TACIT COLLUSION IN AUCTIONS AND CONDITIONS FOR ITS ... plott tacit... · transform tacit collusion...

TACIT COLLUSION IN AUCTIONS AND CONDITIONS FOR ITSFACILITATION AND PREVENTION: EQUILIBRIUM SELECTION IN

LABORATORY EXPERIMENTAL MARKETS

JIN LI and CHARLES R. PLOTT*

The paper studies bidder behavior in simultaneous, continuous, ascending priceauctions. We design and implement a ‘‘collusion incubator’’ environment based ona type of public, symmetrically ‘‘folded’’ and ‘‘item-aligned’’ preferences. Tacitcollusion develops quickly and reliably within the environment. Once tacitcollusion developed, it proved remarkably robust to institutional changes thatweakened it as an equilibrium of a game-theoretic model. The only successfulremedy was a non-public change in the preference of participants that destroyedthe symmetrically, ‘‘folded’’ and ‘‘item aligned’’ patterns of preferences, creatinghead-to-head competition between two agents reminiscent of the concept ofa ‘‘maverick.’’ (JEL L50, L94, D43)

I. INTRODUCTION

The research reported here explores bothconditions under which tacit collusion devel-ops and the remedies that might be taken totransform tacit collusion into a competitivesolution. The issue is approached from thepoint of view of four broad questions. Canwe create economic environments withinwhich tacit collusion emerges (in the absenceof conspiracy)? Such an environment can beviewed as a ‘‘tacit collusion incubator.’’ A suc-cessful incubator would provide an opportu-nity to study the details of phenomena thathave difficulty surviving in other environ-ments. Can the evolving behavior be under-stood in terms of game-theoretic models andif so, can the behavior be understood in termsof equilibrium selection? Is the underlyingmodel robust in the sense of predicting behav-

ior when possibly collusion breaking, institu-tional perturbations are imposed? The overallgoal is to direct focus on the principles thatoperate in the hope that an understandingof the principles will help us understand thephenomenon in the numerous, different, com-plex environments found naturally occurring.

The institutional setting is an auction inwhich game-theoretic models have a naturalinterpretation within which multiple equilibriaexist. In the context of the game-theoreticmodels, the question is focused on the severalequilibria of non-repeated game models andthe conditions under which some equilibriaare favored by the data and others are not.Specifically, one equilibrium is favorable tothe buyers (labeled ‘‘tacit collusion’’) andanother is favorable to the seller (labeled‘‘competitive’’). If the system has a tendencyto go to one, under what conditions can thesystem be made to naturally gravitate to theother and what role can theory play in identi-fying the conditions?

Collusion among several (five or more)agents has never been observed in an experi-mental market environment in the absence

*The support of the National Science Foundation andthe Caltech Laboratory for Experimental Economics isgratefully acknowledged. Comments from Katerina Sher-styuk, Joseph Cook, and participants in the Caltech sem-inar on laboratory methods in economics and politicalscience were very helpful.

Li: Kellogg School of Management, NorthwesternUniversity, Leverone Hall, 6th Floor, 2001 SheridanRoad, Evanston, IL 60208. Phone 847 467 0306, Fax847 467 1777, E-mail [email protected]

Plott: California Institute of Technology, Division ofHumanities and Social Sciences, Pasadena, CA91125. Phone 626395-4209, Fax 626 793 8580,E-mail [email protected]

ABBREVIATIONS

NE: Nash Equilibrium

SAA: Simultaneous Ascending Auction

SPE: Subgame Perfect Equilibrium

Economic Inquiry doi:10.1111/j.1465-7295.2008.00152.x

(ISSN 0095-2583) Online Early publication July 9, 2008

Vol. 47, No. 3, July 2009, 425–448 � 2008 Western Economic Association International

425

of conspiracy and/or special facilitating devi-ces. Consequently, few experimental studieshave addressed the issue of how to stop tacitcollusion once it has started. This paperaddresses the issue in two steps. First, we iden-tify an experimental environment in whichtacit collusive-type behavior evolves naturallyand quickly and does so without the aid of ver-bal communication and without the aid of sidepayments. We call it the ‘‘collusion incubator’’environment. Second, we explore the use andeffectiveness of vehicles that hold a potentialfor disrupting or terminating collusion.Because of the complexity of the issues, thelarge set of potential collusion ‘‘remedies’’and necessarily sparse data, we use an explor-atory methodology. Institutional changes areimplemented in sequences that depend uponwhat has been observed. The exploratoryapproach is frequently used under suchcircumstances.

Two different experimental environmentswere developed and explored. One environ-ment is labeled ‘‘collusion conducive’’ or ‘‘col-lusion incubator.’’ It was designed with thepurpose of creating an environment in whichtacit collusion would evolve. This environ-ment necessarily abstracts from parametersthat might exist naturally and differs dramat-ically from the parameters studied in otherexperiments. The purpose is to create andstudy phenomenon that is difficult to find inany form. Indeed, tacit collusion in a multipleitem auction has never been (convincingly)observed. The reliable creation of phenome-non that is thought to exist is a first step tounderstand the conditions under which itmight be found, observe the basic principlesat work in its evolution, and identify how itmight be detected if it exists in more complexenvironments. Thus, as is the case with exper-imental methods in all science, the extremedivorce of the experimental environment fromthe naturally occurring environment is veryuseful, if not necessary.

The second environment, labeled ‘‘compet-itive conducive,’’ involves changes in the firstenvironment, implemented for the purpose ofstudying the stability of (tacit) collusivebehavior. For convenience, from time to timewe may refer to these two environments as the‘‘collusive’’ environment and the ‘‘competi-tive’’ environment even though the latter willtake various forms as we study how collusioncan be extinguished.

Both the collusive and competitive environ-ments have common structural elements andsome common institutional elements. Thecommon environment has ‘‘repeated game’’features in that agents participated in a seriesof auctions. Except as otherwise noted, thenumber of items equaled the number of partic-ipants. Preferences of agents were additive, inthe sense that no synergies existed. Exceptunder the case of special treatments that willbe discussed later, the preferences overthe items had two important symmetries thatwe thought would be supportive of (tacit)collusive behavior.

The first property of preferences we will call‘‘strong ordinal symmetry’’—if on Item Xagent i had the m-th highest value and agentj had the n-th highest value, then agent ihad the n-th highest value on the item whereindividual j had the m-th highest value. As willbe discussed, this pattern of symmetry hasa type of ‘‘folded’’ property in that it hasthe potential for simultaneously placing anytwo individuals in two, exactly opposite con-flicts. This pattern of relationships holds thepossibility that competition can ‘‘unfold’’ intotacit collusion, as pairs are able to find a mutu-ally beneficial equilibrium. If i and j do notcompete, then both face a ‘‘next in line’’ witha similar conflict. The sequential resolutionof these conflicts can theoretically result inan ‘‘unfolding’’ from one equilibrium to acompletely different one.

The second feature of preferences we willcall ‘‘item-aligned’’ preferences. For any itemthat an individual preferred the most, thatindividual also had the highest value amongall agents. That is, each had his or her ‘‘ownitem’’ in the sense that it was clear that inany bidding contest an individual wouldalways ‘‘win’’ the item that the individual pre-ferred the most. In that sense, we can label anitem as ‘‘individual i’s item’’ as if it is clear whowould get it. Thus, except in the special casesdiscussed later, each agent had the highestvalue among all agents for one item and thatwas also the item that was most valuable to theindividual agent.

The issue is how the institutions mightinteract with these basic structural featuresto determine equilibrium. Institutional issuesaside, how the two structural features mightwork together suggests why they might be sup-portive of collusion. Suppose agent i’s item isA. If j has the second highest value then i has

426 ECONOMIC INQUIRY

the second highest value on B where j has thehighest value. If they compete then i will payj’s value for A and j will pay i’s value for B. Ifthey do not compete, then they face competi-tion from agents whose values are third fromthe top, and if competition can be removed atthat level through the same mechanism thencompetition is encountered at still a lowerlevel. This ‘‘unfolding’’ of competition is whatthe environment was designed to facilitate.The issue is how the unfolding might be facil-itated by institutions and more importantly,what institutions might ‘‘reverse’’ the processif a (tacit) collusive equilibrium evolved. Thus,partial success would be the creation ofan environment in which collusion wouldnaturally evolve.

The basic institutional framework studiedis the simultaneous, continuous, ascendingprice auction. The collusive conducive envi-ronment operated under conditions of fullinformation. All preferences, payoffs, andbids and the identification numbers of bidderswere public information. By contrast, thecompetitive conducive environment operatedwith less public information and with slightlydifferent rules that operated within theauction.

The primary objective was to learn ifthe collusive conducive environment couldfacilitate the (tacit) collusive equilibrium.1

As mentioned in the introductory paragraph,the creation of the tacit collusive outcome wasdefinitely not a foregone conclusion becausecollusion with a large number of agents hadnever before been observed. Early experimentshad demonstrated that symmetry is an impor-tant feature in moving solutions away fromsimple competitive outcomes toward morecooperative ones (Plott, 1982) and more recentresearch by Sherstyuk and her co-authors suc-cessfully produced tacit collusion in auctionswith no more than three bidders, but in morecomplex environments prices almost alwaysconverge to near the competitive equilibrium.2

The first basic result is that within the col-lusive conducive environment the systemequilibrated quickly to the (tacit) collusiveequilibrium. Several measures and character-istics of this process are chronicled in theresults section of the paper. The process doesnot seem to have the sequential property assuggested by the unfolding property butinstead is discontinuous, with an almost‘‘jump’’ toward the equilibrium. Thus, the firstpart of the study was very successful. Collusiveequilibria emerged and did so reliably undercollusive conducive conditions.

