent

ive

B in

cent

ive

(c)







5 Discussion





Baseline


B ret (prop) 009 001 016 001 056 1370 0059CI 050 000 051 000 098 3190 0118

No incentive


B ret (prop) 024 001 014 001 171 1890 0006CI 063 001 054 000 117 3960 0000

Fixed inc


B ret (prop) 017 001 010 001 170 1500 0153CI 054 001 052 000 104 2640 0319

A incentive


B ret (prop) 023 001 021 001 110 1340 0333CI 060 001 056 001 107 3650 0002

B incentive


B ret (prop) 019 001 018 001 106 1300 0387CI 058 001 055 000 105 3070 0071







1509lowastlowast







1012dagger



0024lowast



lowastlowast



lowast





lowastlowastlowast

























Appendices










B Instructions











































Acknowledgments


References


















































Sarah A Tulman







1 Introduction





















Candidate 119884 119902 minus 120579 119902









119901lowast

119894119895= int

infin

minusinfin

120591119894119895(119888) 119891 (119888) 119889119888 = int

119888lowast

119894119895

minusinfin

119891 (119888) 119889119888 = 119865 (119888lowast

119894119895) (1)




119888lowast

119894119895=

(119881119894119895minus 119881119894119898)


119894119895+ Pr (break tie)

119894119895)

(2)




119894119895) is




119894119895minus 119888119896 (3)





119896119894119895= 120587119896119894119895+120576119896119894119895

where 120576119896119894119895






119894119890120582120587119894 where 120587








119901119896vote119894119895 =

119890120582120587119896119894119895

119890120582120587119896119894119883 + 119890

120582120587119896119894119884 + 119890

120582120587119896119894119886

(4)


119901119896119894119895=

119890120582120587119896119894119895

1 + 119890120582120587119896119894119883 + 119890

120582120587119896119894119884

(5)

0

01

02

03

04

05

06

07

08

09

1

Pr (v

ote)

clowast





Candidate 119884 119902 119902


119894119883 119901119894119884 1 minus



likelihood function

log 119871 = sum119897

(1198731119883119897times ln (119901

1119883119897) + 1198731119884119897

times ln (1199011119884119897) + (num obs

1119897minus 1198731119883119897minus 1198731119884119897)


times ln (1199012119883119897) + 1198732119884119897times ln (119901

2119884119897)



(6)

where 119873119894119895119897












1119884minus 119888119896+ 120572 (7)





1119884minus 119888119896 (8)




1119884minus 119888119896+ 120572 (9)












119894119895























(a)


02996 03069 033331199011119884

0 00793 03241199012119883

0 00793 03241199012119884

02996 03069 033331198881119883

01221198881119884

01198882119883

01198882119884

0122

(b)


02608 03069 033331199011119884

0 00793 03241199012119883

0 00793 03241199012119884

02608 03069 033331198881119883

01221198881119884

01198882119883

01198882119884

0122





119894119883minus119901119894119884)




For 120582 = 7 120572 = 0 120572 = 01 120572 = 025 120572 = 05 120572 = 075 120572 = 1















(a)


(b)


(c)














1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1Pr

(vot

e)


(a)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)


(b)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)


(c)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)






119894119905


119894119905+ 1205733cost119894119905


119894119905

+ 1205737round


119894119905

+ 1205739( same type)

119894119905


119894119905


119894119905

+ 119908turnout119894 + 119906turnout119894119905


119894119905


119894119905+ 1205743cost119894119905


119894119905

+ 1205747round


119894119905

+ 1205749( same type)

119894119905


119894119905


119894119905

+ 119908favored119894 + 119906favored119894119905


119894119905


119894119905+ 1205793cost119894119905


119894119905









(006) (006) (001) (006) (006) (000) (002)



(000) (000) (000) (000) (000) (000) (000)


(008) (008) (001) (009) (009) (000) (002)



(002) (002) (001) (003) (003) (000) (001)




+ 1205797round


119894119905

+ 1205799( same type)

119894119905


119894119905


119894119905


(10)


















120582














0

02

04

06

08

1


Pr (v

ote)

0

02

04

06

08

1

Pr (v

ote)






0

02

04

06

08

1


Pr (v

ote)

0

02

04

06

08

1

Pr (v

ote)
































4 Conclusions









Appendices



119894119895) is the






Pr (pivotal)1119883

=1

2(

min(11987311198732)

sum

119899119884=1

119899119884

sum

1198992119884=0

119899119884minus1

sum

1198992119883=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198731minus 1

119899119884minus 1 minus 119899

2119883 119899119884minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119884minus1minus1198992119883

1119883119901119899119884minus1198992119884

1119884


+

min(11987311198732)minus1

sum

119899119884=0

119899119884

sum

1198992119884=0

119899119884

sum

1198992119883=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198732minus 1

119899119884minus 1198992119883 119899119884minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119884minus1198992119883

1119883119901119899119884minus1198992119884

1119884



)

(A1)


119895








119883= 119899119884




of 1198992119884

and 1198992119883


1119884and 119899

1119883mdashthe


119884minus 1198992119884) and (119899

119883minus 1198992119883)


and 1198991119883

In the first term ( 119873211989921198831198992119884

) 1199011198992119883

21198831199011198992119884



and 1198992119884




) 119901119899119884minus1minus1198992119883

1119883119901119899119884minus1198992119884

1119884(1 minus


be exactly 1198991119883

and 1198991119884




1119884

=1

2(

min(1198731 1198732)

sum

119899119883=1

119899119883

sum

1198992119883=0

119899119909minus1

sum

1198992119884=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198731minus 1


2119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119883minus1198992119883

1119883119901119899119883minus1minus1198992119884

1119884


+

min(1198731 1198732)minus1

sum

119899119883=0

119899119883

sum

1198992119883=0

119899119909

sum

1198992119884=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198732minus 1

119899119883minus 1198992119883 119899119883minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119883minus1198992119883

1119883119901119899119883minus1198992119884

1119884



)


Pr (pivotal)2119883

=1

2(

min(1198731 1198732)

sum

119899119884=1

119899119884

sum

1198991119884=0

119899119884minus1

sum

1198991119883=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119884minus 1 minus 119899

1119883 119899119884minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119884minus1minus1198991119883

2119883119901119899119884minus1198991119884

2119884


+

min(1198731 1198732)minus1

sum

119899119884=0

119899119884

sum

1198991119884=0

119899119884

sum

1198991119883=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119884minus 1198991119883 119899119884minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119884minus1198991119883

2119883119901119899119884minus1198991119884

2119884



)

Pr (pivotal)2119884

=1

2(

min(1198731 1198732)

sum

119899119883=1

119899119883

sum

1198991119883=0

119899119909minus1

sum

1198991119884=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1


1119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119883minus1198991119883

2119883119901119899119883minus1minus1198991119884

2119884


+

min(1198731 1198732)minus1

sum

119899119883=0

119899119883

sum

1198991119883=0

119899119909

sum

1198991119884=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119883minus 1198991119883 119899119883minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119883minus1198991119883

2119883119901119899119883minus1198991119884

2119884



)

(A2)



0

02

04

06

08

1

1 3 5 7 9 11 13 15 17 19

Pr (v

ote)

Round

MaleFemale























D Instructions


Page 1








Page 2
















Submit answers





Page 5





Continue













Disclaimer




Acknowledgments


References
























1 Introduction













2 The Experiment










119894isin 119873+ and 0 le 119892

119894le 119908119894 The


given by

120587119894(119908119894 119892119894) = 2 (119908

119894minus 119892119894) + 120 + 119892

119894+ 119866minus119894 (1)



119906119894(119908119894 119892119894 120572119894) = 120587119894(119908119894 119892119894) minus 120572119894

(119908119894minus 119892119894)2

119908119894

(2)




119892NE119894

=2120572119894minus 1

2120572119894

119908119894 (3)








119902119894(119908119894 119892119894 120572119894 120583) =

exp 119906119894(119908119894 119892119894 120572119894) 120583

sum119908119894

119892119895=0exp 119906

119894(119908119894 119892119895 120572119894) 120583

(4)



119894-




0 119908119894 Let q

119894(119908119894 g119894 120572119894 120583)119894isin119875



119894will also



119894(119908119894 g119894 120572119894 120583)119894isin119875




NE119894 120572119894 0) = 1 and 119902

119894(119908119894 119892119894 120572119894 0) = 0 forall119892

119894= 119892

NE119894)



119871119894(119908119894 g119894 120572 120583) =

119899

prod119908119894

119892119895=0119899 (119892119895)

119908119894

prod

119892119896=0

119902QRE119894

(119908119894 119892119896 120572 120583)

119899(119892119896)

(5)










119908119894= 160 2453 3500 3503 5568 3767

119908119894= 200 4750 4375 4376 6557 3948

119908119894= 240 4557 5250 5250 7486 4036



119871119894(119908119894 g119894 1205721 1205722 120572

119878 1205741 1205742 120574

119878 120583)

=

119878

sum

119904=1

120574119904

119899

prod119908119894

119892119895=0119899 (119892119895)

119908119894

prod

119892119896=0

119902QRE119894

(119908119894 119892119896 120572119904 120583)119899(119892119896)

(6)

where 1205741 1205742 120574

119878 with sum

119878

119904=1120574119904




119878



119904le

120572119904+1



4 Results




+(LR =









119908119894= 160 1763 2346 2404 4168 2601

119908119894= 200 3528 2932 2971 4830 2630

119908119894= 240 2970 3519 3545 5460 2639



120583 1205721 and 120572

2(119899 = 20) 120583 120572

1 and 120572

2(119899 = 5) 120583 120572

1 1205722 and 120572

3(119899 = 20) 120583 120572

1 1205722 and 120572

3(119899 = 5)

120583 2850 [2588 3126] 1507 [1290 1764] 2214 [2056 2395] 1425 [1204 1685]1205721

0 [0 001] 0 [0 002] 0 [0 001] 0 [0 002]1205722

053 [046 060] 054 [047 061] 043 [039 046] 048 [040 056]1205723

104 [092 116] 076 [053 101]1205741

066 [053 078] 063 [050 075] 041 [033 046] 059 [049 064]1205742



119908119894= 160 3604 2463 3678 2468

119908119894= 200 3901 2677 4083 2699

119908119894= 240 4148 2873 4445 2913










1= 0 120572

2= 038 120572

3= 061 120572

4= 104

1205741= 039 120574

2= 042 and 120574



Table 4








3= 104






5 Conclusions







Appendix






Acknowledgments


References
















































1 Introduction














2 Literature Review






















119909 (119888) = 120574minus1(

3

sum

119895=1

120587119895 int

1

119888

minus

1

119886

1198651015840119895 (119886) 119889119886) (1)

where

1198651015840119895 (119886) =

(3 minus 1)

(119895 minus 1) (3 minus 119895)


119865 (119886)119895minus2

sdot 1198651015840(119886) ((1 minus 3) 119865 (119886) + (119895 minus 1))

(2)


119865119895 (119886) =(3 minus 1)

(119895 minus 1) (3 minus 119895)

(1 minus 119865 (119886))3minus119895

119865 (119886)119895minus1

(3)








119904120572119894 = 120572120587119894 + (1 minus 120572) (

2

3


(4)






120572 = 0

120572 = 05

120572 = 1

AR

E

D

0

12

26 33 100

120587120572

B

120587120572

C

120587120572

A



119863= (26 12 12)


119877= 119904119877119860 119904119877119861 119904119877119862 where 119904

119877119894 = 120587119894 forall119894 isin













120597119904120572119894

120597120572

= 2 (120587119894 minus 120587) (5)

























119860 119861 119862Coefficient of









5 Results








signBaseline


Constant 3170 0200 00001198772 00132

119865 376 0015Veto

Mean minus0313 0208 0137 +



Constant 2729 0180 00001198772 0008


915

0

2

4

6

8

Mea

n gr

oup

effor

t

10


749 763

875



0

003001

021

016

005

01

015

02

Def

ault

rate










029 029

042

050

003

010

039047

SkewedMajority

RandomContest

Unanimity

0

01

02

03

04

05

Mea

n of120572









6 Conclusion





Symmetric (0057) (0035)119875 le 001 119875 = 0022

0079 0131Right-skewed (0029) (0031)