The second result is that collusive equilib-ria, once established, are stable in the sensethat removal of central properties of the col-lusive conducive environment did not forcethe auction away from the collusive equilib-rium. Public information about preferencesand about bidder identification did not changethe equilibrium. Several changes in auctionrules had no effect. Disruptions of the strong,ordinal symmetry by removing some of theitems did not disrupt the equilibrium. Onlywhen a ‘‘maverick’’ preference was introducedunder conditions of lack of public informationabout preferences, the collusive equilibriumwas quickly disrupted and the system evolvedto the competitive solution. The ‘‘maverick’’was an agent who had the same most preferreditem as one of the other agents.

The context of these results needs emphasis.While the study answers some questions, itcertainly does not answer all questions.Exactly what theory might be applied remainsopen, including more detailed analysis of ‘‘oneshot’’ game models, the application ofrepeated game concepts, or whether or notgame theory itself is an appropriate tool forunderstanding the phenomenon. How theresults might be translated to complex field-work remains unaddressed. The questionsposed here are primarily empirical with onlycrude steps toward adequate theory and theresults are reported in the hope that theoristswill be challenged to provide convincing the-ory of what is observed.

The paper is outlined as follows. In SectionII, the background experimental work is dis-cussed. Our experiments build on that litera-ture. Section III contains the details of theexperimental environments. The details ofthe preferences and institutions are foundthere. Section IV is the experimental designthat explains the number of experiments and

1. The paper provides a sufficient condition/environ-ment for the emergence of stable tacit collusion, but it doesnot investigate what features of the collusion incubator aremost critical. Instead, it investigates what remedies can beapplied to break the collusion once it has emerged.

2. The posted price effect discovered by Plott andSalmon (2004) shows that a ‘‘sealed-bid’’ like institutioncan have the effect to improve the payoff of the side ofthe market tendering the bid.

LI & PLOTT: TACIT COLLUSION IN AUCTIONS 427

the conditions that were in place for eachexperiment. Section V discusses models andtheory. The correspondence between thegame-theoretic solutions and the experimentalprocedures in the auctions are discussed indetail. Section VI contains the results, whichare divided into two sections. The first sectionof the results explains the nature of the collu-sive equilibrium and how it is reached. The sec-ond section of the results explains the changesin the environment that we implemented asattempts to make the collusive equilibriumswitch itself to the competitive equilibrium.Section VII is a summary of conclusions.

II. BACKGROUND EXPERIMENTAL WORK

There are very few empirical examples ofcollusion in auctions in the absence of conspir-acy or without the aid of facilitating devices.Hendricks and Porter (1988) study the biddingof drainage leases on the Outer ContinentalShelf. Their findings suggest that there arecoordination in bids among firms that owntracts adjacent to the lease for sale. Cramtonand Schwarz (2000) report what appear to beattempts to collude in FCC spectrum auctionbut have no evidence of actual, price influenc-ing collusion. In particular, they point out howbidders used the last several digits of the bidsto signal their intents.

Of course, since bidder values are typicallyunknown in the field, collusion is hard to doc-ument. This difficulty can be easily overcomeby laboratory experiments, since values of thebidders can be chosen by the experimenters.Isaac and Plott (1981) are the first to studyconspiracy experimentally. Isaac and Walker(1985) allow explicit communications amongbidders in auctions, and they observe conspir-acies in 7 out of 12 auction series. Tacit collu-sion in auctions with standard procedure,however, is rarely observed in earlier experi-mental studies. Even when it is observed, tacitcollusion is unstable and the prices easily con-verge back to competitive levels; see, for exam-ple, Burns (1985) and Clauser and Plott(1993). In a related industrial organizationcontext, Isaac and Smith (1985) search forpredatory behavior in constraint of competi-tion and do not find any. Kagel (1995) surveysearlier experiments on collusions in auctions.

Recently, a sequence of experiments suc-cessfully observed robust tacit collusion in

special auction institutions. Sherstyuk (1999)studies ascending auctions under common val-ues. In her experiments, each auction consistsof two units of items and three bidders, whodemand one unit only. She introduces in theauction a particular bid improvement rule:bidders are allowed to submit a bid that equalsthe highest outstanding bid. If there are mul-tiple highest bidders on the items, a lottery willbe run to decide the winner. With this bidimprovement rule, Sherstyuk reports persis-tent and stable tacit collusions: bidders matcheach other’s low bids in most of the auctions.Sherstyuk (2002) obtains similar results underprivate values.

Robust tacit collusion in auctions with‘‘standard institutions and procedures’’ is firstobserved by Kwasnica and Sherstyuk (2007).In their environment, there are two items forsale in an ascending auction with either two orfive bidders, who value both items. The exper-imental environment conforms to the theoret-ical conditions of Brusco and Lopomo’s(2002), and the auction results are broadlyconsistent with the theoretical predictions:when there are two bidders only, they oftentacitly collude through splitting the two items.No collusion is found in auctions with fivebidders. Kwasnica and Sherstyuk (2007) alsoexamine the role of complementarity in collu-sions. They find that complementarities be-tween items reduce collusion. However, whenthe level of complementarity is moderate, bid-ders still tacitly collude by taking turns to winthe auction. A survey of recent results on col-lusion in auction can be found in Sherstyuk(forthcoming).

III. COLLUSION INCUBATOR/COLLUSION-CONDUCIVE ENVIRONMENTS

Each experiment starts with a collusion-conducive environment. In this environment,each auction consists of eight subjects and eightitems. The basic market architecture has allitems offered in simultaneously functioning,continuous, ascending first-price auctions.Within that basic structure, the experimentalenvironments have three major parts. The cen-tral feature, the ‘‘folded’’ and ‘‘item-aligned’’properties of valuations pattern will be discussedbelow in the subsection of Items and Preferen-ces. The details of the auction architecture willbe in the institution environment subsection.


The information structure is discussed in a sep-arate subsection.

Once perfect tacit collusion3 has emerged andhas persisted, the experimental environmentis transitioned into a competition-conduciveenvironment through various treatments.These treatments are exploratory in natureand occasionally differ in experiments. Wedocument these treatments in detail in theexperimental procedure and design section.

A. Items and Preferences

Each experiment has eight subjects, num-bered from 101 to 108 anonymously. In thecollusive environments, there are eight itemsfor sale, numbered from 1 to 8. The subjects’valuations for each item range from 50 to 900francs, where one franc can be exchanged afterthe experiment for 1 cent. There is no comple-mentarity between items: a subject’s valuationof a group of items is the sum of his valuationsof each item in the group.

The valuations are kept identical acrossexperiments to facilitate comparisons. Withineach experiment, the valuations differ roundby round and are designed to facilitate tacitcollusion in earlier rounds. Figure 1 belowgives a sample valuation sheet that has beenused in the experiments.

In Figure 1, subject 107 has the highest val-uation on Item 1 at 824 and subject 102 has thesecond highest valuation at 784. The relativeposition of the two is reversed on Item 8,where subject 102 now has the highest valua-tion at 802 and 107 has the second highest oneat 728. This illustrates a symmetry in the val-

uations: if subject A has the highest valuationon an item where subject B has the secondhighest valuation, then A has the second high-est valuation on the item where B has the high-est valuation.

In Figure 1, if 102 and 107 do not ‘‘com-pete’’ with each other on these two items, thenthe competition they face (on these items) willbe considerably lower. To encourage tacit col-lusion as much as possible, in the first fiverounds of the experiment, we design the valu-ation sheet so that the third-highest valuationon each item is lower than 400 francs while thetop 2 are above 700. This gap in valuationsincreases the gain of tacit collusion fromthe top.

The symmetry of valuations runs deep intothe preference. For convenience, we call a sub-ject n-th on an item if he has the n-th highestvaluations on the item. We design the valua-tions such that, if subject i is m-th and subjectj is n-th on an item, then j is m-th on the itemwhere i is n-th. In Figure 1, 101 is second and105 is seventh on Item 4. Then on Item 5,where 105 is second, 101 is seventh. For thepurpose of collusion, it is enough to note thatfor any subject i and j, if j is n-th on an Item i isfirst, then i is n-th on the item where j is first.

The general ordinal structure of foldedpreference remains throughout the collusion-incubator environments, while the exact valu-ations of subjects differ from round to round.In general, the pairs of subjects who share thehighest two valuations on the same items inany given round are typically not ‘‘pairedtogether’’ in the next round.

Other than strong ordinal symmetry,another crucial feature in the design of prefer-ence is ‘‘item-alignedness.’’ This says that forany bidder, the item on which he has the

FIGURE 1

A Sample Valuation Sheet

Subject\Item Item 1 Item 2 Item 3 Item 4 Item 5 Item 6 Item 7 Item 8

101 514 123 446 730 263 869 388 615

102 784 312 244 585 427 649 116 802

103 153 584 889 349 671 476 707 211

104 387 732 658 153 879 298 548 430

105 492 873 566 206 787 124 618 302

106 636 217 361 853 160 738 494 506

107 824 403 179 626 352 561 202 728

108 253 645 755 443 513 363 899 181

3. We define an auction outcome as perfect tacit col-lusion if all items are sold efficiently at their reservationprice.


highest valuation among the bidders also hap-pens to be the one he values most. The generalproperty of ‘‘item-aligned’’ feature is also clearfrom Figure 1: subject 108 has the highest val-uations among all subjects for Item 7. Conse-quently, 108 by design values Item 7 mostamong all items. This ‘‘item-aligned’’ featuresuggests that 108 would be a natural winnerof Item 7. In this sense, we denote Item 7 as‘‘108’s item.’’ From now on, we also denotethe item on which subject i has the highest val-uation as subject i’s item.

Finally, it is worthwhile to point out thatthere are no ties in valuations: no two subjectsor more share the same valuation on any item;no subject has the same valuations on any twoor more items.

The total payoff of a subject in any givenround in an experiment is calculated by sum-ming his payoffs from all the items he wins.The subject’s payoff on each of his winningitem is equal to his valuation of the item minushis winning bid price. In particular, the subjectcan incur a loss on an item if his winning priceexceeds his valuation. The subject’s total pay-off in the experiment is the sum of his payoffsin all rounds.

B. Institutional Environment

The basic institutional environment isa computerized simultaneous ascending auc-tion (SAA). SAA is an auction in which allthe items are open for sale simultaneouslyusing an English-auction like format untilsome ending criteria is reached. Subjectsplace their bids through computers. Eachbid specifies an item to bid for and the asso-ciated bid price, while satisfying two restric-tions. First, there is a reservation price of allitems: the smallest bid that can be entered onany item is 100 francs. Second, the minimalbid increment is 10 francs so that a new bidon an item is required to be at least 10 francshigher than the existing highest bid. Forexample, if the current highest bid for an itemis 680, then the lowest possible new bid on it is680 + 10 5 690. The auction ends when nonew bid is entered for any item for consecu-tive 30 sec. That is, all items are open for biduntil the auction closes. The highest bidder ofan item wins it and pays his bidding price. Abidder can win multiple items in a round ofauction.

Information Structure

In the collusion incubator stage of the envi-ronment, the subjects have almost completeinformation of the auctions: they know thevaluation of all subjects through a valuationsheet passed out by the experimenter at thebeginning of each auction; they can observefrom the computer screen who (given by theID number) have bid what price on whichitem. The only information missing for thesubjects is that they do not know the totalnumber of rounds, except in two experimentswhere the experimenters announce immedi-ately before the final round that it is the finalround.

When the experiment transitions into thecompetition-conducive phase, various treat-ments are applied to reduce information.We discuss the details in the next section.

IV. EXPERIMENTAL PROCEDURES AND DESIGN

A. Experimental Procedures

The subjects in the experiments are mostlyCaltech undergraduates, with a few graduatestudents in non-economics departments insome experiments. All of the experiments lastbetween 1½ to 2½ h and take place in the Lab-oratory of Experimental Economics and Polit-ical Science. Upon entry into the experimentallaboratory, subjects are randomly assigned anID number and a seat. They also receive aninstruction sheet and a sample valuation sheetand are asked to read them carefully. Severalminutes after all subjects have arrived, theexperimenter reads the instruction sheet outloud in the laboratory. To make sure all sub-jects understand the rules, the experimenterencourages the subjects to raise their handsand ask questions about the experiment.The experimenter answers the questions inpublic until no more questions are asked.All subjects then go through a practice round,which is identical to a real round except itsoutcome is not counted in the total payoffs.After the practice round is over, subjects areagain encouraged to ask questions, and theexperimenter answers them in public. The firstreal round starts only when no further ques-tions are asked.

In the first several rounds of the experi-ments until perfect tacit collusion has emergedand persisted, i.e. in the collusion incubator


stage of the experiment, all subjects receivea valuation sheet from the experimenter atthe beginning of each round. The sheet con-tains information about the valuation of allsubjects on every item in that round. The sub-jects are asked to read the valuation sheet care-fully and raise their hands once they are readyfor the auction. The experimenter waits for allof the subjects to raise their hands before start-ing the auction. The auction is conducted inthe format described in Section III. Oncethe auction ends, the subjects are asked torecord their earnings on the sheet at the endof their instruction sheet. The experimenterwaits for 1–2 min for the subjects to recordtheir earnings before starting another roundby passing out new valuation sheets.

Once the experiment transitions to the com-petition-conducive phase, the experimentersstop passing out valuation sheets under condi-tions to be discussed later, and the subjectsobtain their own valuation through a com-puter screen. In some experiments, the exper-imenter also announces before the final roundthat it is the final round. Once all auctionsare finished, subjects receive their payoffsfrom the experimenter. This concludes theexperiment.

B. Experimental Design

A total of six experiments are conducted.With the exception of institutional and infor-mational changes that will be described below,all experiments are conducted in the same way.In addition, three pilot experiments are con-ducted as part of the process of developingprocedures and debugging instructions andsoftware.

The design calls for all experiments to beginwith the collusion incubator environment. Thecollusion incubator stage of the experimentstarts from the first round and lasts until thesubjects reach two consecutive rounds of per-fect collusion. The experiment then transitionsinto the competition-conducive stage. In thisstage, the experimenter applies several treat-ments sequentially in an attempt to breakthe collusion.4

The treatments are motivated by the sixexperimental design features, which are partic-ularly conducive to tacit collusion. We discuss

these features in Section V. For each of the fea-tures, we design one treatment to eliminate it.These treatments include (i) removing publicdisplays of the subject’s ID associated withthe bids; (ii) removing public information ofvaluations by not passing out the valuationsheet; (iii) changing the ending rule of the auc-tion to fixed duration, (iv) removing severalitems for sale to remove the ‘‘folded prefer-ence’’ feature; (v) destroying expectations ofsubjects by unexpectedly changing the struc-ture of the preferences to eliminate the fold-ing and item-alignedness features; and (vi)announcing the final round.

These treatments are applied in roughly thesame order as above in most experiments whereoccasional differences in orders come from theexploratory nature of the treatments. We fol-low the general rule that a new treatment isimplemented only if the previous treatmentfails to sustain a competitive outcome and tacitcollusion returns. When applying the new treat-ment, we retain all the previous treatments. Inthis way, we examine the combined effects ofthe treatments. Since the speed to reach perfecttacit collusion after each treatment varies in dif-ferent experiments, the number of rounds ineach experiment also differs. Table 1 belowdocuments the details of these treatments.

Four of the experimental treatments arestraightforward, including blanking the bid-der ID, taking away valuation sheets, chang-ing the ending rule to fixed duration, andannouncing the final rounds. Removing theitems for sale and destroying expectationsare more complicated. When we remove theitems for sale, we start from the same structureof valuations and block several items for sale.Typically, three items are taken out of theeight items, but in one experiment five itemsare removed. When we destroy the expecta-tions, we create multiple subjects with valua-tions above 800 francs on some items.5 Inthis way, all subjects with valuations above800 francs on an item might want to bidfor it, and this could lead to a destructionof expectations. Details on how items are

4. Of course, this design is unknown to the subjects.

5. In the preplanned valuation sheet (that is unknownto the subjects), on each item there is exactly one bidderwith valuation above 800 francs. To create multiple bid-ders with valuations above 800 francs on the same item, wetypically switch the valuations of two items for some sub-jects. For example, if we switch the preference of 101 onItems 4 and 6 in Figure 1, we create two bidders (101 and106) with valuations above 800 francs on Item 4.


removed and how valuations of items areswitched are documented in detail in SectionVI and can be found in Table 10 and Table 11.

V. MODELS

As is well known, game theory lacks preci-sion when the environment applied has multi-ple Nash Equilibria (NE). In some situations,the prediction can be so broad that almost anypattern of outcomes can be described as a NE.The challenge, of course, is to supplement thetheory with additional principles to yield moreprecise predictions. Auction theory is a goodexample in this regard. Since Robinson(1985), many papers have demonstrated thetheoretical existence of low-price NE. See,for example, Milgrom (1987) and Menzies(1996). Recently, Brusco and Lopomo(2002) and Engelbrechet-Wiggans and Kahn(2005) show that as long as there are nottoo many more bidders than the number ofobjects sold, there is a Perfect Bayesian Equi-librium resulting in low prices even underincomplete information.

While our environment has complete infor-mation, the multiple equilibria issue also exists.In particular, there is a continuum of SubgamePerfect Equilibrium (SPE) when each round ofauction is treated as a separate game. In termsof predicting outcomes, there are two equilibriathat deserve special attention.

The first equilibrium corresponds to com-petitive bidding. The behaviors of the subjectsthat reach this equilibrium can be character-ized by two principles. The first is the principleof surplus maximization, which says that whenchoosing a bid, each individual bidder willplace his bid on the item that maximizes his

surplus given the current prices in the auction.The second is a minimum bid principle, whichsays that the individuals choose the minimalpossible price allowed by the system whenchoosing an amount to bid.6

In the environments studied here, if eachsubject uses a strategy implied by these twoprinciples, these strategy profiles can be shownto form an SPE. Moreover, the equilibriumprice of each item is equal to the second high-est valuations on it, at which the individualwith the second highest value stops bidding.

The second equilibrium corresponds to per-fect tacit collusion.7 If we treat all buyers asa group, perfect tacit collusion occurs whenthe equilibrium prices and allocations maxi-mize their own total surplus, defined as thesum of profits across all buyers. In a perfecttacit collusion, each item goes to the bidderwith the highest valuation while the pricesof all items equal the minimum prices set bythe auction (the auction reservation price).

To describe intermediate levels of tacit collu-sion, we define cðTotal ProfitÞ 5 ðMaximumSurplus � Total ProfitÞ=ðMaximum Surplus �Competitive ProfitÞ as a measure of the com-petitiveness of the market. In this definition,Maximum Surplus is the largest possiblesum of profits of the buyers, obtained whenperfect tacit collusion arises. Competitive

TABLE 1

Summary of Experimental Environment

Experiments E1 E2 E3 E4 E5 E6

Date February 18, 2002 April 19, 2002 May 10, 2002 June 12, 2002 June 21, 2002 June 29, 2002

Total rounds 11 14 18 23 26 17

Collusion incubatorrounds

1– 1–5 1–7 1–6 1–3

Blank ID 9–11 N/A 6–18 8–23 7–26 4–6

Remove valuations N/A 9–14 3–18 12–23 10–26 7–8

Change end-rules N/A 11–12 N/A N/A N/A N/A

Remove items N/A N/A 4–18 14–23 13–26 9–17

Destroy expectations N/A N/A 12–13 17–18 22–23 12–13

Announce final round Yes No No Yes No No

6. These two principles were first described and testedin Brewer and Plott (1996) and later were combined andtermed ‘‘straightforward bidding’’ by Milgrom (2000).The principles were tested again in Plott and Salmon(2004).