119875 le 001 119875 le 001





Appendices

A Further Tables


B Instructions









Prizescheme



payoff119894 119888119894 1st 2nd 3rd 119864(120587119894)

119909119894 119895 120587119895

1 04 562 375 62 10 38 A 33 15 181 2 06 250 500 250 4 21 B 17 13 4

3 08 62 375 562 1 9 C 0 8 minus81 04 562 375 62 19 77 A 67 31 36

2 2 06 250 500 250 8 42 B 33 25 83 08 62 375 562 2 18 C 0 15 minus151 04 562 375 62 13 20 A 25 8 17

3 2 06 250 500 250 10 11 B 17 7 103 08 62 375 562 8 5 C 8 4 41 04 562 375 62 27 38 A 50 15 35

4 2 06 250 500 250 21 21 B 33 12 213 08 62 375 562 18 9 C 17 7 101 04 562 375 62 12 29 A 36 11 25

5 2 06 250 500 250 8 10 B 7 6 13 08 62 375 562 7 2 C 7 2 51 04 562 375 62 24 57 A 72 23 49

6 2 06 250 500 250 16 20 B 14 12 23 08 62 375 562 14 4 C 14 3 111 04 562 375 62 14 14 A 26 6 20

7 2 06 250 500 250 13 5 B 12 3 93 08 62 375 562 12 1 C 12 1 111 04 562 375 62 29 28 A 52 11 41

8 2 06 250 500 250 25 10 B 24 6 183 08 62 375 562 24 2 C 24 2 22









Effort 120572

119860 119861 119862 119860 119861 119862


Symmetric 439 314 163 059 051 041(022) (020) (032) (003) (003) (003)

Right-skewed 426 238 085 040 010 010(017) (020) (030) (004) (002) (002)

Unanimity rule

Symmetric 392 255 116 048 049 047(021) (021) (031) (002) (002) (002)

Right-skewed 424 280 172 043 041 043(026) (022) (035) (002) (003) (002)








+


(B2)



(3 + 5 + 10) sdot 05

3

= 15 + 3 = 45

(B3)







(4 + 6 + 10) sdot 045

3

= 22 + 3 = 52

(B4)





Acknowledgments




Endnotes



















References














































































Editorial Board













Contents





















Acknowledgment



Jens Groszliger

References














1 Introduction















Player 2119889 ℎ

Player 1 119889 1 1 119909 119910

ℎ 119910 119909 025 025












(119904lowast

1 119904lowast

2) (119880

lowast

1 119880lowast

2) Pareto optimal

C1 (119889 119889) (1 1) Yes

C2 (119889 119889) (1 1) Yes(ℎ ℎ) (025 025) No

C3 (119889 ℎ) (119909 119910) Yes(ℎ 119889) (119910 119909) Yes

C4 (ℎ ℎ) (025 025) No










(a) C1 119910 minus 119909 = 0

2119889 ℎ

1 119889 1 1 05 05ℎ 05 05 025 025

(b) C2 119910 minus 119909 = 05

2119889 ℎ

1 119889 1 1 0 05ℎ 05 0 025 025

(c) C3 119910 minus 119909 = 1

2119889 ℎ

1 119889 1 1 05 15ℎ 15 05 025 025

(d) C4 119910 minus 119909 = 15

2119889 ℎ

1 119889 1 1 0 15ℎ 15 0 025 025







d h

d h d h

1

1

x

y

y

x

025

025

P1

P2P2

u1

u2







2= 11990421198891199042ℎrepresents







119894(1199041 1199042)

is defined by 119906119894(119889 11990421198891199042ℎ) = 119906

119894(119889 1199042119889) and 119906

119894(ℎ 11990421198891199042ℎ) =




2) is a SPNE


2) ge 1199062(1199041 1199042) for every 119904


1isin 1198781

and 1199061(119904lowast

1 119904lowast

2) ge 1199061(1199041 119904lowast

2) for every 119904

1isin 1198781


2119889119904lowast






2119889119904lowast




(119904lowast

1 119904lowast

2) (119906

lowast

1 119906lowast

2) Pareto optimal




2) is a SPNE in


1 119904lowast

2) = 119909

and 1199062(119904lowast

1 119904lowast

2) = 119910









1119898 119898lowast

2) ge 119906

2(119904lowast

1 119904lowast

2) where

1199061(119904lowast

1119898 119898lowast

2) ge 119906

1(1199041 119898lowast

2) for every 119904


1119898is






1 119906lowast

2) Threat (119906

1 1199062) Promise (119906

1 1199062)




2= 119904lowast

2


only if 1199062(119889119898lowast

2) = 1199062(119889 119904lowast

2) and 119906

2(ℎ119898lowast

2) lt 1199062(ℎ 119904lowast

2)


only if 1199062(119889119898lowast

2) lt 1199062(119889 119904lowast

2)



119894(119904119894 119889) gt





1= 119889




2such that 119906

2(119904lowast

1119898 119898lowast

2) gt 119906

2(119904lowast

1 119904lowast

2) if and

only if 119910 gt 1
















1isin 119889 ℎ





1205792isinΘ2

119901(1205792) = 1 In the second





2= (119898119889119889 119898119889ℎ 119898ℎ119889 119898ℎℎ) is a


2isin






1205792

= 11990421198891199042ℎ


2 and119898

2= (1198981205792

119898minus1205792


2




2isin 1205792 where 119904

1119889119889is




1199041(minus1198981205792) 1198981205792

119898minus1205792

) = 119906119894(1199041 1199042) for 119904

11198981205792

= 1199041and

1205792= 1199042


1205792|1198982




1205792isinΘ2

1205721205792|1198982



2= 119889119889 In the






2= (119898

lowast

119889119889 119898lowast

119889ℎ 119898lowast

ℎ119889 119898lowast

ℎℎ) a


1119889119889 119904lowast

1119889ℎ 119904lowast

1ℎ119889 119904lowast

1ℎℎ) and a





1198981205792isinΘ2

1199061205792

(119904lowast

1119898 1198981205792

119898minus1205792

)

(2) 119904lowast1119898


1isin1198781

sum1205792isinΘ2

1205721205792|1198982

sdot 1199061(11990411198981205792

1199041(minus1198981205792)

119898lowast

2)

(3) 120572lowast1205792|1198982


2 For all 120579

2who play the


1205792|119898lowast

2

=

1199011205792


2


1205792




1205792|1198982



u1

u2

d

h

d

h

d

h

d

h

x

y

x

y x

y

x

y

N

1 1

1

1

11

1

1

1

1

1 1

x y

x y

025 025

025

025

025

025

025 025

120572hh120572dh

120572dh

120572dh

120572dh

120572hd

120572hd

120572hd

120572hd

120572dd

120572dd

120572dd

120572dd

120579hh

120579dh

120579hd

120579dd

mhh

mdh

mhd

mdd

d

h

d

h

d

h

d

h

d h d h d h d h

d hd hd hd h

120572hh

120572hh120572hh

u1 u2

y x

y x

1 1

1 1

x y

x y

025 025

025 025

u1 u2

y x

y x

u1

u2 x

y

x

y x

y

x

y

1

1

1

1

025

025

025

025



2



2= 1205792 then 120572

1205792|2



1205792|1198982

= 1





2 then 119904lowast

11198981205792

= 119889


1205792


1205792






119889119889 119898lowast

119889ℎ 119898lowast

ℎ119889 119898lowast

ℎℎ) (119904

lowast

1119889119889 119904lowast

1119889ℎ 119904lowast

1ℎ119889 119904lowast

1ℎℎ) (120572

lowast

119889119889|119889119889 120572lowast

119889ℎ|119889ℎ 120572lowast

ℎ119889|ℎ119889 120572lowast

ℎℎ|ℎℎ)

C1 (119889119889 119889ℎ 119889119889 ℎℎ) (119889 119889 ℎ 119889) (119901119889119889

(119901119889119889+ 119901ℎ119889) 1 1 1)

C2 (119889119889 119889ℎ 119889119889 119889ℎ) (119889 119889 ℎ ℎ) (119901119889119889

(119901119889119889+ 119901ℎ119889)119901119889ℎ

(119901119889ℎ+ 119901ℎℎ) 1 1)

C3 (119889ℎ 119889ℎ ℎℎ ℎℎ) (ℎ 119889 ℎ 119889) (1119901119889ℎ

(119901119889119889+ 119901119889ℎ) 1119901ℎℎ

(119901ℎ119889+ 119901ℎℎ))

C4 (119889ℎ 119889ℎ 119889ℎ 119889ℎ) (ℎ 119889 ℎ ℎ) (1119901119889ℎ

(119901119889119889+ 119901119889ℎ+ 119901ℎ119889+ 119901ℎℎ) 1 1)









1205792|lowast

2(1205792)= 1 lowast

1205792


1205792

(119904lowast

1119898 lowast

1205792

lowast

minus1205792

) ge 1199061205792

(119904lowast

1119898

1198981205792

lowast

minus1205792

) for every1198981205792

isin Θ2




lowast

1205792



1205792




= 2is a








C1 120572lowast

119889119889|119889119889ge

120572lowast

ℎ119889|119889119889(119910 minus 119909)


C2 120572lowast

119889ℎ|119889ℎge

120572lowast

ℎℎ|119889ℎ(025 minus 119909)

075120572lowast

119889119889|119889119889ge

120572lowast

ℎ119889|119889119889(119910 minus 119909)

(1 minus 119910)

C3 Lie 120572lowast

ℎℎ|ℎℎge

120572lowast

ℎ119889|ℎℎ(119910 minus 119909)

(119909 minus 025)120572lowast

119889ℎ|119889ℎge

120572lowast

119889119889|119889ℎ(119910 minus 1)

075

C4 Lie 120572lowast

119889ℎ|119889ℎge

120572lowast

119889119889|119889ℎ(119910 minus 1) + 120572

lowast

ℎ119889|119889ℎ(119910 minus 119909) + 120572

lowast

ℎℎ|119889ℎ(025 minus 119909)

075






only if (1minus119910)120572119889119889|119898(120579

2)+(075)120572

119889ℎ|119898(1205792)+(119909minus119910)120572

ℎ119889|119898(1205792)+(119909minus

025)120572ℎℎ|119898(120579

2)ge 0





5 Conclusions










Appendix


1 119904lowast

2) = 119909 and

1199062(119904lowast

1 119904lowast

2) = 119910 then 119910 ge 1 If 119904lowast

2= ℎ119889 then 119906

1(119889 ℎ119889) ge


2= ℎℎ

then 1199061(119889 ℎℎ) ge 119906


1199062(ℎ ℎℎ) ge 119906



1 119904lowast

2) = 119909 and

1199062(119904lowast

1 119904lowast

2) = 119910



1 lowast

2) ge 119906

2(119904lowast

1 119904lowast

2)


2= 119904lowast

2


2) ge 1199062(119904lowast

1 119904lowast

2) If lowast

2= 119904lowast


2is


2) ge 1199062(119904lowast

1 119904lowast

2) then

119904lowast

1= 119889 or 119904lowast



2(ℎ lowast

2) lt

1199062(119889 119904lowast

2) If 119904lowast

1= ℎ and


2(ℎ lowast

2) lt

1199062(119904lowast

1 119904lowast

2) If 119904lowast

1= ℎ and


1199062(ℎ lowast

2) ge 119906

2(ℎ 119904lowast

2) and 119906

2(119889 lowast

2) lt 119906

2(119889 119904lowast

2) The C1



1= 119889



1 119904lowast

2) = 119910 and


1(ℎ 119889ℎ) then 119898

2= 119889ℎ is a


2(119904lowast

1 119904lowast

2)