7. The definition here refers to the collusive equilib-rium outcome. There are many strategies that lead tothe same tacit collusion outcome. One such strategy is out-lined in the next page when we discuss the continuum ofequilibrium.


Profit is the sum of the profits obtained by thebuyers when each item is sold to the highest-valued buyer at the price of the second highestvaluation, obtained when the competitiveequilibrium arises. A higher c reflects a greatercompetitiveness of the market. In particular,c5 1 indicates the market is perfectly compet-itive and c 5 0 corresponds to perfect tacitcollusion.

While the tacit collusion is defined in termsof prices and allocations, it is also useful todescribe collusion in terms of the actions ofthe bidders. A subject is defined as a tacit col-luder only if he only bids on items that satisfyone of the following three criteria:

(1) The item is the subject’s own item.(2) The item remains at the reservation

price for more than 60 sec.8

(3) The item ‘‘belongs’’ to a bidder who haspreviously bid on the subject’s item.

Otherwise, the subject is considered a non-colluder. We say ‘‘an item belongs to a subject’’if the subject has the highest valuation on theitem.

Although this definition does not containthe full complexities of bidding strategies,which will be discussed further in the SectionVI, it does capture the key characteristics ofthe collusive behaviors. The definition reflectsthe idea that a tacit collusive bidder is one whomostly bids only on his own item. The tacitcolluder bids on another subject’s item ifand only if either (1) the item has not beenbid on at all for a long time, or (2) the itembelongs to a non-collusive subject who hasbid on another bidder’s item. In the first case,the tacit colluder tries to exploit the possiblenegligence of some bidder. In the second case,some other subject has bid on the tacit col-luder’s item. The tacit colluder bids on theother subject’s item to punish him and possiblyto push him away from competitive bidding.Under this definition, perfect tacit collusionarises if every subject is a tacit colluder.

Other than perfect collusion and the com-petitive equilibrium, there is also a continuumof equilibrium in between. In particular, anyefficient allocation with a price vector betweenthe reservation price and the second highest

valuations can be supported as an SPE. Tosee this, we construct a strategy profile as fol-lows. In his first bid, each subject bids on hisitem at the specified equilibrium price. If anysubject deviates, all the subjects then revert tothe strategy given by the Principle of SurplusMaximization and Minimum Bid Principle.The strategy constructed above can bechecked as an SPE. In other words, all inter-mediate equilibrium can be thought of asa combination of the competitive strategyand the collusive strategy.

The discussions above suggest that ifwe simply look for NE or even SPE in ourcollusion-conducive environment, little canbe said about the final outcome. Without fur-ther equilibrium refinement, any price vectorbetween the reservation price and the secondhighest-valuations of the items is a possibleoutcome consistent with an SPE. However,since the collusive and competitive equilibriacorresponds to clear bidding behaviors andthe other equilibria are combinations of thetwo, it is natural to conjecture that the biddingoutcome is likely to converge to one of thethese two polar equilibria.

On the one hand, one might think the com-petitive equilibrium is the natural outcomebecause it can be supported by natural andsimple bidding strategies. Moreover, the com-petitive equilibrium has been observed in vir-tually all previous experimental studies onascending auctions. On the other hand, theprofits of the subjects under the competitiveequilibrium are very low compared to thatunder perfect tacit collusion. Furthermore,the typical difficulties to obtain a collusiveequilibrium, which requires a grand collusionamong all subjects, are drastically reduced inour environment. Our symmetrical and ‘‘itemaligned’’ feature of the preferences helpsreplace a grand collusion with many subjectsinto a sequence of smaller collusions withtwo subjects. Since there are no a priori rea-sons for which types of equilibrium will beselected, the experimental designs helpuncover institutional features that favor onetype of equilibrium over the other.

Together with the symmetrical and ‘‘item-aligned’’ feature, there are six key featuresof the collusion-conducive environment ofthis auction that facilitate the tacit collusionmodel as an equilibrium to be selected. Wefocus on these because their removal or pertur-bation could eliminate important support for

8. The ‘‘at least one bid in the 60 seconds’’ is to requirethat the subjects are not ‘‘negligent.’’ We pick 60 secondsarbitrarily: any change from 30 to 100 sec will not makeany difference in our definition of tacit colluders.


the collusive equilibrium and thus lead to theidentification of tools that help establish andenforce the competitive outcome.

(1) There is common knowledge of valua-tions for all items: before each auction, a val-uation sheet with every subject’s valuations ispassed to all subjects. This facilitates perfecttacit collusion by helping the subjects coordi-nate on the allocation of the items.

(2) The bidding behavior of subjects can beidentified: all subjects know the ID numberassociated with each bid. In other words, theyknow who has entered which bid. This makesperfect tacit collusion easier to implementbecause it facilitates the punishments neces-sary to enforce the collusive equilibrium.For example, if subject i enters a bid on subjectj’s item, j knows i has made the bid and cantarget his punishment at i. Increasing the pun-ishment power of j helps discourage i frombidding on j’s item.

(3) Every bid can be followed by a reactionof other bidders: The ending rule in this exper-iment states that the auction ends if and only ifno new bids are entered for a consecutive 30sec. In contrast to auctions with fixed endingtimes, this rule deters ‘‘last second deviation’’and in turn prevents competitive bidding infear of ‘‘last second deviation.’’ Related dis-cussions on ending rules are in, for example,Roth and Ockenfels (2002) and Ockenfelsand Roth (2006).

(4) The environment is characterized bystrong ordinal symmetry. Symmetry of the val-uations is known to increase the likelihood ofcollusive outcomes. Exactly why this occursis still unknown. It could be related to the easewith which a subject can identify and under-stand the behavior of others. Alternatively, itcould be related to a focal point argument.The fact that each item has a distinct highestvalued subject makes focal the tacit collusiveequilibrium in which every subject bids reserva-tion price on the item he has the highest valu-ation. Moreover, the symmetry implies in thistacit equilibrium the division of profits for thebuyers is natural: each subject wins his ownitem. Without this property, a tacit collusioncan be sustained only through some ‘‘repeatedgame’’ arguments: bidders who do not win anyitem in one round will be compensated in thefuture. The symmetry supports tacit collusionas an equilibrium within each round withoutrelying on such dynamic tradeoffs.

(5) The valuations of the bidders satisfy theitem aligned property. In other words, the sub-ject who has the highest valuation on an itemalso prefers the item most. As with symmetry,this greatly facilitates the division of profitsamong the subjects. In particular, the itemaligned property implies, in the collusive equi-librium, no subject prefers the allocation (theitems won and the total price paid) of anothersubject over his own. In other words, the col-lusive equilibrium is the unique Pareto equilib-rium and it is also envy-free.9 With itemalignedness, what is best for the group is alsobest for all individual subjects. In contrast, ifwe destroy this property by having severalsubjects prefer the same item most, there willbe several Pareto-collusive equilibria. Sinceeach subject prefers the equilibrium that giveshim the highest profit, this conflict in equilib-rium preference might push the subjects to thecompetitive equilibrium.

The above observations make clear a centraltheoretical property of the situation. With itemalignedness property, together with symmetry,there is a unique Pareto-optimal SPE from thepoint of view of the buyers. The followingproposition states the property formally.

PROPOSITION 1. Consider a simultaneousascending auction with n bidders and n items.If each bidder has the highest valuation onexactly one item, which also is his most valueditem, then the unique buyer Pareto-optimalSPE outcome with undominated strategy is thateach bidder wins his item (the item he has thehighest valuation) at the reservation price.

Proof. See Appendix.

(6) Finally, the theory of repeated gamescan be applied to the experiment. The repeatednature of the auctions in the experiment makesthe tacit collusion more likely to occur. As men-tioned in (4), perfect tacit collusion cannot besupported as an SPE if there are more buyersthan the items and the buyers play undomi-nated strategy. However, if we allow forrepeated auctions, perfect tacit collusion out-come can be enforceable as an SPE outcomeif the subjects have sufficiently high discount

9. While the competitive equilibrium is envy free, thisproperty is not true for collusive equilibrium in general.Another feature of this collusive equilibrium is that itsallocation is efficient.


rates. One possible strategy that supports theperfect tacit collusion is that if any subject everbids on another subject’s item, then all of thesubjects revert back to competitive biddingsimmediately in all future rounds. In summary,the repeated structure brings extra instrumentsto sustain the tacit collusion outcome.

Together these six features in our experi-mental design create a presumption in favorof the perfect tacit collusion as opposed toother outcomes. Even if perfect tacit collusionhappens in this setting, however, it is unclearwhich feature or the combinations of themfoster the occurrence of perfect tacit collusion.Since each of the six features can be associatedwith the institutional features above, we applytreatments designed to remove one of the fea-tures at a time to develop insights about how itinteracted with other features to promote thecollusive or the competitive outcome.

VI. RESULTS

The Results section is divided into two sub-sections. The first subsection summarizes thepatterns of behavior within the collusion incu-bator environment. The second subsectionfocuses on treatments that examine the robust-ness of collusion and the institutional perturba-tions that hold a potential for mitigating orstopping collusion once it has evolved.

A. Collusion and Features of the CollusionFormation Process

Result 1. In the collusion incubator environ-ments, perfect tacit collusion is reached.

Support. We use the competitiveness of themarket to measure the degree of tacit collusion.Recall that ðcðTotal ProfitÞ 5 ðMaximumSurplus � Total ProfitÞ=ðMaximum Surplus�Competitive ProfitÞÞ is a normalized measurefor market competitiveness: c 5 1 (so the auc-tion market is perfectly competitive) whenitems are sold on average at a price that equalsthe second highest valuation. When c 5 0,every item is sold to the highest-valued bidderat its reservation price, and perfect tacit collu-sion is reached. Figure 2 below shows theevolution of c in all six experiments.