If 1199062(119889 119889ℎ) gt 119906

2(119904lowast

1 119904lowast

2) then 1 gt 119906

2(119904lowast

1 119904lowast

2) to satisfy this


1 119904lowast

2) = (ℎ ℎ119904

lowast


2= ℎ119904lowast


then 1199062(119889 ℎ119904lowast

2ℎ) gt 1199062(119889 119889119904lowast



2= ℎ119904lowast


1(119904lowast

1 119904lowast

2) =


2(119904lowast

1 119904lowast






1119898 119898lowast

2) gt 1199062(119904lowast

1 119904lowast

2)


119889ℎ and 120579ℎ119889= ℎ119889 If 119898lowast




1199061(119889 1199041(minus119889ℎ) 119889ℎ119898

minus119889ℎ) = 1 and 119906

1(ℎ 1199041(minus119889ℎ) 119889ℎ119898

minus119889ℎ) =


1(119889 1199041(minusℎ119889) ℎ119889119898

minusℎ119889) =

119909 and 1199061(ℎ 1199041(minusℎ119889) ℎ119889119898




minus119889ℎ) = 1

and 119906ℎ119889(119889 1199041(minusℎ119889) ℎ119889119898






1205792


1119898lowast(1205792)= 119889 If lowast

1205792




(119889 1199041(minus119898lowast(120579

2)) lowast

1205792

119898lowast

minus1205792

) lt 1199061205792

(119904lowast

1119898 1198981205792

119898lowast

minus1205792


1205792

(119889 1199041(minus119898lowast(120579

2)) lowast

1205792

119898lowast

minus1205792

) ge 1199061205792

(119904lowast

1119898

119898lowast

1205792

119898lowast

minus1205792



1205792

= 2is a self-

committing message



1199061(1198891198981205792

) = 1120572119889119889|119898(120579

2)+ 1120572119889ℎ|119898(120579

2)+ 119909120572ℎ119889|119898(120579

2)+

119909120572ℎℎ|119898(120579

2)

1199061(ℎ1198981205792

) = 119910120572119889119889|119898(120579

2)+ 025120572

119889ℎ|119898(1205792)+ 119910120572ℎ119889|119898(120579

2)+

025120572ℎℎ|119898(120579

2)

therefore 1199061(1198891198981205792

) ge 1199061(ℎ1198981205792


2)+ (075)120572

119889ℎ|119898(1205792)+ (119909 minus 119910)120572

ℎ119889|119898(1205792)+ (119909 minus

025)120572ℎℎ|119898(120579

2)ge 0



Acknowledgments


References
























1 Introduction











2 Hypotheses



119875A = 119889A minus 119868 + 119877

119875B = 119889B + 119898119868 minus 119877(1)








max [119901 (119891) (119887 + 119890) + 119889 (119890 119891) minus 119896] (2)


119891gt 0)








119891(119887 + 119890) + 119889

1015840

119891= 0 (3)
























3 Methods





4 Results




A investments

2

3

4

5

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(a)

B returns

1

2

3

4

5

6

7

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(b)

Cooperation index

045

05

055

06

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(c)









A investments

1

2

3

4

5

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(a)

B returns

1

2

3

4

5

6

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(b)

Cooperation index

045

05

055

06

MU

Base

line

No

ince

ntiv

e

Fixe

d in

cent

ive

A in

cent

ive

B in

cent

ive

(c)







5 Discussion





Baseline


B ret (prop) 009 001 016 001 056 1370 0059CI 050 000 051 000 098 3190 0118

No incentive


B ret (prop) 024 001 014 001 171 1890 0006CI 063 001 054 000 117 3960 0000

Fixed inc


B ret (prop) 017 001 010 001 170 1500 0153CI 054 001 052 000 104 2640 0319

A incentive


B ret (prop) 023 001 021 001 110 1340 0333CI 060 001 056 001 107 3650 0002

B incentive


B ret (prop) 019 001 018 001 106 1300 0387CI 058 001 055 000 105 3070 0071







1509lowastlowast







1012dagger



0024lowast



lowastlowast



lowast





lowastlowastlowast

























Appendices










B Instructions











































Acknowledgments


References


















































Sarah A Tulman







1 Introduction





















Candidate 119884 119902 minus 120579 119902









119901lowast

119894119895= int

infin

minusinfin

120591119894119895(119888) 119891 (119888) 119889119888 = int

119888lowast

119894119895

minusinfin

119891 (119888) 119889119888 = 119865 (119888lowast

119894119895) (1)




119888lowast

119894119895=

(119881119894119895minus 119881119894119898)


119894119895+ Pr (break tie)

119894119895)

(2)




119894119895) is




119894119895minus 119888119896 (3)





119896119894119895= 120587119896119894119895+120576119896119894119895

where 120576119896119894119895






119894119890120582120587119894 where 120587








119901119896vote119894119895 =

119890120582120587119896119894119895

119890120582120587119896119894119883 + 119890

120582120587119896119894119884 + 119890

120582120587119896119894119886

(4)


119901119896119894119895=

119890120582120587119896119894119895

1 + 119890120582120587119896119894119883 + 119890

120582120587119896119894119884

(5)

0

01

02

03

04

05

06

07

08

09

1

Pr (v

ote)

clowast





Candidate 119884 119902 119902


119894119883 119901119894119884 1 minus



likelihood function

log 119871 = sum119897

(1198731119883119897times ln (119901

1119883119897) + 1198731119884119897

times ln (1199011119884119897) + (num obs

1119897minus 1198731119883119897minus 1198731119884119897)


times ln (1199012119883119897) + 1198732119884119897times ln (119901

2119884119897)



(6)

where 119873119894119895119897












1119884minus 119888119896+ 120572 (7)





1119884minus 119888119896 (8)




1119884minus 119888119896+ 120572 (9)












119894119895























(a)


02996 03069 033331199011119884

0 00793 03241199012119883

0 00793 03241199012119884

02996 03069 033331198881119883

01221198881119884

01198882119883

01198882119884

0122

(b)


02608 03069 033331199011119884

0 00793 03241199012119883

0 00793 03241199012119884

02608 03069 033331198881119883

01221198881119884

01198882119883

01198882119884

0122





119894119883minus119901119894119884)




For 120582 = 7 120572 = 0 120572 = 01 120572 = 025 120572 = 05 120572 = 075 120572 = 1















(a)


(b)


(c)














1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1Pr

(vot

e)


(a)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)


(b)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)


(c)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)






119894119905


119894119905+ 1205733cost119894119905


119894119905

+ 1205737round


119894119905

+ 1205739( same type)

119894119905


119894119905


119894119905

+ 119908turnout119894 + 119906turnout119894119905


119894119905


119894119905+ 1205743cost119894119905


119894119905

+ 1205747round


119894119905

+ 1205749( same type)

119894119905


119894119905


119894119905

+ 119908favored119894 + 119906favored119894119905


119894119905


119894119905+ 1205793cost119894119905


119894119905









(006) (006) (001) (006) (006) (000) (002)



(000) (000) (000) (000) (000) (000) (000)


(008) (008) (001) (009) (009) (000) (002)



(002) (002) (001) (003) (003) (000) (001)




+ 1205797round


119894119905

+ 1205799( same type)

119894119905


119894119905


119894119905


(10)


















120582














0

02

04

06

08

1


Pr (v

ote)

0

02

04

06

08

1

Pr (v

ote)






0

02

04

06

08

1


Pr (v

ote)

0

02

04

06

08

1

Pr (v

ote)
































4 Conclusions









Appendices



119894119895) is the






Pr (pivotal)1119883

=1

2(

min(11987311198732)

sum

119899119884=1

119899119884

sum

1198992119884=0

119899119884minus1

sum

1198992119883=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198731minus 1

119899119884minus 1 minus 119899

2119883 119899119884minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119884minus1minus1198992119883

1119883119901119899119884minus1198992119884

1119884


+

min(11987311198732)minus1

sum

119899119884=0

119899119884

sum

1198992119884=0

119899119884

sum

1198992119883=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198732minus 1

119899119884minus 1198992119883 119899119884minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119884minus1198992119883

1119883119901119899119884minus1198992119884

1119884



)

(A1)


119895








119883= 119899119884




of 1198992119884

and 1198992119883


1119884and 119899

1119883mdashthe


119884minus 1198992119884) and (119899

119883minus 1198992119883)


and 1198991119883

In the first term ( 119873211989921198831198992119884

) 1199011198992119883

21198831199011198992119884



and 1198992119884




) 119901119899119884minus1minus1198992119883

1119883119901119899119884minus1198992119884

1119884(1 minus


be exactly 1198991119883

and 1198991119884




1119884

=1

2(

min(1198731 1198732)

sum

119899119883=1

119899119883

sum

1198992119883=0

119899119909minus1

sum

1198992119884=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198731minus 1


2119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119883minus1198992119883

1119883119901119899119883minus1minus1198992119884

1119884


+

min(1198731 1198732)minus1

sum

119899119883=0

119899119883

sum

1198992119883=0

119899119909

sum

1198992119884=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198732minus 1

119899119883minus 1198992119883 119899119883minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119883minus1198992119883

1119883119901119899119883minus1198992119884

1119884



)


Pr (pivotal)2119883

=1

2(

min(1198731 1198732)

sum

119899119884=1

119899119884

sum

1198991119884=0

119899119884minus1

sum

1198991119883=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119884minus 1 minus 119899

1119883 119899119884minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119884minus1minus1198991119883

2119883119901119899119884minus1198991119884

2119884


+

min(1198731 1198732)minus1

sum

119899119884=0

119899119884

sum

1198991119884=0

119899119884

sum

1198991119883=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119884minus 1198991119883 119899119884minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119884minus1198991119883

2119883119901119899119884minus1198991119884

2119884



)

Pr (pivotal)2119884

=1

2(

min(1198731 1198732)

sum

119899119883=1

119899119883

sum

1198991119883=0

119899119909minus1

sum

1198991119884=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1


1119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119883minus1198991119883

2119883119901119899119883minus1minus1198991119884

2119884


+

min(1198731 1198732)minus1

sum

119899119883=0

119899119883

sum

1198991119883=0

119899119909

sum

1198991119884=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119883minus 1198991119883 119899119883minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119883minus1198991119883

2119883119901119899119883minus1198991119884

2119884



)

(A2)



0

02

04

06

08

1

1 3 5 7 9 11 13 15 17 19

Pr (v

ote)

Round

MaleFemale























D Instructions


Page 1








Page 2






Page 3








Please select the best answer Your answer will bechecked for you on the page that follows



Submit answers





Page 5





Continue













Disclaimer




Acknowledgments


References
























1 Introduction













2 The Experiment










119894isin 119873+ and 0 le 119892

119894le 119908119894 The


given by

120587119894(119908119894 119892119894) = 2 (119908

119894minus 119892119894) + 120 + 119892

119894+ 119866minus119894 (1)



119906119894(119908119894 119892119894 120572119894) = 120587119894(119908119894 119892119894) minus 120572119894

(119908119894minus 119892119894)2

119908119894

(2)




119892NE119894

=2120572119894minus 1

2120572119894

119908119894 (3)








119902119894(119908119894 119892119894 120572119894 120583) =

exp 119906119894(119908119894 119892119894 120572119894) 120583

sum119908119894

119892119895=0exp 119906

119894(119908119894 119892119895 120572119894) 120583

(4)



119894-




0 119908119894 Let q

119894(119908119894 g119894 120572119894 120583)119894isin119875



119894will also



119894(119908119894 g119894 120572119894 120583)119894isin119875




NE119894 120572119894 0) = 1 and 119902

119894(119908119894 119892119894 120572119894 0) = 0 forall119892

119894= 119892

NE119894)



119871119894(119908119894 g119894 120572 120583) =

119899

prod119908119894

119892119895=0119899 (119892119895)

119908119894

prod

119892119896=0

119902QRE119894

(119908119894 119892119896 120572 120583)

119899(119892119896)

(5)










119908119894= 160 2453 3500 3503 5568 3767

119908119894= 200 4750 4375 4376 6557 3948

119908119894= 240 4557 5250 5250 7486 4036



119871119894(119908119894 g119894 1205721 1205722 120572

119878 1205741 1205742 120574

119878 120583)

=

119878

sum

119904=1

120574119904

119899

prod119908119894

119892119895=0119899 (119892119895)

119908119894

prod

119892119896=0

119902QRE119894

(119908119894 119892119896 120572119904 120583)119899(119892119896)

(6)

where 1205741 1205742 120574

119878 with sum

119878

119904=1120574119904




119878



119904le

120572119904+1



4 Results




+(LR =









119908119894= 160 1763 2346 2404 4168 2601

119908119894= 200 3528 2932 2971 4830 2630

119908119894= 240 2970 3519 3545 5460 2639



120583 1205721 and 120572

2(119899 = 20) 120583 120572

1 and 120572

2(119899 = 5) 120583 120572

1 1205722 and 120572

3(119899 = 20) 120583 120572

1 1205722 and 120572

3(119899 = 5)

120583 2850 [2588 3126] 1507 [1290 1764] 2214 [2056 2395] 1425 [1204 1685]1205721

0 [0 001] 0 [0 002] 0 [0 001] 0 [0 002]1205722

053 [046 060] 054 [047 061] 043 [039 046] 048 [040 056]1205723

104 [092 116] 076 [053 101]1205741

066 [053 078] 063 [050 075] 041 [033 046] 059 [049 064]1205742



119908119894= 160 3604 2463 3678 2468

119908119894= 200 3901 2677 4083 2699

119908119894= 240 4148 2873 4445 2913










1= 0 120572

2= 038 120572

3= 061 120572

4= 104

1205741= 039 120574

2= 042 and 120574



Table 4








3= 104






5 Conclusions







Appendix






Acknowledgments


References
















































1 Introduction














2 Literature Review






















119909 (119888) = 120574minus1(

3

sum

119895=1

120587119895 int

1

119888

minus

1

119886

1198651015840119895 (119886) 119889119886) (1)

where

1198651015840119895 (119886) =

(3 minus 1)

(119895 minus 1) (3 minus 119895)


119865 (119886)119895minus2

sdot 1198651015840(119886) ((1 minus 3) 119865 (119886) + (119895 minus 1))

(2)


119865119895 (119886) =(3 minus 1)

(119895 minus 1) (3 minus 119895)

(1 minus 119865 (119886))3minus119895

119865 (119886)119895minus1

(3)








119904120572119894 = 120572120587119894 + (1 minus 120572) (

2

3


(4)






120572 = 0

120572 = 05

120572 = 1

AR

E

D

0

12

26 33 100

120587120572

B

120587120572

C

120587120572

A



119863= (26 12 12)


119877= 119904119877119860 119904119877119861 119904119877119862 where 119904

119877119894 = 120587119894 forall119894 isin













120597119904120572119894

120597120572

= 2 (120587119894 minus 120587) (5)

























119860 119861 119862Coefficient of









5 Results








signBaseline


Constant 3170 0200 00001198772 00132

119865 376 0015Veto

Mean minus0313 0208 0137 +



Constant 2729 0180 00001198772 0008


915

0

2

4

6

8

Mea

n gr

oup

effor

t

10


749 763

875



0

003001

021

016

005

01

015

02

Def

ault

rate










029 029

042

050

003

010

039047

SkewedMajority

RandomContest

Unanimity

0

01

02

03

04

05

Mea

n of120572









6 Conclusion





Symmetric (0057) (0035)119875 le 001 119875 = 0022

0079 0131Right-skewed (0029) (0031)

119875 le 001 119875 le 001





Appendices

A Further Tables


B Instructions









Prizescheme



payoff119894 119888119894 1st 2nd 3rd 119864(120587119894)

119909119894 119895 120587119895

1 04 562 375 62 10 38 A 33 15 181 2 06 250 500 250 4 21 B 17 13 4

3 08 62 375 562 1 9 C 0 8 minus81 04 562 375 62 19 77 A 67 31 36

2 2 06 250 500 250 8 42 B 33 25 83 08 62 375 562 2 18 C 0 15 minus151 04 562 375 62 13 20 A 25 8 17

3 2 06 250 500 250 10 11 B 17 7 103 08 62 375 562 8 5 C 8 4 41 04 562 375 62 27 38 A 50 15 35

4 2 06 250 500 250 21 21 B 33 12 213 08 62 375 562 18 9 C 17 7 101 04 562 375 62 12 29 A 36 11 25

5 2 06 250 500 250 8 10 B 7 6 13 08 62 375 562 7 2 C 7 2 51 04 562 375 62 24 57 A 72 23 49

6 2 06 250 500 250 16 20 B 14 12 23 08 62 375 562 14 4 C 14 3 111 04 562 375 62 14 14 A 26 6 20

7 2 06 250 500 250 13 5 B 12 3 93 08 62 375 562 12 1 C 12 1 111 04 562 375 62 29 28 A 52 11 41

8 2 06 250 500 250 25 10 B 24 6 183 08 62 375 562 24 2 C 24 2 22









Effort 120572

119860 119861 119862 119860 119861 119862


Symmetric 439 314 163 059 051 041(022) (020) (032) (003) (003) (003)

Right-skewed 426 238 085 040 010 010(017) (020) (030) (004) (002) (002)

Unanimity rule

Symmetric 392 255 116 048 049 047(021) (021) (031) (002) (002) (002)

Right-skewed 424 280 172 043 041 043(026) (022) (035) (002) (003) (002)








+


(B2)



(3 + 5 + 10) sdot 05

3

= 15 + 3 = 45

(B3)







(4 + 6 + 10) sdot 045

3

= 22 + 3 = 52

(B4)





Acknowledgments




Endnotes



















References












































































Editorial Board













Contents





















Acknowledgment



Jens Groszliger

References














1 Introduction















Player 2119889 ℎ

Player 1 119889 1 1 119909 119910

ℎ 119910 119909 025 025












(119904lowast

1 119904lowast

2) (119880

lowast

1 119880lowast

2) Pareto optimal

C1 (119889 119889) (1 1) Yes

C2 (119889 119889) (1 1) Yes(ℎ ℎ) (025 025) No

C3 (119889 ℎ) (119909 119910) Yes(ℎ 119889) (119910 119909) Yes

C4 (ℎ ℎ) (025 025) No










(a) C1 119910 minus 119909 = 0

2119889 ℎ

1 119889 1 1 05 05ℎ 05 05 025 025

(b) C2 119910 minus 119909 = 05

2119889 ℎ

1 119889 1 1 0 05ℎ 05 0 025 025

(c) C3 119910 minus 119909 = 1

2119889 ℎ

1 119889 1 1 05 15ℎ 15 05 025 025

(d) C4 119910 minus 119909 = 15

2119889 ℎ

1 119889 1 1 0 15ℎ 15 0 025 025







d h

d h d h

1

1

x

y

y

x

025

025

P1

P2P2

u1

u2







2= 11990421198891199042ℎrepresents







119894(1199041 1199042)

is defined by 119906119894(119889 11990421198891199042ℎ) = 119906

119894(119889 1199042119889) and 119906

119894(ℎ 11990421198891199042ℎ) =




2) is a SPNE


2) ge 1199062(1199041 1199042) for every 119904


1isin 1198781

and 1199061(119904lowast

1 119904lowast

2) ge 1199061(1199041 119904lowast

2) for every 119904

1isin 1198781


2119889119904lowast






2119889119904lowast




(119904lowast

1 119904lowast

2) (119906

lowast

1 119906lowast

2) Pareto optimal




2) is a SPNE in


1 119904lowast

2) = 119909

and 1199062(119904lowast

1 119904lowast

2) = 119910









1119898 119898lowast

2) ge 119906

2(119904lowast

1 119904lowast

2) where

1199061(119904lowast

1119898 119898lowast

2) ge 119906

1(1199041 119898lowast

2) for every 119904


1119898is






1 119906lowast

2) Threat (119906

1 1199062) Promise (119906

1 1199062)




2= 119904lowast

2


only if 1199062(119889119898lowast

2) = 1199062(119889 119904lowast

2) and 119906

2(ℎ119898lowast

2) lt 1199062(ℎ 119904lowast

2)


only if 1199062(119889119898lowast

2) lt 1199062(119889 119904lowast

2)



119894(119904119894 119889) gt





1= 119889




2such that 119906

2(119904lowast

1119898 119898lowast

2) gt 119906

2(119904lowast

1 119904lowast

2) if and

only if 119910 gt 1
















1isin 119889 ℎ





1205792isinΘ2

119901(1205792) = 1 In the second





2= (119898119889119889 119898119889ℎ 119898ℎ119889 119898ℎℎ) is a


2isin






1205792

= 11990421198891199042ℎ


2 and119898

2= (1198981205792

119898minus1205792


2




2isin 1205792 where 119904

1119889119889is




1199041(minus1198981205792) 1198981205792

119898minus1205792

) = 119906119894(1199041 1199042) for 119904

11198981205792

= 1199041and

1205792= 1199042


1205792|1198982




1205792isinΘ2

1205721205792|1198982



2= 119889119889 In the






2= (119898

lowast

119889119889 119898lowast

119889ℎ 119898lowast

ℎ119889 119898lowast

ℎℎ) a


1119889119889 119904lowast

1119889ℎ 119904lowast

1ℎ119889 119904lowast

1ℎℎ) and a





1198981205792isinΘ2

1199061205792

(119904lowast

1119898 1198981205792

119898minus1205792

)

(2) 119904lowast1119898


1isin1198781

sum1205792isinΘ2

1205721205792|1198982

sdot 1199061(11990411198981205792

1199041(minus1198981205792)

119898lowast

2)

(3) 120572lowast1205792|1198982


2 For all 120579

2who play the


1205792|119898lowast

2

=

1199011205792


2


1205792




1205792|1198982



u1

u2

d

h

d

h

d

h

d

h

x

y

x

y x

y

x

y

N

1 1

1

1

11

1

1

1

1

1 1

x y

x y

025 025

025

025

025

025

025 025

120572hh120572dh

120572dh

120572dh

120572dh

120572hd

120572hd

120572hd

120572hd

120572dd

120572dd

120572dd

120572dd

120579hh

120579dh

120579hd

120579dd

mhh

mdh

mhd

mdd

d

h

d

h

d

h

d

h

d h d h d h d h

d hd hd hd h

120572hh

120572hh120572hh

u1 u2

y x

y x

1 1

1 1

x y

x y

025 025

025 025

u1 u2

y x

y x

u1

u2 x

y

x

y x

y

x

y

1

1

1

1

025

025

025

025



2



2= 1205792 then 120572

1205792|2



1205792|1198982

= 1





2 then 119904lowast

11198981205792

= 119889


1205792


1205792






119889119889 119898lowast

119889ℎ 119898lowast

ℎ119889 119898lowast

ℎℎ) (119904

lowast

1119889119889 119904lowast

1119889ℎ 119904lowast

1ℎ119889 119904lowast

1ℎℎ) (120572

lowast

119889119889|119889119889 120572lowast

119889ℎ|119889ℎ 120572lowast

ℎ119889|ℎ119889 120572lowast

ℎℎ|ℎℎ)

C1 (119889119889 119889ℎ 119889119889 ℎℎ) (119889 119889 ℎ 119889) (119901119889119889

(119901119889119889+ 119901ℎ119889) 1 1 1)

C2 (119889119889 119889ℎ 119889119889 119889ℎ) (119889 119889 ℎ ℎ) (119901119889119889

(119901119889119889+ 119901ℎ119889)119901119889ℎ

(119901119889ℎ+ 119901ℎℎ) 1 1)

C3 (119889ℎ 119889ℎ ℎℎ ℎℎ) (ℎ 119889 ℎ 119889) (1119901119889ℎ

(119901119889119889+ 119901119889ℎ) 1119901ℎℎ

(119901ℎ119889+ 119901ℎℎ))

C4 (119889ℎ 119889ℎ 119889ℎ 119889ℎ) (ℎ 119889 ℎ ℎ) (1119901119889ℎ

(119901119889119889+ 119901119889ℎ+ 119901ℎ119889+ 119901ℎℎ) 1 1)