Perfect collusions are reached in all sixexperiments. Furthermore, the speed theyare reached is remarkable. By Round 4, thecompetitiveness has fallen below 0.1 in allbut two experiments. By Round 7, all sixexperiments have reached perfect tacit collu-sion. It is also worthwhile to note that oncesubjects have perfectly tacitly colluded, theycontinue to do so (for at least one more round)before any treatments are applied to the exper-imental environment.

Result 2. Tacit collusion develops over timeand is indicated by (i) falling average prices,(ii) decreasing number of bids, and (iii)decreasing duration of the auction.

Support. The support for the result isdivided into three parts. Each addresses a sep-arate feature of the process.

(i) Prices fall over rounds. Figure 3 plots theaverage winning prices per round until perfect.All six experiments display solid decreases inprices and convergence toward perfect tacit

FIGURE 2

Competitiveness of the Market

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 3 4 5 6 7Rounds

Co

mp

etit

iven

ess e1

e2

e3

e4

e5

e6

2


collusion. The average prices in all but fourrounds are lower than the previous round.To get a sense of how fast the prices decreaseon average, we ran a regression of average pri-ces on their round numbers. The results arereported in Table 2. The average price inthe first round is 534.14, and it drops on aver-age per round by 71.33. We also ran log priceon the number of rounds and find that theaverage price decreases in each round by28%. Although the coefficients are highly sig-nificant despite the small sample size, they aresimply descriptive statistics and have no struc-tural interpretation contents. In fact, that pat-tern of decrease in the average prices is farfrom linear, which is an important fact thatwe will discuss in Result 3.

(ii) The number of bids decreases overrounds. The total number of bids in the roundsof the six experiments is plotted in Figure 4.Although there are a few ‘‘rebounds’’ in thenumber of bids, it is clear that the numberof bids falls over rounds. The rate of decreasein bid number can be found in Table 2 above.

On average, the number of bids drops by 11bids per round. Furthermore, the number ofbids decreases by 41% per round on average.Just as the decrease in prices, the decrease inbids is also not linear and will be discussed fur-ther in Result 3.

(iii) The duration of the auction decreasesover rounds. The duration of the auctionsare plotted in Figure 5. Although there areoccasional rebounds in the duration of somerounds, the downward trend toward perfectcollusion is very clear. To get a quantitativesense about the decrease in duration, we findin Table 2 that on average the durationdecreased by 54 sec per round. In addition,the duration of the auctions decreases by39% each round on average. As in pricesand bids, the decline in durations is again non-linear and will be discussed in Result 3.

Result 3. The convergence toward perfecttacit collusion depends on the behavior ofthe subjects in the first round. In experiments

TABLE 2

Price, Bids, and Duration on Number of Rounds Played*

Dependent Variables Constant Rounds R2 Number of Observations

Prices 534.14** (70.48) �71.33** (12.86) .23 30

Log Prices 6.39** (0.26) �.28** (.05) .24 30

Bids 81.42** (15.80) �11.15** (3.40) .19 30

Log Bids 4.35** (.48) �.41** (.07) .07 30

Duration 456.29** (111.93) �54.37** (24.5) .12 30

Log Duration 6.07** (.56) �.39** (.11) .07 30

*Estimated from random effects model. Standard errors in parentheses.

**Means significant at 5% level.

FIGURE 3

The Figure Shows the Average Price Series for the Six Experiments UntilPerfect Collusion

Average Prices

0

100

200

300

400

500

600

700

1 2 3 4 5 6Rounds

Fra

ncs

e1

e2

e3

e4

e5

e6


with less than two competitive bidders in thefirst round, collusion is immediate. In otherexperiments, subjects switch sequentially fromcompetitive bidding to collusive bidding. Oncea subject becomes a collusive bidder, he rarelyreverts back.

Support. Recall that a subject is classifiedas a tacit colluder if he only bids on items thatsatisfy one of the following three criteria:

(1) The item ‘‘belongs’’ to the subject.(2) The item remains at the reservation

price for more than 60 sec.(3) The item ‘‘belongs’’ to a subject who

has previously bid on the subject’s item.Otherwise, the subject is considered a non-

colluder.

Table 3 below classifies the subjects aseither tacit colluders or non-colluders accord-ing to the definition above. Every subject is

denoted as 0 if he is a colluder in a given roundand is denoted as 1 otherwise.

From Table 3, we see that in Experiments 3and 6, seven out of the eight subjects are tacitcolluders in the first round. In these twoexperiments, perfect tacit collusions are bothreached by Round 3. In other experiments,there are between three to five tacit colludersin the first round, and it takes significantlymore rounds and longer rounds to reach per-fect tacit collusion. The correlation betweenthe number of tacit colluders in the first roundand the number of rounds to reach perfect col-lusion is �.80.

The most striking feature of the table is thatthe evolution of bidding behavior moves inonly one direction: once a subject becomesa tacit colluder, he almost always remains a tacitcolluder. In the 184 transitions in Table 3, thereare only six cases where bidders switch backto competitive bidding from collusive behavior.

FIGURE 4

Number of Bids Until Perfect Collusion

Number of Bids

0

20

40

60

80

100

120

140

1 2 3 4 5 6Rounds

Bid

se1

e2

e3

e4

e5

e6

FIGURE 5

Duration of the Auctions Until Perfect Collusion

Duration

0100200300400500600700800900

1 2 3 4 5 6

Rounds

Sec

on

ds

e1

e2

e3

e4

e5

e6


Furthermore, most of these cases are due tothe insufficiencies of our method of classifyingcollusive behavior. In particular, if the price ofa collusive bidder’s item has been raised toa very high level through repeated biddingby a competitive bidder, then the collusive bid-der might retaliate in the next round by open-ing the bidding on the competitive bidder’sitems. By our definition, this bidder will thenbe classified as non-colluder in that round.Such situation happens several times in ourexperiments.

Second, a competitive bidder might be acci-dentally classified as a colluder. For example,when two non-collusive bidders i and j sharethe highest two valuations on two items and ifi bids on both items before j does, then j is clas-sified as a colluder as long as he only bids onthese two items. This may cause a switch fromtacit colluder to non-colluder for j. We alsoobserved a few such cases in our experiments.

Another dimension that the definition doesnot capture is the variation of bidding behav-iors of subjects within the same group. Even ifthe colluders bid only on their own items, theydo not always start bidding at the reservationprice: some bidders start out at a bid of 500francs on their own items. Furthermore,among the non-colluders, some subjects area lot more competitive than others. This var-iation in the intensive margin requires a moredetailed look at the dynamics of biddingbehavior, which leads us to Result 4.

Result 4. In experiments where collusions arenot immediate, prices and bidding behaviorshave a tendency for discontinuous develop-ment, similar to a regime shift, toward perfectcollusion.

Support. We list our results separately forbids and prices.

Distribution of Bids and Bidding Wars. It isclear from Figure 4 that the number of bidsdid not decrease at a constant speed. In mostrounds, the bid number decreases by less than20 bids with an average of 11 bids per round.However, after Round 2 of Experiment 1,Round 5 of Experiment 2, Round 1 of Exper-iment 4, and Round 1 of Experiment 5, thenumber of bids drops by more than 65 bids.These four rounds also have the most bidsin their respective experiments (except inExperiment 2 in which Round 5 is only twobids less than the round with the most bids).

Interestingly, perfect tacit collusionemerges rapidly after these rounds. In thissense, the dynamics of the auctions experiencea ‘‘regime shift’’ in these rounds. Table 4below reports the average prices and numberof bids at and after ‘‘the regime shift.’’ Thedecreases in the number of bids in theserounds are six times of the average decrease(11 bids per round); the decrease in averageprices in these rounds is three times of theaverage (71 francs per round).

TABLE 3

Summary Statistics for Colluders

Experiment/Round 101 102 103 104 105 106 107 108

E1 (February 18, 2002)

Round 1 0* 1 1 1 0 0 0 0

Round 2 0 1 1 1 0 0 0 0

Round 3 0 0 1 1 0 0 0 0

Round 4 1 0 1 0 0 0 0 0

Round 5 0 0 1 0 0 0 0 0

Round 6 0 0 0 0 0 0 1 0

Round 7 0 0 0 0 0 0 0 0

E2 (April 19, 2002)

Round 1 0 0 1 1 1 1 1 0

Round 2 0 1 1 0 1 1 0 1

Round 3 1 0 1 0 1 0 0 1

Round 4 0 0 1 0 1 0 0 0

Round 5 0 0 1 0 0 0 0 0

Round 6 0 0 0 0 0 0 0 0

E3 (May 10, 2002)

Round 1 0 0 0 0 0 0 1 0

Round 2 0 0 1 0 0 0 0 0

Round 3 0 0 0 0 0 0 0 0

E4 (June 12, 2002)

Round 1 1 1 0 0 0 1 1 0

Round 2 1 0 0 0 0 1 0 0

Round 3 1 0 0 0 0 0 0 0

Round 4 1 1 0 0 0 0 0 0

Round 5 0 0 0 0 0 0 0 0

E5 (June 21, 2002)

Round 1 0 1 0 1 0 1 1 0

Round 2 0 0 0 1 0 1 1 0

Round 3 0 0 0 1 0 1 0 0

Round 4 0 0 0 1 0 1 0 0

Round 5 0 0 0 1 0 1 0 0

Round 6 0 0 0 0 0 0 0 0

E6 (June 29, 2002)

Round 1 1 0 0 0 0 0 0 0

Round 2 0 0 0 0 0 0 0 0

*In the table, a bidder is marked as a 0 if he is classifiedas a colluder and 1 otherwise.


Some detailed look at the bidding behaviorright before the regime change can help under-stand why the regime shift takes place. TakeExperiment 2 (April 19, 2002) as an example;there are 86 bids in Round 5 and only eightbids in Round 6. In Round 5, 39 bids of the86 come from Subject 103, and another 23 bidsfrom 107.

Subject 103 has always bid competitivelyand entered largest number of bids in all pre-vious rounds. Subject 107, however, has dis-covered the opportunity for collusion ratherearly: it enters only two bids in both Round3 and Round 4. To understand why 107 sud-denly entered 23 bids in Round 7, we look atTable 5, which shows the valuation of 103 and107 on Items 1 and 3.

Table 5 shows that 103 and 107 share highvaluations on Items 1 and 3. A price war canarise if 103 attempts to win both items. Wereport in Table 6 the sequence of bids madeon Item 1 and Item 3 in Round 5. Subject107 started by bidding the reservation priceon ‘‘his item,’’ Item 3. Soon after, however,103 also bid on Item 3. In response to this,107 initially added the minimal incrementonto Item 3, possibly hoping that 103 can stopbidding. This did not happen and 103 contin-ued to raise bids on Item 3. Soon after, 107

retaliated and bid on Item 1, which is 103’sitem. A bidding war broke out between 103and 107 on Items 1 and 3 and a total of 43 bidswere placed onto these two items. WhenRound 6 ended, 103 won Item 1 at a price

TABLE 4

Number of Bids and Average Prices at and

after the Regime Shift

Experiment RoundNumber of

BidsAveragePrices

E1 (February 18, 2002) 2 101 558

3 34 310

E2 (April 19, 2002) 5 86 368

6 8 100

E4 (June 12, 2002) 1 108 427

2 33 210

E5 (June 21, 2002) 1 128 651

2 28 269

TABLE 5

Valuation of 103 and 107 on Items 1 and 3

Subject/Items 1 3

103 867 773

107 771 861

TABLE 6

A Bidding War

Time Item ID Price

3,055 3 107 100

3,069 1 103 200

3,077 3 103 200

3,099 3 107 210

3,120 3 103 220

3,132 3 107 230

3,144 3 103 300

3,160 3 107 310

3,173 3 103 400

3,178 1 107 300

3,185 3 107 410

3,199 1 103 310

3,211 1 107 350

3,226 1 103 360

3,237 1 107 400

3,247 1 103 450

3,257 1 107 500

3,264 3 103 500

3,276 1 107 510

3,282 3 107 510

3,291 1 103 520

3,302 3 103 520

3,309 3 107 530

3,326 3 103 550

3,334 3 107 560

3,341 1 107 550

3,344 3 103 600

3,356 3 107 610

3,367 1 103 600

3,376 3 103 650

3,387 1 107 650

3,392 3 107 700

3,417 1 103 761

3,429 3 103 710

3,436 1 103 771

3,445 3 107 720

3,453 1 103 781

3,472 3 103 730

3,482 3 107 740

3,507 3 103 752

3,523 3 107 773

3,529 1 107 791

3,546 1 103 801


of 801 francs and 107 won Item 3 with 773francs.

It is worthwhile noting that 107 placed onItem 1 a bid of 791 francs, 20 francs higherthan his valuation. If 107 won the item at thatprice, it would be at a loss. Since biddingabove one’s valuation is a weakly dominatedaction, one might think 107 is irrational. In theexperiment, however, this bid may have sur-prised 103 and caused him to revise his biddingbehavior. Indeed, only eight bids are cast thenext round. We believe that these ‘‘spiteful’’actions are helpful in changing the behaviorof other subjects. A similar incidence happensin Round 5 of Experiment 1, in which 102 (acollusive bidder) suddenly raises the price by240 on the item of 103, who has been biddingcompetitively. After the ‘‘spiteful’’ behavior of102, 103 stops bidding competitively in thenext round.

B. Remedies to Prevent a Collusive Equilibrium

Once perfect tacit collusion has emergedand persisted, various treatments are appliedto break it. These methods include: a) forcedanonymous bidding, b) removal of commonknowledge of preference, c) change of the end-ing rule of the auction, d) removal of severalitems for sale, e) destruction of the expecta-tions of the subjects, and f) announcementof the final round.

In the first three experiments, we examinethe effects of these treatments in a very explor-atory manner. As our understanding increased,we consistently applied these treatments in thesame order as listed in the previous paragraph.New treatments are applied only if the previoustreatment has failed to cause or sustain compe-tition and the bidding returns to almost perfecttacit collusion. When new treatments areapplied, we keep all the previous treatments.In this way, we are able to measure the com-bined effects of all the treatments.

Result 5. Forced anonymity in bidding has noeffect.

Support. Once perfect tacit collusion haspersisted for two rounds, the first step we mostoften take to disrupt it is to force anonymousbidding. As mentioned in Section 5, anony-mous bidding weakens the monitoring tech-nology of the subjects, so they cannot target

their punishment on the particular deviator.This makes deviation from collusive equilib-rium more tempting and the collusion lesssustainable.

To implement anonymous bidding, weblank the ID associated with the bids, so sub-jects only observes that a bid is entered and notwhich subject made the bid. Forced anonym-ity has virtually no effect in breaking the tacitcollusion.

Table 7 documents the various statistics ofprices, bids, and durations of the auctionsonce the IDs are blanked. The average pricesare below 110 francs in 12 out of 15 roundsand never exceed 160 francs. Occasionally,there are a few attempts that moved the out-come away from equilibrium. But, theseattempts almost never generate a price above200, and the average prices fall below 105francs within four rounds.

Result 6. Removal of common knowledge ofpreferences by taking away the valuation sheethas little effect on breaking the tacit collusion.

Support. Once the subjects return to theperfect tacit collusion with their IDs blanked,the next step we most often take is to removecommon knowledge of preferences. Lack ofcommon knowledge forces the subjects toform their own expectations of the valuationsof other subjects and complicates the coordi-nation of tacit collusion. Since the IDs of thesubjects remain blanked, it is even more entic-ing for the subjects to bid on more than oneitems and destroy the tacit collusion.

We remove common knowledge of prefer-ences by stop passing out the valuation sheets.Instead, subjects learned about their own val-uations from the computer screens. The re-moval of common knowledge of preferenceshas very small effects in breaking the collusion.

Table 8 documents the effects. After thevaluation sheets are removed, some subjectsfight over a few items in some rounds. Butthe fights are uncommon and in general ofsmall scale. The winning prices are rarelyabove 200. Moreover, bidding wars never lastmore than one round: the average winning pri-ces fall below 105 francs in one round after thevaluation sheets are first taken away. In otherwords, within two rounds after the valuationsheets are taken away, almost perfect tacitcollusion reappears.


It is interesting to note that in Table 8 thefinal winning prices are not always 100. Forexample, in Round 7 and 8 of Experiment6, subject 104 bid 104 instead of 100. This sug-gests that some bidders have tried to use thelast digit number of the bid to signal their iden-tities, possibly trying to enforce the collusiveequilibrium.

Result 7. Changes in the structure of the gameto eliminate the collusive equilibrium byswitching the ending rule to fixed length resultsin some inefficiencies in allocations but has lit-tle immediate effect in prices.

Support. When removal of information failsto break tacit collusion, we change the ending

TABLE 7

Summary Statistics after IDs are Blanked

Experiment /Round Highest Price Lowest Price Average Price Duration Time Total Bids


Round 9 100 100 100 22 8

Round 10 200 100 113.5 29 8

Round 11 100 100 100 14 8

E3 (May, 10, 2002)

Round 6 100 100 100 14 5

Round 7 100 100 100 15 5

E4 (June 12, 2002)

Round 8 121 100 108 115 17

Round 9 150 100 106.25 35 9

Round 10 255 100 135.13 134 16

Round 11 100 100 100 16 8

E5 (June 21, 2002)

Round 7 160 120 151.25 214 45

Round 8 160 100 107.5 15 8

Round 9 110 100 101.25 37 9

E6 (June 29, 2002)

Round 4 100 100 100 19 8

Round 5 120 100 102.63 44 9

Round 6 101 100 101.13 18 8

TABLE 8

Summary Statistics after Valuation Sheets are Taken Away

Experiment/Round Highest Price Lowest Price Average Price Duration Time Total Bids

E2 (April 19, 2002)

Round 9 807 100 263.25 165 20

Round 10 100 100 100 21 8

E3 (May 10, 2002)

Round 3 100 100 100 20 8

E4 (June 12, 2002)

Round 12 170 151 160.88 228 37

Round 13 100 100 100 17 8

E5 (June 21, 2002)

Round 10 310 100 127.5 40 10

Round 11 110 100 101.25 23 8

Round 12 120 100 102.5 19 8

E6 (June 29, 2002)

Round 7 104 100 100.5 18 8

Round 8 104 100 100.63 19 8


rule in Experiment 2. In the collusive conduciveenvironment, an auction ends only if no newbids are entered for a consecutive 30 sec. Byturning the variable ending rule into a fixedduration rule, we change the structure of theauction by transforming it from a possiblyinfinite horizon game to one with a finite hori-zon. Theoretically, the new ending rule encour-ages subjects to deviate from the collusiveequilibrium immediately before a round ends.Anticipating this, the subjects might engagein competitive bidding earlier on.

We change the ending rule in Round 11 and12 in Experiment 2 (April 19, 2002) to a fixedduration of 30 sec, so all legal bids have to beentered within 30 sec after the auctions start.

Table 9 documents the effect of fixed endingrule. The prices of the winning items remainlow, although the allocations of the items areno longer efficient. There are some deviationsat the end of the auction. In Round 11, subject107 bids 120 francs at the last second on Item 2,which is 102’s item. In Round 12, subject 107bids on Item 8, which is 106’s item. Subject107’s behavior is consistent with what isreported in Ockenfels and Roth (2007), whofinds that various EBay bidders enter their bidsin the last second. In our environment, we haveeight subjects, and 107 is the only deviator inboth rounds. After these two rounds, we restorethe old ending rules. We do not observe retal-iations from subjects whose items are bid awayby others at the last second.