1205792|lowast

2(1205792)= 1 lowast

1205792


1205792

(119904lowast

1119898 lowast

1205792

lowast

minus1205792

) ge 1199061205792

(119904lowast

1119898

1198981205792

lowast

minus1205792

) for every1198981205792

isin Θ2




lowast

1205792



1205792




= 2is a








C1 120572lowast

119889119889|119889119889ge

120572lowast

ℎ119889|119889119889(119910 minus 119909)


C2 120572lowast

119889ℎ|119889ℎge

120572lowast

ℎℎ|119889ℎ(025 minus 119909)

075120572lowast

119889119889|119889119889ge

120572lowast

ℎ119889|119889119889(119910 minus 119909)

(1 minus 119910)

C3 Lie 120572lowast

ℎℎ|ℎℎge

120572lowast

ℎ119889|ℎℎ(119910 minus 119909)

(119909 minus 025)120572lowast

119889ℎ|119889ℎge

120572lowast

119889119889|119889ℎ(119910 minus 1)

075

C4 Lie 120572lowast

119889ℎ|119889ℎge

120572lowast

119889119889|119889ℎ(119910 minus 1) + 120572

lowast

ℎ119889|119889ℎ(119910 minus 119909) + 120572

lowast

ℎℎ|119889ℎ(025 minus 119909)

075






only if (1minus119910)120572119889119889|119898(120579

2)+(075)120572

119889ℎ|119898(1205792)+(119909minus119910)120572

ℎ119889|119898(1205792)+(119909minus

025)120572ℎℎ|119898(120579

2)ge 0





5 Conclusions










Appendix


1 119904lowast

2) = 119909 and

1199062(119904lowast

1 119904lowast

2) = 119910 then 119910 ge 1 If 119904lowast

2= ℎ119889 then 119906

1(119889 ℎ119889) ge


2= ℎℎ

then 1199061(119889 ℎℎ) ge 119906


1199062(ℎ ℎℎ) ge 119906



1 119904lowast

2) = 119909 and

1199062(119904lowast

1 119904lowast

2) = 119910



1 lowast

2) ge 119906

2(119904lowast

1 119904lowast

2)


2= 119904lowast

2


2) ge 1199062(119904lowast

1 119904lowast

2) If lowast

2= 119904lowast


2is


2) ge 1199062(119904lowast

1 119904lowast

2) then

119904lowast

1= 119889 or 119904lowast



2(ℎ lowast

2) lt

1199062(119889 119904lowast

2) If 119904lowast

1= ℎ and


2(ℎ lowast

2) lt

1199062(119904lowast

1 119904lowast

2) If 119904lowast

1= ℎ and


1199062(ℎ lowast

2) ge 119906

2(ℎ 119904lowast

2) and 119906

2(119889 lowast

2) lt 119906

2(119889 119904lowast

2) The C1



1= 119889



1 119904lowast

2) = 119910 and


1(ℎ 119889ℎ) then 119898

2= 119889ℎ is a


2(119904lowast

1 119904lowast

2)

If 1199062(119889 119889ℎ) gt 119906

2(119904lowast

1 119904lowast

2) then 1 gt 119906

2(119904lowast

1 119904lowast

2) to satisfy this


1 119904lowast

2) = (ℎ ℎ119904

lowast


2= ℎ119904lowast


then 1199062(119889 ℎ119904lowast

2ℎ) gt 1199062(119889 119889119904lowast



2= ℎ119904lowast


1(119904lowast

1 119904lowast

2) =


2(119904lowast

1 119904lowast






1119898 119898lowast

2) gt 1199062(119904lowast

1 119904lowast

2)


119889ℎ and 120579ℎ119889= ℎ119889 If 119898lowast




1199061(119889 1199041(minus119889ℎ) 119889ℎ119898

minus119889ℎ) = 1 and 119906

1(ℎ 1199041(minus119889ℎ) 119889ℎ119898

minus119889ℎ) =


1(119889 1199041(minusℎ119889) ℎ119889119898

minusℎ119889) =

119909 and 1199061(ℎ 1199041(minusℎ119889) ℎ119889119898




minus119889ℎ) = 1

and 119906ℎ119889(119889 1199041(minusℎ119889) ℎ119889119898






1205792


1119898lowast(1205792)= 119889 If lowast

1205792




(119889 1199041(minus119898lowast(120579

2)) lowast

1205792

119898lowast

minus1205792

) lt 1199061205792

(119904lowast

1119898 1198981205792

119898lowast

minus1205792


1205792

(119889 1199041(minus119898lowast(120579

2)) lowast

1205792

119898lowast

minus1205792

) ge 1199061205792

(119904lowast

1119898

119898lowast

1205792

119898lowast

minus1205792



1205792

= 2is a self-

committing message



1199061(1198891198981205792

) = 1120572119889119889|119898(120579

2)+ 1120572119889ℎ|119898(120579

2)+ 119909120572ℎ119889|119898(120579

2)+

119909120572ℎℎ|119898(120579

2)

1199061(ℎ1198981205792

) = 119910120572119889119889|119898(120579

2)+ 025120572

119889ℎ|119898(1205792)+ 119910120572ℎ119889|119898(120579

2)+

025120572ℎℎ|119898(120579

2)

therefore 1199061(1198891198981205792

) ge 1199061(ℎ1198981205792


2)+ (075)120572

119889ℎ|119898(1205792)+ (119909 minus 119910)120572

ℎ119889|119898(1205792)+ (119909 minus

025)120572ℎℎ|119898(120579

2)ge 0



Acknowledgments


References
























1 Introduction











2 Hypotheses



119875A = 119889A minus 119868 + 119877

119875B = 119889B + 119898119868 minus 119877(1)








max [119901 (119891) (119887 + 119890) + 119889 (119890 119891) minus 119896] (2)


119891gt 0)








119891(119887 + 119890) + 119889

1015840

119891= 0 (3)
























3 Methods





4 Results




A investments

2

3

4

5

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(a)

B returns

1

2

3

4

5

6

7

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(b)

Cooperation index

045

05

055

06

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(c)









A investments

1

2

3

4

5

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(a)

B returns

1

2

3

4

5

6

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(b)

Cooperation index

045

05

055

06

MU

Base

line

No

ince

ntiv

e

Fixe

d in

cent

ive

A in

cent

ive

B in

cent

ive

(c)







5 Discussion





Baseline


B ret (prop) 009 001 016 001 056 1370 0059CI 050 000 051 000 098 3190 0118

No incentive


B ret (prop) 024 001 014 001 171 1890 0006CI 063 001 054 000 117 3960 0000

Fixed inc


B ret (prop) 017 001 010 001 170 1500 0153CI 054 001 052 000 104 2640 0319

A incentive


B ret (prop) 023 001 021 001 110 1340 0333CI 060 001 056 001 107 3650 0002

B incentive


B ret (prop) 019 001 018 001 106 1300 0387CI 058 001 055 000 105 3070 0071







1509lowastlowast







1012dagger



0024lowast



lowastlowast



lowast





lowastlowastlowast

























Appendices










B Instructions











































Acknowledgments


References


















































Sarah A Tulman







1 Introduction





















Candidate 119884 119902 minus 120579 119902









119901lowast

119894119895= int

infin

minusinfin

120591119894119895(119888) 119891 (119888) 119889119888 = int

119888lowast

119894119895

minusinfin

119891 (119888) 119889119888 = 119865 (119888lowast

119894119895) (1)




119888lowast

119894119895=

(119881119894119895minus 119881119894119898)


119894119895+ Pr (break tie)

119894119895)

(2)




119894119895) is




119894119895minus 119888119896 (3)





119896119894119895= 120587119896119894119895+120576119896119894119895

where 120576119896119894119895






119894119890120582120587119894 where 120587








119901119896vote119894119895 =

119890120582120587119896119894119895

119890120582120587119896119894119883 + 119890

120582120587119896119894119884 + 119890

120582120587119896119894119886

(4)


119901119896119894119895=

119890120582120587119896119894119895

1 + 119890120582120587119896119894119883 + 119890

120582120587119896119894119884

(5)

0

01

02

03

04

05

06

07

08

09

1

Pr (v

ote)

clowast





Candidate 119884 119902 119902


119894119883 119901119894119884 1 minus



likelihood function

log 119871 = sum119897

(1198731119883119897times ln (119901

1119883119897) + 1198731119884119897

times ln (1199011119884119897) + (num obs

1119897minus 1198731119883119897minus 1198731119884119897)


times ln (1199012119883119897) + 1198732119884119897times ln (119901

2119884119897)



(6)

where 119873119894119895119897












1119884minus 119888119896+ 120572 (7)





1119884minus 119888119896 (8)




1119884minus 119888119896+ 120572 (9)












119894119895























(a)


02996 03069 033331199011119884

0 00793 03241199012119883

0 00793 03241199012119884

02996 03069 033331198881119883

01221198881119884

01198882119883

01198882119884

0122

(b)


02608 03069 033331199011119884

0 00793 03241199012119883

0 00793 03241199012119884

02608 03069 033331198881119883

01221198881119884

01198882119883

01198882119884

0122





119894119883minus119901119894119884)




For 120582 = 7 120572 = 0 120572 = 01 120572 = 025 120572 = 05 120572 = 075 120572 = 1















(a)


(b)


(c)














1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1Pr

(vot

e)


(a)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)


(b)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)


(c)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)






119894119905


119894119905+ 1205733cost119894119905


119894119905

+ 1205737round


119894119905

+ 1205739( same type)

119894119905


119894119905


119894119905

+ 119908turnout119894 + 119906turnout119894119905


119894119905


119894119905+ 1205743cost119894119905


119894119905

+ 1205747round


119894119905

+ 1205749( same type)

119894119905


119894119905


119894119905

+ 119908favored119894 + 119906favored119894119905


119894119905


119894119905+ 1205793cost119894119905


119894119905









(006) (006) (001) (006) (006) (000) (002)



(000) (000) (000) (000) (000) (000) (000)


(008) (008) (001) (009) (009) (000) (002)



(002) (002) (001) (003) (003) (000) (001)




+ 1205797round


119894119905

+ 1205799( same type)

119894119905


119894119905


119894119905


(10)


















120582














0

02

04

06

08

1


Pr (v

ote)

0

02

04

06

08

1

Pr (v

ote)






0

02

04

06

08

1


Pr (v

ote)

0

02

04

06

08

1

Pr (v

ote)
































4 Conclusions









Appendices



119894119895) is the






Pr (pivotal)1119883

=1

2(

min(11987311198732)

sum

119899119884=1

119899119884

sum

1198992119884=0

119899119884minus1

sum

1198992119883=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198731minus 1

119899119884minus 1 minus 119899

2119883 119899119884minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119884minus1minus1198992119883

1119883119901119899119884minus1198992119884

1119884


+

min(11987311198732)minus1

sum

119899119884=0

119899119884

sum

1198992119884=0

119899119884

sum

1198992119883=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198732minus 1

119899119884minus 1198992119883 119899119884minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119884minus1198992119883

1119883119901119899119884minus1198992119884

1119884



)

(A1)


119895








119883= 119899119884




of 1198992119884

and 1198992119883


1119884and 119899

1119883mdashthe


119884minus 1198992119884) and (119899

119883minus 1198992119883)


and 1198991119883

In the first term ( 119873211989921198831198992119884

) 1199011198992119883

21198831199011198992119884



and 1198992119884




) 119901119899119884minus1minus1198992119883

1119883119901119899119884minus1198992119884

1119884(1 minus


be exactly 1198991119883

and 1198991119884




1119884

=1

2(

min(1198731 1198732)

sum

119899119883=1

119899119883

sum

1198992119883=0

119899119909minus1

sum

1198992119884=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198731minus 1


2119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119883minus1198992119883

1119883119901119899119883minus1minus1198992119884

1119884


+

min(1198731 1198732)minus1

sum

119899119883=0

119899119883

sum

1198992119883=0

119899119909

sum

1198992119884=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198732minus 1

119899119883minus 1198992119883 119899119883minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119883minus1198992119883

1119883119901119899119883minus1198992119884

1119884



)


Pr (pivotal)2119883

=1

2(

min(1198731 1198732)

sum

119899119884=1

119899119884

sum

1198991119884=0

119899119884minus1

sum

1198991119883=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119884minus 1 minus 119899