We restore the old ending rule mainlybecause there is a technical problem withthe computer system in Round 12, as one sub-ject complains that his last-second bid fails togo through. As a result, there is no winner forItem 5. We are concerned that if many subjectsenter their bids in the last several seconds,many bids might not go through. This can leadto confusion, making the experimental out-come hard to interpret. Because of this, wedo not test the effects of fixed duration endingrule in later experiments.10

Result 8. Destruction of short-term symmetryhas some effects, but the experiment still con-verges to perfect tacit collusion in the end.

TABLE

9

Su

mm

ary

Sta

tist

ics

for

the

Fix

edD

ura

tio

nR

ule

Experim

ent/Round

Item

1Price

(Winner)

Item

2Price

(Winner)

Item

3Price

(Winner)

Item

4Price

(Winner)

Item

5Price

(Winner)

Item

6Price

(Winner)

Item

7Price

(Winner)

Item

8Price

(Winner)

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10. On the other hand, it is possible that, as pointedout by a referee, if subject 107 continued to win morethan one item in the future periods, the collusion mightbreak up.


Support. Two crucial design features thatcould facilitate tacit collusion are the ordinalsymmetry and ‘‘item alignedness’’ propertiesof preference. Theoretically, symmetry and‘‘item alignedness’’ makes the tacit collusionthe unique Pareto Equilibrium. In experimen-tal literature, symmetry can also help facilitatethe collusion equilibrium by making it focal.Our expectation was that the removal of thesefeatures would break tacit collusion.

To implement this, we remove several itemsfor sale in some of the experiments.

Experiment 3 is the first time we removeitems for sale. In that experiment, we rotatethe items to be removed. From Rounds 4 to7, three items are removed for sale. FromRounds 8 to 11, five items are removed in eachround. In later experiments, we have standard-ized our method to always remove Items 1 to3. Surprisingly, the destruction of these twofeatures does not move the equilibrium to

competitive outcome, and almost perfect tacitcollusion returns in all experiments.11

Table 10 reports the price, bid, and dura-tion of the auctions with items removed forsale. Removing the items for sale has greater(incremental) effect in pushing the subjectsaway from collusive equilibrium than the pre-vious treatments. In Experiment 5, the averageprice jumps to 680 francs immediately after thefirst three items are blocked for sale. Further-more, prices stay at relatively high levels foranother two rounds and even attain an aver-age price of 743 francs in Round 15, a roundwith the most competitive scenarios in the

TABLE 10

Summary Statistics after Some Items are Removed for Sale

Experiment/Round Highest Price Lowest Price Average Price Duration Time Total Bids Items Blocked

E3 (May 10, 2002)

Round 4 100 100 100 14 5 1,4,5

Round 5 100 100 100 14 5 3,4,7

Round 6 100 100 100 14 5 2,5,8

Round 7 100 100 100 15 5 2,3,6

Round 8 100 100 100 6 3 2,3,4,6,7

Round 9 120 100 110 48 6 2,3,4,5,8

Round 10 100 100 100 6 3 1,2,6,7,8

Round 11 100 100 100 3 3 4,5,6,7,8

E4 (June 12, 2002)

Round 14 280 160 196 233 37 1,2,3

Round 15 180 140 160 202 27 1,2,3

Round 16 122 100 104.38 30 7 1,2,3

E5 (June 21, 2002)

Round 13 710 660 682 391 68 1,2,3

Round 14 780 410 596 174 38 1,2,3

Round 15 800 710 743 682 117 1,2,3

Round 16 130 100 110 38 9 1,2,3

Round 17 610 310 428 480 51 1,2,3

Round 18 610 100 229 149 27 1,2,3

Round 19 110 100 102 20 6 1,2,3

Round 20 120 100 104 43 7 1,2,3

Round 21 120 100 106 36 7 1,2,3

E6 (June 29, 2002)

Round 9 110 100 103.38 23 6 1,2,3

Round 10 155 100 113.25 47 9 1,2,3

Round 11 107 100 101.38 9 5 1,2,3

11. One possibility is that subjects treat each round ofauction as a stage game in a repeated game and believethat refraining from competing in the current period willbe rewarded by collusive behaviors in the future. However,this logic will not go through if the total number of roundsof auctions are known to be finite.


experiment: 117 bids are entered and theauction lasts for 682 sec, the third-longestround in the entire experiment series.

Although removing the items for sale hasmore impact than the previous treatments,its effects remain modest and diminish overtime. In Experiment 3, perfect collusion per-sists even when five items are blocked in somerounds. In Experiments 4 and 6, although theprices rise somewhat after the items areblocked, their levels are low and the auctionsare still filled with collusive behavior. Even inExperiment 5, where removal of items has ledto the most competitive behaviors among theexperiments, the prices converge to the perfectcollusive level after six rounds.

Result 9. Destruction of common expecta-tions by surprise competitive entry: a) leadsto competitive biddings on the item with entry,(b) spreads the competitive behavior to itemswith single high-valued bidder, and (c) causesprice wars in future rounds of auctions withsymmetry.

Support. In the collusive equilibrium, a sub-ject with a valuation above 800 francs on anitem wins it at the reservation price. Once col-lusion has persisted for more than 10 periods,it is natural for the subjects to expect that thiswill continue to happen. To destroy this com-mon expectation, we typically ‘‘switch’’ thevaluations of some subjects to generate multi-ple bidders with valuations above 800 on someitems. Take Figure 1 as an example, subject103’s original valuation is 707 on Item 7and 889 on Item 3. In this treatment, we switchhis valuations on these two items, so subject103’s new valuations become 707 on Item 3and 889 on Item 7. This change destroys thesymmetry of valuation by adding a surpriseentry: there are now two bidders, 103 and108, both having valuations above 800 on Item3. It also destroys the ‘‘item aligned’’ propertyof the preference: 103 no longer has the highestvaluation on the Item 7, the item he valuesmost. Of course, since the subjects only knowabout their own valuations at this stage of theexperiment, their expectations of the valua-tions of the other subjects come probably frompast experience. Therefore, the subjects prob-ably do not expect this change of valuations tohave happened. In each experiment, this sur-prise entry takes place in two consecutiverounds. After these two rounds, we remove

this treatment so that each item has again onlyone subject with valuation above 800 francs.

Table 11 reports the details of the changeand the winning prices of all items for sale.Three patterns emerge out of destruction ofcommon expectation. First, the prices of itemswith multiple high-valued subjects rise to veryhigh levels except in Round 17 of Experiment4 and in Round 12 of Experiment 6. Further-more, the prices on some items are even abovethe competitive levels. For example, in Round13 of Experiment 6, Items 4, 5, and 6 are soldat 834, 863, and 862 respectively, which arehigher than 746, 834, and 808, the secondhighest valuations on these items. This sug-gests that subjects have used spiteful behavioreither as pure retaliation or as an urge forreturn to collusive equilibrium.

Second, there is a contagion of price warfrom initially contested items to items with sin-gle high-valued subject. In Round 13 of Exper-iment 6, only Items 5 and 6 have more than onesubject with valuations above 800 francs. Con-sequently, the only initial bid wars are between101 and 102 on Item 6 and between 106 and 107on Item 5. The price of Item 4 remains at 104francs even when the prices of Items 5 and 6have both risen above 800. Subject 101, whois potentially frustrated by his loss in the pricewar on Item 6, soon starts to bid on Item 4. Theprice of Item 4 rises quickly, and 101’s final bidon it is 800, more than 100 above his valuation.Similar patterns of bidding wars exist in otherexperiments as well.

Third, once the common expectation is bro-ken, price wars appear in later rounds whereall items have a single subject with valuationabove 800 francs. Some of the bidding warsseem to result from frustrations and angerin the previous round. For example, subject104 doesn’t win any item in Round 13 ofExperiment 4, even though his valuation onItem 2 is above 800. In Round 14, 104 startedout by bidding 749 on Item 1 and 769 on Item3, even though his valuations for these twoitems are only 249 and 190 francs respectively.

The spiteful behaviors gradually die out andprices fall down to collusive levels in mostexperiments after more rounds of auction oncethe treatment is removed. However, the conver-gence toward collusive equilibrium appears dif-ficult after expectations have been destroyed. InExperiment 3, for example, the prices never fallto the perfect collusive level after commonexpectations are destroyed. There are two items


sold at above 500 francs five rounds after thetreatment is removed. Furthermore, even ifthe prices approach the collusive level, the col-lusive equilibrium does not seem to be veryrobust. This can be seen from the ‘‘last round’’behaviors of the subjects.

Result 10. Destruction of the repeated natureof the game creates competition if there arefewer items than bidders and common expec-tations are destroyed before.

Support. In Experiments 1 and 4, the exper-imenter announced before the last round that‘‘the next round will be the last round of theexperiment.’’ Table 12 reports the effects ofannouncing the final round on the final winningprices. Two dramatically different behaviors areobserved. In Experiment 1, the experimenterannounced that Round 11 is the last round.The subjects respond very little to thisannouncement. The prices remain low, as thehighest price of the items is only 120 and theaverage price is 103.75. This is very close tothe perfect tacit collusion.

In Experiment 4, however, the subjectsresponded dramatically to the announcementthat Round 23 was the last round. Before that,

the outcome of Round 22 is fairly collusive:the highest price is only 200 and the averageprice is 143. In Round 23, subjects bid aggres-sively on all of the five items. The average priceis 694.75, and there are three items pricedabove 700. Furthermore, item 3 is sold at824, 25 francs higher than the second highest-valuation on this item.12

There are several possible reasons for thecompetitive bidding behaviors. For example,there are only five items for sale in this round.Therefore, perfect collusion cannot be supportedas a NE if the subjects follow undominatedstrategies. It is also possible that collusive equi-librium is less stable after the common expect-ations have been destroyed. In general, thelast-round behavior may be affected by any his-tory of previous competitive outbreaks. It willbe interesting to explore further about the keyfactors that affect the final round behavior.