1119883 119899119884minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119884minus1minus1198991119883

2119883119901119899119884minus1198991119884

2119884


+

min(1198731 1198732)minus1

sum

119899119884=0

119899119884

sum

1198991119884=0

119899119884

sum

1198991119883=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119884minus 1198991119883 119899119884minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119884minus1198991119883

2119883119901119899119884minus1198991119884

2119884



)

Pr (pivotal)2119884

=1

2(

min(1198731 1198732)

sum

119899119883=1

119899119883

sum

1198991119883=0

119899119909minus1

sum

1198991119884=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1


1119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119883minus1198991119883

2119883119901119899119883minus1minus1198991119884

2119884


+

min(1198731 1198732)minus1

sum

119899119883=0

119899119883

sum

1198991119883=0

119899119909

sum

1198991119884=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119883minus 1198991119883 119899119883minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119883minus1198991119883

2119883119901119899119883minus1198991119884

2119884



)

(A2)



0

02

04

06

08

1

1 3 5 7 9 11 13 15 17 19

Pr (v

ote)

Round

MaleFemale























D Instructions


Page 1








Page 2






Page 3











Submit answers









Continue













Disclaimer




Acknowledgments


References
























1 Introduction













2 The Experiment










119894isin 119873+ and 0 le 119892

119894le 119908119894 The


given by

120587119894(119908119894 119892119894) = 2 (119908

119894minus 119892119894) + 120 + 119892

119894+ 119866minus119894 (1)



119906119894(119908119894 119892119894 120572119894) = 120587119894(119908119894 119892119894) minus 120572119894

(119908119894minus 119892119894)2

119908119894

(2)




119892NE119894

=2120572119894minus 1

2120572119894

119908119894 (3)








119902119894(119908119894 119892119894 120572119894 120583) =

exp 119906119894(119908119894 119892119894 120572119894) 120583

sum119908119894

119892119895=0exp 119906

119894(119908119894 119892119895 120572119894) 120583

(4)



119894-




0 119908119894 Let q

119894(119908119894 g119894 120572119894 120583)119894isin119875



119894will also



119894(119908119894 g119894 120572119894 120583)119894isin119875




NE119894 120572119894 0) = 1 and 119902

119894(119908119894 119892119894 120572119894 0) = 0 forall119892

119894= 119892

NE119894)



119871119894(119908119894 g119894 120572 120583) =

119899

prod119908119894

119892119895=0119899 (119892119895)

119908119894

prod

119892119896=0

119902QRE119894

(119908119894 119892119896 120572 120583)

119899(119892119896)

(5)










119908119894= 160 2453 3500 3503 5568 3767

119908119894= 200 4750 4375 4376 6557 3948

119908119894= 240 4557 5250 5250 7486 4036



119871119894(119908119894 g119894 1205721 1205722 120572

119878 1205741 1205742 120574

119878 120583)

=

119878

sum

119904=1

120574119904

119899

prod119908119894

119892119895=0119899 (119892119895)

119908119894

prod

119892119896=0

119902QRE119894

(119908119894 119892119896 120572119904 120583)119899(119892119896)

(6)

where 1205741 1205742 120574

119878 with sum

119878

119904=1120574119904




119878



119904le

120572119904+1



4 Results




+(LR =









119908119894= 160 1763 2346 2404 4168 2601

119908119894= 200 3528 2932 2971 4830 2630

119908119894= 240 2970 3519 3545 5460 2639



120583 1205721 and 120572

2(119899 = 20) 120583 120572

1 and 120572

2(119899 = 5) 120583 120572

1 1205722 and 120572

3(119899 = 20) 120583 120572

1 1205722 and 120572

3(119899 = 5)

120583 2850 [2588 3126] 1507 [1290 1764] 2214 [2056 2395] 1425 [1204 1685]1205721

0 [0 001] 0 [0 002] 0 [0 001] 0 [0 002]1205722

053 [046 060] 054 [047 061] 043 [039 046] 048 [040 056]1205723

104 [092 116] 076 [053 101]1205741

066 [053 078] 063 [050 075] 041 [033 046] 059 [049 064]1205742



119908119894= 160 3604 2463 3678 2468

119908119894= 200 3901 2677 4083 2699

119908119894= 240 4148 2873 4445 2913










1= 0 120572

2= 038 120572

3= 061 120572

4= 104

1205741= 039 120574

2= 042 and 120574



Table 4








3= 104






5 Conclusions







Appendix






Acknowledgments


References
















































1 Introduction














2 Literature Review






















119909 (119888) = 120574minus1(

3

sum

119895=1

120587119895 int

1

119888

minus

1

119886

1198651015840119895 (119886) 119889119886) (1)

where

1198651015840119895 (119886) =

(3 minus 1)

(119895 minus 1) (3 minus 119895)


119865 (119886)119895minus2

sdot 1198651015840(119886) ((1 minus 3) 119865 (119886) + (119895 minus 1))

(2)


119865119895 (119886) =(3 minus 1)

(119895 minus 1) (3 minus 119895)

(1 minus 119865 (119886))3minus119895

119865 (119886)119895minus1

(3)








119904120572119894 = 120572120587119894 + (1 minus 120572) (

2

3


(4)






120572 = 0

120572 = 05

120572 = 1

AR

E

D

0

12

26 33 100

120587120572

B

120587120572

C

120587120572

A



119863= (26 12 12)


119877= 119904119877119860 119904119877119861 119904119877119862 where 119904

119877119894 = 120587119894 forall119894 isin













120597119904120572119894

120597120572

= 2 (120587119894 minus 120587) (5)

























119860 119861 119862Coefficient of









5 Results








signBaseline


Constant 3170 0200 00001198772 00132

119865 376 0015Veto

Mean minus0313 0208 0137 +



Constant 2729 0180 00001198772 0008


915

0

2

4

6

8

Mea

n gr

oup

effor

t

10


749 763

875



0

003001

021

016

005

01

015

02

Def

ault

rate










029 029

042

050

003

010

039047

SkewedMajority

RandomContest

Unanimity

0

01

02

03

04

05

Mea

n of120572









6 Conclusion





Symmetric (0057) (0035)119875 le 001 119875 = 0022

0079 0131Right-skewed (0029) (0031)

119875 le 001 119875 le 001





Appendices

A Further Tables


B Instructions









Prizescheme



payoff119894 119888119894 1st 2nd 3rd 119864(120587119894)

119909119894 119895 120587119895

1 04 562 375 62 10 38 A 33 15 181 2 06 250 500 250 4 21 B 17 13 4

3 08 62 375 562 1 9 C 0 8 minus81 04 562 375 62 19 77 A 67 31 36

2 2 06 250 500 250 8 42 B 33 25 83 08 62 375 562 2 18 C 0 15 minus151 04 562 375 62 13 20 A 25 8 17

3 2 06 250 500 250 10 11 B 17 7 103 08 62 375 562 8 5 C 8 4 41 04 562 375 62 27 38 A 50 15 35

4 2 06 250 500 250 21 21 B 33 12 213 08 62 375 562 18 9 C 17 7 101 04 562 375 62 12 29 A 36 11 25

5 2 06 250 500 250 8 10 B 7 6 13 08 62 375 562 7 2 C 7 2 51 04 562 375 62 24 57 A 72 23 49

6 2 06 250 500 250 16 20 B 14 12 23 08 62 375 562 14 4 C 14 3 111 04 562 375 62 14 14 A 26 6 20

7 2 06 250 500 250 13 5 B 12 3 93 08 62 375 562 12 1 C 12 1 111 04 562 375 62 29 28 A 52 11 41

8 2 06 250 500 250 25 10 B 24 6 183 08 62 375 562 24 2 C 24 2 22









Effort 120572

119860 119861 119862 119860 119861 119862


Symmetric 439 314 163 059 051 041(022) (020) (032) (003) (003) (003)

Right-skewed 426 238 085 040 010 010(017) (020) (030) (004) (002) (002)

Unanimity rule

Symmetric 392 255 116 048 049 047(021) (021) (031) (002) (002) (002)

Right-skewed 424 280 172 043 041 043(026) (022) (035) (002) (003) (002)








+


(B2)



(3 + 5 + 10) sdot 05

3

= 15 + 3 = 45

(B3)







(4 + 6 + 10) sdot 045

3

= 22 + 3 = 52

(B4)





Acknowledgments




Endnotes



















References





















































































Contents





















Acknowledgment



Jens Groszliger

References














1 Introduction















Player 2119889 ℎ

Player 1 119889 1 1 119909 119910

ℎ 119910 119909 025 025












(119904lowast

1 119904lowast

2) (119880

lowast

1 119880lowast

2) Pareto optimal

C1 (119889 119889) (1 1) Yes

C2 (119889 119889) (1 1) Yes(ℎ ℎ) (025 025) No

C3 (119889 ℎ) (119909 119910) Yes(ℎ 119889) (119910 119909) Yes

C4 (ℎ ℎ) (025 025) No










(a) C1 119910 minus 119909 = 0

2119889 ℎ

1 119889 1 1 05 05ℎ 05 05 025 025

(b) C2 119910 minus 119909 = 05

2119889 ℎ

1 119889 1 1 0 05ℎ 05 0 025 025

(c) C3 119910 minus 119909 = 1

2119889 ℎ

1 119889 1 1 05 15ℎ 15 05 025 025

(d) C4 119910 minus 119909 = 15

2119889 ℎ

1 119889 1 1 0 15ℎ 15 0 025 025







d h

d h d h

1

1

x

y

y

x

025

025

P1

P2P2

u1

u2







2= 11990421198891199042ℎrepresents







119894(1199041 1199042)

is defined by 119906119894(119889 11990421198891199042ℎ) = 119906

119894(119889 1199042119889) and 119906

119894(ℎ 11990421198891199042ℎ) =




2) is a SPNE


2) ge 1199062(1199041 1199042) for every 119904


1isin 1198781

and 1199061(119904lowast

1 119904lowast

2) ge 1199061(1199041 119904lowast

2) for every 119904

1isin 1198781


2119889119904lowast






2119889119904lowast




(119904lowast

1 119904lowast

2) (119906

lowast

1 119906lowast

2) Pareto optimal




2) is a SPNE in


1 119904lowast

2) = 119909

and 1199062(119904lowast

1 119904lowast

2) = 119910









1119898 119898lowast

2) ge 119906

2(119904lowast

1 119904lowast

2) where

1199061(119904lowast

1119898 119898lowast

2) ge 119906

1(1199041 119898lowast

2) for every 119904


1119898is






1 119906lowast

2) Threat (119906

1 1199062) Promise (119906

1 1199062)




2= 119904lowast

2


only if 1199062(119889119898lowast

2) = 1199062(119889 119904lowast

2) and 119906

2(ℎ119898lowast

2) lt 1199062(ℎ 119904lowast

2)


only if 1199062(119889119898lowast

2) lt 1199062(119889 119904lowast

2)



119894(119904119894 119889) gt





1= 119889




2such that 119906

2(119904lowast

1119898 119898lowast

2) gt 119906

2(119904lowast

1 119904lowast

2) if and

only if 119910 gt 1
















1isin 119889 ℎ





1205792isinΘ2

119901(1205792) = 1 In the second





2= (119898119889119889 119898119889ℎ 119898ℎ119889 119898ℎℎ) is a


2isin






1205792

= 11990421198891199042ℎ


2 and119898

2= (1198981205792

119898minus1205792


2




2isin 1205792 where 119904

1119889119889is




1199041(minus1198981205792) 1198981205792

119898minus1205792

) = 119906119894(1199041 1199042) for 119904

11198981205792

= 1199041and

1205792= 1199042


1205792|1198982




1205792isinΘ2

1205721205792|1198982



2= 119889119889 In the






2= (119898

lowast

119889119889 119898lowast

119889ℎ 119898lowast

ℎ119889 119898lowast

ℎℎ) a


1119889119889 119904lowast

1119889ℎ 119904lowast

1ℎ119889 119904lowast

1ℎℎ) and a





1198981205792isinΘ2

1199061205792

(119904lowast

1119898 1198981205792

119898minus1205792

)