VII. SUMMARY OF CONCLUSIONS

The fundamental results reflect the discov-ery of a collusion incubator environment in

TABLE 11

Price Statistics after Common Expectation Destruction

Experiment/Round Chosen/ItemsItem 1Price

Item 2Price

Item 3Price

Item 4Price

Item 5Price

Item 6Price

Item 7Price

E3 (May 10, 2002)

Round 12 1,2 870 873 730

Round 13 1,2 843 840 669

E4 (June 12, 2002)

Round 17 6,8 410 530 460 420 460

Round 18 6,8 860 770 876 751 631

E5 (June 21, 2002)

Round 22 5,* 750 877 760 440 430

Round 23 5,6 600 580 843 711 800

E6 (June 29, 2002)

Round 12 5,6 100 200 302 100 100

Round 13 5,6 834 863 862 100 100

Notes: On May 10, 2002, in Round 12, 101’s valuation on Items 2 and 5 was switched; 106’s valuation on Item 1 andItem 8 was switched. In Round 13, 102’s valuation on Items 2 and 6 was switched; 105’s valuation on Items 1 and 3 wasswitched. On June 12, 2002, in Round 17, 102’s valuation on Item 6 was switched to 822; 104’s valuation on Item 8 wasswitched to 821. In Round 18, 102’s valuation on Item 3 and Item 6 was switched; 108’s valuation on Item 8 and Item 4was switched. On June 21, 2002, in Round 22, 104’s valuation on Item 2 and Item 5 was switched; 107’s valuation on Item 1and Item 5 was switched. In Round 23, 101’s valuation on Item 2 and Item 6 was switched; 106’s valuation on Item 1 andItem 5 was switched. On June 29, 2002, in Round 12, 104’s valuation on Items 2 and 6 was switched; 107’s valuation onItems 1 and 5 was switched. In Round 13, 101’s valuation on Items 2 and 6 was switched; 106’s valuation on Items 1 and 5was switched.

*There is an unintended switching, so 101, 104, and 107 all had high values in Item 5.

12. However, there is no spiteful behavior here, as thehigh price results from a jump bid of the winner.


which tacit collusion evolves naturally withoutconspiracy and without intervention orencouragement by the experimenter and with-out any special facilitating device. The alloca-tion, including prices, is predicted accuratelyby a specific solution to a game-theoreticmodel of the auction process. In this solution,which we call the collusive outcome, the allo-cation maximizes the joint surplus of buyers.13

In the collusion incubator environment, thefolding patterns of preferences are knownand opportunity of coordination into mutu-ally beneficial patterns of behavior can be eas-ily identified. In addition, the institutionalenvironment supplies opportunities for retali-ation for unwanted competitive behavior. Inthis environment, perfectly collusive strategiesdevelop quickly.

Our experiment provides the first environ-ment where tacit collusion arises naturallyamong more than two subjects. In our designof the environment, we have been conservativeand included many potentially collusion-conducive features because past experimentssuggest that the competitive force is verystrong. Although combination of the featuresmakes it highly special, our environment is animportant step toward a better understandingof tacit collusion if not collusion itself. First,our environment generates rapid and robustcollusion, so it can serve as the basis for futureresearch on how tacit collusion may be bro-ken. Second, the structural and institutionalfeatures in our environment form a sufficientcondition to obtain collusion. This providesa starting point for future research to identifya minimalist set of features that can generatecollusion. Finally, our environment helps cau-tion the practitioners that collusion is likely to

occur if several of our features appear in theauction design.

The dynamics of adjustment in the collu-sion incubator environment exhibit distinctpatterns. In some cases, the tacit collusion isimmediate. That is, from the structure of theenvironment alone agents deduce and imple-ment a commonly held strategy of tacit collu-sion. In these cases, there is no learning,retaliations, or adjustments. The advantagesof tacit collusion result from cognition aloneand are implemented. In the cases in whichthe tacit collusion does not occur immediately,the system typically starts with the competitiveequilibrium outcome. As the rounds proceed,prices decline gradually, until there is oneround that appears as a "regime shift" and per-fect tacit collusion is reached immediatelyafterwards.

Because tacit collusion is successfully cre-ated, an opportunity is created to conducta few experiments using the ‘‘exploratorymethodology.’’ The fact that a tacit collusionexists enables us to tentatively explore some ofthe many dimensions that can be imagined as‘‘remedies.’’ Rather than choose one ‘‘rem-edy’’ and collect many observations on it, sev-eral remedies are explored and done so insequence. Thus, we have produced a prelimi-nary ‘‘map’’ of a varied and complex land-scape of institutions together with hintsabout where different remedies might lead.While the results are ‘‘hints’’ rather than firmconclusions, it is hoped that this map willguide researchers using more surgically preciseexperimental designs through this complexterrain. Our summary that follows shouldbe read from that perspective.

Once collusion has developed, it is difficultto disrupt and appears to be held in place bya pure system of belief as opposed to institu-tions and information that enable the main-tenance of a Perfect Bayesian Equilibrium.

TABLE 12

Winning Prices after the Announcement of Final Round

Experiment /RoundItem 1Price

Item 2Price

Item 3Price

Item 4Price

Item 5Price

Item 6Price

Item 7Price

Item 8Price


Round 11 100 100 100 100 100 100 100 120

E4 (June 12, 2002)

Round 23 N/A N/A N/A 750 510 824 610 780

13. In the collusive outcome, the final allocation is alsoenvy free.


The sequential removal of informational andinstitutional features that are prominent inthe creation of equilibria in the models con-sistent with the collusive pattern of behav-iors do nothing to change the behavior.Tacit collusion remains, or if disrupted bythe environmental change, returns as theprominent pattern of behavior.

The treatment or ‘‘remedy’’ that effectivelyeliminates the collusion is a change that cre-ates competition in one or two of the markets.If an agent finds himself or herself with a com-petition for his/her item, the competitionspreads to other markets. In a sense, theunexpected introduction of a ‘‘maverick’’under circumstances in which almost all ofthe features of the collusive compatible envi-ronment have been removed destroys the col-lusive behavior. The dramatic change inbehavior could be the result of destroyedbeliefs about the behavior of others. Thischange nudges competition from the con-tested market to other markets. An under-standing of this process of contagion isneeded.

A closing comment about methodology isin order. Some feel that a proper experimen-tal methodology has the experimental envi-ronment mirroring some aspect of thenaturally occurring world as closely as possi-ble, satisfying some abstract quality of‘‘external validity.’’ While that philosophymight be useful for some questions, it isnot useful as a general guide to experimentaldesign and is not applied here. Experimentalmethods have important uses as tools thatallow us to study the principles at work ingeneral. It is not unusual for those principlesto be most clearly seen under conditions thatnature has not and perhaps never will pro-vide for us. While the collusion incubatorenvironment may never be found occurringnaturally, it nevertheless allows us to createand study the development and evolutionof tacit collusion in multiple item auctions,which has not been successfully created orstudied otherwise.

Clearly the results presented here are pri-marily empirical. While we are able to pro-vide some theoretical insights, a world ofquestions remains and even new questionsare stimulated by what we report. Such ques-tions are about the detail of the strategies thatmight be at work, alternative solution con-cepts that might be applicable, theories of

repeated games that might be applied or ana-lyzed with new or extended experiments,learning models that might be applicable,etc. In addition, the simultaneous, increasingprice auction has been applied in field settingsand there are open questions about how thedata from these experiments might helpexplain field data and whether or not tacitcollusion was operating. These are all inter-esting questions that were not addressed herebut we hope that the tools provided here willhelp answering them.

APPENDIX: PROOF OF PROPOSITION 1

Lemma 1. Let m be the minimal bid increment. Assumethat for every Item k, we have 2m,Vk � V 2

k , whereVk is the highest valuation on Item k and V 2

k is its secondhighest valuation. If bidder i does not win any item ina subgame perfect equilibrium outcome, the final priceon i’s item must be greater than or equal to Vi � m.

Proof. Suppose the contrary. Then bidder i’s profit is zero,and the price on his item, pi , is larger than or equal toVi � m. Now suppose i bids MaxfVi � m; pi þ mg on Itemi and bids Vj � m� e on items such that Vj � 2m. pj forj 6¼ i, where e 5 Minj6¼i;Vj�2m. pjfVj � 2m� pj;mg. In thisway, the price of each item is larger than its highest valua-tion minus twice of the minimal bid increment, which islarger than the second highest valuation. Therefore, anyNash Equilibrium of this subgame must have that no bidderwill want to bid on item that he doesn’t have the highestvaluation and that each bidder j whose item has been bidby i will bid on his own item. Therefore, bidder I can guar-antee himself positive payoff in this subgame, which is a con-tradiction. Q.E.D.

Proof of Proposition 1

First, the discussion above indicates that every bidderwinning his item at the reservation price can be supportedas a SPE. This implies that for a subject, call it subject A,to acquire a higher level of profit than in perfect collusion,he must obtain positive profits from at least two items.This implies that A wins at least two items. Now we par-tition the set of items into W and L, where W includes allof items whose highest-valued bidder is a winner (wins atleast one item) and L includes items whose highest-valuedbidder does not win any item. Since A wins at least twoitems, at least one bidder is itemless, so L is not empty.Now by lemma 1, the prices of items in L must be higherthan the second highest valuations on these items. Thisimplies that the winners of these items in L must sufferlosses from them. Because A has positive profits fromat least two items in W, one winner must win zero itemfrom W. The only items this winner wins must come fromL and thus his total profit from the auction is negative.This leads to a contradiction because in equilibriumany bidder can at least guarantee himself nonnegativeprofits. Q.E.D.


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