(2) 119904lowast1119898


1isin1198781

sum1205792isinΘ2

1205721205792|1198982

sdot 1199061(11990411198981205792

1199041(minus1198981205792)

119898lowast

2)

(3) 120572lowast1205792|1198982


2 For all 120579

2who play the


1205792|119898lowast

2

=

1199011205792


2


1205792




1205792|1198982



u1

u2

d

h

d

h

d

h

d

h

x

y

x

y x

y

x

y

N

1 1

1

1

11

1

1

1

1

1 1

x y

x y

025 025

025

025

025

025

025 025

120572hh120572dh

120572dh

120572dh

120572dh

120572hd

120572hd

120572hd

120572hd

120572dd

120572dd

120572dd

120572dd

120579hh

120579dh

120579hd

120579dd

mhh

mdh

mhd

mdd

d

h

d

h

d

h

d

h

d h d h d h d h

d hd hd hd h

120572hh

120572hh120572hh

u1 u2

y x

y x

1 1

1 1

x y

x y

025 025

025 025

u1 u2

y x

y x

u1

u2 x

y

x

y x

y

x

y

1

1

1

1

025

025

025

025



2



2= 1205792 then 120572

1205792|2



1205792|1198982

= 1





2 then 119904lowast

11198981205792

= 119889


1205792


1205792






119889119889 119898lowast

119889ℎ 119898lowast

ℎ119889 119898lowast

ℎℎ) (119904

lowast

1119889119889 119904lowast

1119889ℎ 119904lowast

1ℎ119889 119904lowast

1ℎℎ) (120572

lowast

119889119889|119889119889 120572lowast

119889ℎ|119889ℎ 120572lowast

ℎ119889|ℎ119889 120572lowast

ℎℎ|ℎℎ)

C1 (119889119889 119889ℎ 119889119889 ℎℎ) (119889 119889 ℎ 119889) (119901119889119889

(119901119889119889+ 119901ℎ119889) 1 1 1)

C2 (119889119889 119889ℎ 119889119889 119889ℎ) (119889 119889 ℎ ℎ) (119901119889119889

(119901119889119889+ 119901ℎ119889)119901119889ℎ

(119901119889ℎ+ 119901ℎℎ) 1 1)

C3 (119889ℎ 119889ℎ ℎℎ ℎℎ) (ℎ 119889 ℎ 119889) (1119901119889ℎ

(119901119889119889+ 119901119889ℎ) 1119901ℎℎ

(119901ℎ119889+ 119901ℎℎ))

C4 (119889ℎ 119889ℎ 119889ℎ 119889ℎ) (ℎ 119889 ℎ ℎ) (1119901119889ℎ

(119901119889119889+ 119901119889ℎ+ 119901ℎ119889+ 119901ℎℎ) 1 1)









1205792|lowast

2(1205792)= 1 lowast

1205792


1205792

(119904lowast

1119898 lowast

1205792

lowast

minus1205792

) ge 1199061205792

(119904lowast

1119898

1198981205792

lowast

minus1205792

) for every1198981205792

isin Θ2




lowast

1205792



1205792




= 2is a








C1 120572lowast

119889119889|119889119889ge

120572lowast

ℎ119889|119889119889(119910 minus 119909)


C2 120572lowast

119889ℎ|119889ℎge

120572lowast

ℎℎ|119889ℎ(025 minus 119909)

075120572lowast

119889119889|119889119889ge

120572lowast

ℎ119889|119889119889(119910 minus 119909)

(1 minus 119910)

C3 Lie 120572lowast

ℎℎ|ℎℎge

120572lowast

ℎ119889|ℎℎ(119910 minus 119909)

(119909 minus 025)120572lowast

119889ℎ|119889ℎge

120572lowast

119889119889|119889ℎ(119910 minus 1)

075

C4 Lie 120572lowast

119889ℎ|119889ℎge

120572lowast

119889119889|119889ℎ(119910 minus 1) + 120572

lowast

ℎ119889|119889ℎ(119910 minus 119909) + 120572

lowast

ℎℎ|119889ℎ(025 minus 119909)

075






only if (1minus119910)120572119889119889|119898(120579

2)+(075)120572

119889ℎ|119898(1205792)+(119909minus119910)120572

ℎ119889|119898(1205792)+(119909minus

025)120572ℎℎ|119898(120579

2)ge 0





5 Conclusions










Appendix


1 119904lowast

2) = 119909 and

1199062(119904lowast

1 119904lowast

2) = 119910 then 119910 ge 1 If 119904lowast

2= ℎ119889 then 119906

1(119889 ℎ119889) ge


2= ℎℎ

then 1199061(119889 ℎℎ) ge 119906


1199062(ℎ ℎℎ) ge 119906



1 119904lowast

2) = 119909 and

1199062(119904lowast

1 119904lowast

2) = 119910



1 lowast

2) ge 119906

2(119904lowast

1 119904lowast

2)


2= 119904lowast

2


2) ge 1199062(119904lowast

1 119904lowast

2) If lowast

2= 119904lowast


2is


2) ge 1199062(119904lowast

1 119904lowast

2) then

119904lowast

1= 119889 or 119904lowast



2(ℎ lowast

2) lt

1199062(119889 119904lowast

2) If 119904lowast

1= ℎ and


2(ℎ lowast

2) lt

1199062(119904lowast

1 119904lowast

2) If 119904lowast

1= ℎ and


1199062(ℎ lowast

2) ge 119906

2(ℎ 119904lowast

2) and 119906

2(119889 lowast

2) lt 119906

2(119889 119904lowast

2) The C1



1= 119889



1 119904lowast

2) = 119910 and


1(ℎ 119889ℎ) then 119898

2= 119889ℎ is a


2(119904lowast

1 119904lowast

2)

If 1199062(119889 119889ℎ) gt 119906

2(119904lowast

1 119904lowast

2) then 1 gt 119906

2(119904lowast

1 119904lowast

2) to satisfy this


1 119904lowast

2) = (ℎ ℎ119904

lowast


2= ℎ119904lowast


then 1199062(119889 ℎ119904lowast

2ℎ) gt 1199062(119889 119889119904lowast



2= ℎ119904lowast


1(119904lowast

1 119904lowast

2) =


2(119904lowast

1 119904lowast






1119898 119898lowast

2) gt 1199062(119904lowast

1 119904lowast

2)


119889ℎ and 120579ℎ119889= ℎ119889 If 119898lowast




1199061(119889 1199041(minus119889ℎ) 119889ℎ119898

minus119889ℎ) = 1 and 119906

1(ℎ 1199041(minus119889ℎ) 119889ℎ119898

minus119889ℎ) =


1(119889 1199041(minusℎ119889) ℎ119889119898

minusℎ119889) =

119909 and 1199061(ℎ 1199041(minusℎ119889) ℎ119889119898




minus119889ℎ) = 1

and 119906ℎ119889(119889 1199041(minusℎ119889) ℎ119889119898






1205792


1119898lowast(1205792)= 119889 If lowast

1205792




(119889 1199041(minus119898lowast(120579

2)) lowast

1205792

119898lowast

minus1205792

) lt 1199061205792

(119904lowast

1119898 1198981205792

119898lowast

minus1205792


1205792

(119889 1199041(minus119898lowast(120579

2)) lowast

1205792

119898lowast

minus1205792

) ge 1199061205792

(119904lowast

1119898

119898lowast

1205792

119898lowast

minus1205792



1205792

= 2is a self-

committing message



1199061(1198891198981205792

) = 1120572119889119889|119898(120579

2)+ 1120572119889ℎ|119898(120579

2)+ 119909120572ℎ119889|119898(120579

2)+

119909120572ℎℎ|119898(120579

2)

1199061(ℎ1198981205792

) = 119910120572119889119889|119898(120579

2)+ 025120572

119889ℎ|119898(1205792)+ 119910120572ℎ119889|119898(120579

2)+

025120572ℎℎ|119898(120579

2)

therefore 1199061(1198891198981205792

) ge 1199061(ℎ1198981205792


2)+ (075)120572

119889ℎ|119898(1205792)+ (119909 minus 119910)120572

ℎ119889|119898(1205792)+ (119909 minus

025)120572ℎℎ|119898(120579

2)ge 0



Acknowledgments


References
























1 Introduction











2 Hypotheses



119875A = 119889A minus 119868 + 119877

119875B = 119889B + 119898119868 minus 119877(1)








max [119901 (119891) (119887 + 119890) + 119889 (119890 119891) minus 119896] (2)


119891gt 0)








119891(119887 + 119890) + 119889

1015840

119891= 0 (3)
























3 Methods





4 Results




A investments

2

3

4

5

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(a)

B returns

1

2

3

4

5

6

7

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(b)

Cooperation index

045

05

055

06

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(c)









A investments

1

2

3

4

5

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(a)

B returns

1

2

3

4

5

6

MU

No

ince

ntiv

e

A in

cent

ive

B in

cent

ive

Base

line

Fixe

d in

cent

ive

(b)

Cooperation index

045

05

055

06

MU

Base

line

No

ince

ntiv

e

Fixe

d in

cent

ive

A in

cent

ive

B in

cent

ive

(c)







5 Discussion





Baseline


B ret (prop) 009 001 016 001 056 1370 0059CI 050 000 051 000 098 3190 0118

No incentive


B ret (prop) 024 001 014 001 171 1890 0006CI 063 001 054 000 117 3960 0000

Fixed inc


B ret (prop) 017 001 010 001 170 1500 0153CI 054 001 052 000 104 2640 0319

A incentive


B ret (prop) 023 001 021 001 110 1340 0333CI 060 001 056 001 107 3650 0002

B incentive


B ret (prop) 019 001 018 001 106 1300 0387CI 058 001 055 000 105 3070 0071







1509lowastlowast







1012dagger



0024lowast



lowastlowast



lowast





lowastlowastlowast

























Appendices










B Instructions











































Acknowledgments


References


















































Sarah A Tulman







1 Introduction





















Candidate 119884 119902 minus 120579 119902









119901lowast

119894119895= int

infin

minusinfin

120591119894119895(119888) 119891 (119888) 119889119888 = int

119888lowast

119894119895

minusinfin

119891 (119888) 119889119888 = 119865 (119888lowast

119894119895) (1)




119888lowast

119894119895=

(119881119894119895minus 119881119894119898)


119894119895+ Pr (break tie)

119894119895)

(2)




119894119895) is




119894119895minus 119888119896 (3)





119896119894119895= 120587119896119894119895+120576119896119894119895

where 120576119896119894119895






119894119890120582120587119894 where 120587








119901119896vote119894119895 =

119890120582120587119896119894119895

119890120582120587119896119894119883 + 119890

120582120587119896119894119884 + 119890

120582120587119896119894119886

(4)


119901119896119894119895=

119890120582120587119896119894119895

1 + 119890120582120587119896119894119883 + 119890

120582120587119896119894119884

(5)

0

01

02

03

04

05

06

07

08

09

1

Pr (v

ote)

clowast





Candidate 119884 119902 119902


119894119883 119901119894119884 1 minus



likelihood function

log 119871 = sum119897

(1198731119883119897times ln (119901

1119883119897) + 1198731119884119897

times ln (1199011119884119897) + (num obs

1119897minus 1198731119883119897minus 1198731119884119897)


times ln (1199012119883119897) + 1198732119884119897times ln (119901

2119884119897)



(6)

where 119873119894119895119897












1119884minus 119888119896+ 120572 (7)





1119884minus 119888119896 (8)




1119884minus 119888119896+ 120572 (9)












119894119895























(a)


02996 03069 033331199011119884

0 00793 03241199012119883

0 00793 03241199012119884

02996 03069 033331198881119883

01221198881119884

01198882119883

01198882119884

0122

(b)


02608 03069 033331199011119884

0 00793 03241199012119883

0 00793 03241199012119884

02608 03069 033331198881119883

01221198881119884

01198882119883

01198882119884

0122





119894119883minus119901119894119884)




For 120582 = 7 120572 = 0 120572 = 01 120572 = 025 120572 = 05 120572 = 075 120572 = 1















(a)


(b)


(c)














1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1Pr

(vot

e)


(a)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)


(b)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)


(c)

1 3 5 7 9 11 13 15 17 19

Round

0

02

04

06

08

1

Pr (v

ote)






119894119905


119894119905+ 1205733cost119894119905


119894119905

+ 1205737round


119894119905

+ 1205739( same type)

119894119905


119894119905


119894119905

+ 119908turnout119894 + 119906turnout119894119905


119894119905


119894119905+ 1205743cost119894119905


119894119905

+ 1205747round


119894119905

+ 1205749( same type)

119894119905


119894119905


119894119905

+ 119908favored119894 + 119906favored119894119905


119894119905


119894119905+ 1205793cost119894119905


119894119905









(006) (006) (001) (006) (006) (000) (002)



(000) (000) (000) (000) (000) (000) (000)


(008) (008) (001) (009) (009) (000) (002)



(002) (002) (001) (003) (003) (000) (001)




+ 1205797round


119894119905

+ 1205799( same type)

119894119905


119894119905


119894119905


(10)


















120582














0

02

04

06

08

1


Pr (v

ote)

0

02

04

06

08

1

Pr (v

ote)






0

02

04

06

08

1


Pr (v

ote)

0

02

04

06

08

1

Pr (v

ote)
































4 Conclusions









Appendices



119894119895) is the






Pr (pivotal)1119883

=1

2(

min(11987311198732)

sum

119899119884=1

119899119884

sum

1198992119884=0

119899119884minus1

sum

1198992119883=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198731minus 1

119899119884minus 1 minus 119899

2119883 119899119884minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119884minus1minus1198992119883

1119883119901119899119884minus1198992119884

1119884


+

min(11987311198732)minus1

sum

119899119884=0

119899119884

sum

1198992119884=0

119899119884

sum

1198992119883=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198732minus 1

119899119884minus 1198992119883 119899119884minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119884minus1198992119883

1119883119901119899119884minus1198992119884

1119884



)

(A1)


119895








119883= 119899119884




of 1198992119884

and 1198992119883


1119884and 119899

1119883mdashthe


119884minus 1198992119884) and (119899

119883minus 1198992119883)


and 1198991119883

In the first term ( 119873211989921198831198992119884

) 1199011198992119883

21198831199011198992119884



and 1198992119884




) 119901119899119884minus1minus1198992119883

1119883119901119899119884minus1198992119884

1119884(1 minus


be exactly 1198991119883

and 1198991119884




1119884

=1

2(

min(1198731 1198732)

sum

119899119883=1

119899119883

sum

1198992119883=0

119899119909minus1

sum

1198992119884=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198731minus 1


2119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119883minus1198992119883

1119883119901119899119883minus1minus1198992119884

1119884


+

min(1198731 1198732)minus1

sum

119899119883=0

119899119883

sum

1198992119883=0

119899119909

sum

1198992119884=0

(

1198732

1198992119883 1198992119884

)

sdot (

1198732minus 1

119899119883minus 1198992119883 119899119883minus 1198992119884

)

times 1199011198992119883

21198831199011198992119884

2119884(1 minus 119901


sdot 119901119899119883minus1198992119883

1119883119901119899119883minus1198992119884

1119884



)


Pr (pivotal)2119883

=1

2(

min(1198731 1198732)

sum

119899119884=1

119899119884

sum

1198991119884=0

119899119884minus1

sum

1198991119883=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119884minus 1 minus 119899

1119883 119899119884minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119884minus1minus1198991119883

2119883119901119899119884minus1198991119884

2119884


+

min(1198731 1198732)minus1

sum

119899119884=0

119899119884

sum

1198991119884=0

119899119884

sum

1198991119883=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119884minus 1198991119883 119899119884minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119884minus1198991119883

2119883119901119899119884minus1198991119884

2119884



)

Pr (pivotal)2119884

=1

2(

min(1198731 1198732)

sum

119899119883=1

119899119883

sum

1198991119883=0

119899119909minus1

sum

1198991119884=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1


1119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119883minus1198991119883

2119883119901119899119883minus1minus1198991119884

2119884


+

min(1198731 1198732)minus1

sum

119899119883=0

119899119883

sum

1198991119883=0

119899119909

sum

1198991119884=0

(

1198731

1198991119883 1198991119884

)

sdot (

1198732minus 1

119899119883minus 1198991119883 119899119883minus 1198991119884

)

times 1199011198991119883

11198831199011198991119884

1119884(1 minus 119901


sdot 119901119899119883minus1198991119883

2119883119901119899119883minus1198991119884

2119884



)

(A2)



0

02

04

06

08

1

1 3 5 7 9 11 13 15 17 19

Pr (v

ote)

Round

MaleFemale























D Instructions


Page 1








Page 2






Page 3











Submit answers





Page 5





Continue





(instructions summary page)








Disclaimer




Acknowledgments


References
























1 Introduction













2 The Experiment










119894isin 119873+ and 0 le 119892

119894le 119908119894 The


given by

120587119894(119908119894 119892119894) = 2 (119908

119894minus 119892119894) + 120 + 119892

119894+ 119866minus119894 (1)



119906119894(119908119894 119892119894 120572119894) = 120587119894(119908119894 119892119894) minus 120572119894

(119908119894minus 119892119894)2

119908119894

(2)




119892NE119894

=2120572119894minus 1

2120572119894

119908119894 (3)








119902119894(119908119894 119892119894 120572119894 120583) =

exp 119906119894(119908119894 119892119894 120572119894) 120583

sum119908119894

119892119895=0exp 119906

119894(119908119894 119892119895 120572119894) 120583

(4)



119894-




0 119908119894 Let q

119894(119908119894 g119894 120572119894 120583)119894isin119875



119894will also



119894(119908119894 g119894 120572119894 120583)119894isin119875




NE119894 120572119894 0) = 1 and 119902

119894(119908119894 119892119894 120572119894 0) = 0 forall119892

119894= 119892

NE119894)



119871119894(119908119894 g119894 120572 120583) =

119899

prod119908119894

119892119895=0119899 (119892119895)

119908119894

prod

119892119896=0

119902QRE119894

(119908119894 119892119896 120572 120583)

119899(119892119896)

(5)










119908119894= 160 2453 3500 3503 5568 3767

119908119894= 200 4750 4375 4376 6557 3948

119908119894= 240 4557 5250 5250 7486 4036



119871119894(119908119894 g119894 1205721 1205722 120572

119878 1205741 1205742 120574

119878 120583)

=

119878

sum

119904=1

120574119904

119899

prod119908119894

119892119895=0119899 (119892119895)

119908119894

prod

119892119896=0

119902QRE119894

(119908119894 119892119896 120572119904 120583)119899(119892119896)

(6)

where 1205741 1205742 120574

119878 with sum

119878

119904=1120574119904




119878



119904le

120572119904+1



4 Results




+(LR =









119908119894= 160 1763 2346 2404 4168 2601

119908119894= 200 3528 2932 2971 4830 2630

119908119894= 240 2970 3519 3545 5460 2639



120583 1205721 and 120572

2(119899 = 20) 120583 120572

1 and 120572

2(119899 = 5) 120583 120572

1 1205722 and 120572

3(119899 = 20) 120583 120572

1 1205722 and 120572

3(119899 = 5)

120583 2850 [2588 3126] 1507 [1290 1764] 2214 [2056 2395] 1425 [1204 1685]1205721

0 [0 001] 0 [0 002] 0 [0 001] 0 [0 002]1205722

053 [046 060] 054 [047 061] 043 [039 046] 048 [040 056]1205723

104 [092 116] 076 [053 101]1205741

066 [053 078] 063 [050 075] 041 [033 046] 059 [049 064]1205742



119908119894= 160 3604 2463 3678 2468

119908119894= 200 3901 2677 4083 2699

119908119894= 240 4148 2873 4445 2913










1= 0 120572

2= 038 120572

3= 061 120572

4= 104

1205741= 039 120574

2= 042 and 120574



Table 4








3= 104






5 Conclusions







Appendix






Acknowledgments


References
















































1 Introduction














2 Literature Review






















119909 (119888) = 120574minus1(

3

sum

119895=1

120587119895 int

1

119888

minus

1

119886

1198651015840119895 (119886) 119889119886) (1)

where

1198651015840119895 (119886) =

(3 minus 1)

(119895 minus 1) (3 minus 119895)


119865 (119886)119895minus2

sdot 1198651015840(119886) ((1 minus 3) 119865 (119886) + (119895 minus 1))

(2)


119865119895 (119886) =(3 minus 1)

(119895 minus 1) (3 minus 119895)

(1 minus 119865 (119886))3minus119895

119865 (119886)119895minus1

(3)








119904120572119894 = 120572120587119894 + (1 minus 120572) (

2

3


(4)






120572 = 0

120572 = 05

120572 = 1

AR

E

D

0

12

26 33 100

120587120572

B

120587120572

C

120587120572

A



119863= (26 12 12)


119877= 119904119877119860 119904119877119861 119904119877119862 where 119904

119877119894 = 120587119894 forall119894 isin













120597119904120572119894

120597120572

= 2 (120587119894 minus 120587) (5)

























119860 119861 119862Coefficient of









5 Results








signBaseline


Constant 3170 0200 00001198772 00132

119865 376 0015Veto

Mean minus0313 0208 0137 +



Constant 2729 0180 00001198772 0008


915

0

2

4

6

8

Mea

n gr

oup

effor

t

10


749 763

875



0

003001

021

016

005

01

015

02

Def

ault

rate










029 029

042

050

003

010

039047

SkewedMajority

RandomContest

Unanimity

0

01

02

03

04

05

Mea

n of120572









6 Conclusion





Symmetric (0057) (0035)119875 le 001 119875 = 0022

0079 0131Right-skewed (0029) (0031)

119875 le 001 119875 le 001





Appendices

A Further Tables


B Instructions









Prizescheme



payoff119894 119888119894 1st 2nd 3rd 119864(120587119894)

119909119894 119895 120587119895

1 04 562 375 62 10 38 A 33 15 181 2 06 250 500 250 4 21 B 17 13 4

3 08 62 375 562 1 9 C 0 8 minus81 04 562 375 62 19 77 A 67 31 36

2 2 06 250 500 250 8 42 B 33 25 83 08 62 375 562 2 18 C 0 15 minus151 04 562 375 62 13 20 A 25 8 17

3 2 06 250 500 250 10 11 B 17 7 103 08 62 375 562 8 5 C 8 4 41 04 562 375 62 27 38 A 50 15 35

4 2 06 250 500 250 21 21 B 33 12 213 08 62 375 562 18 9 C 17 7 101 04 562 375 62 12 29 A 36 11 25

5 2 06 250 500 250 8 10 B 7 6 13 08 62 375 562 7 2 C 7 2 51 04 562 375 62 24 57 A 72 23 49

6 2 06 250 500 250 16 20 B 14 12 23 08 62 375 562 14 4 C 14 3 111 04 562 375 62 14 14 A 26 6 20

7 2 06 250 500 250 13 5 B 12 3 93 08 62 375 562 12 1 C 12 1 111 04 562 375 62 29 28 A 52 11 41

8 2 06 250 500 250 25 10 B 24 6 183 08 62 375 562 24 2 C 24 2 22









Effort 120572

119860 119861 119862 119860 119861 119862


Symmetric 439 314 163 059 051 041(022) (020) (032) (003) (003) (003)

Right-skewed 426 238 085 040 010 010(017) (020) (030) (004) (002) (002)

Unanimity rule

Symmetric 392 255 116 048 049 047(021) (021) (031) (002) (002) (002)

Right-skewed 424 280 172 043 041 043(026) (022) (035) (002) (003) (002)








+


(B2)



(3 + 5 + 10) sdot 05

3

= 15 + 3 = 45

(B3)







(4 + 6 + 10) sdot 045

3

= 22 + 3 = 52

(B4)





Acknowledgments




Endnotes



















References












































































Experimental Game Theory and Its Application in Sociology...

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