Managing Growth to Achieve EfÞcient Coordination in Large...

Managing Growth to Achieve Efficient Coordinationin Large Groups

By ROBERTO A WEBER

Previous experiments using the minimum-effort coordination game reveal a strikingregularitymdashlarge groups never coordinate efficiently Given the frequency withwhich large real-world groups such as firms face similarly difficult coordinationproblems this poses an important question Why do we observe large successfullycoordinated groups in the real world when they are so difficult to create in thelaboratory This paper presents one reason The experiments show that eventhough efficient coordination does not occur in groups that start off large efficientlycoordinated large groups can be ldquogrownrdquo By starting with small groups that findit easier to coordinate we can add entrantsmdashwho are aware of the grouprsquoshistorymdashto create efficiently coordinated large groups This represents the firstexperimental demonstration of large groups tacitly coordinated at high levels ofefficiency (JEL C72 C92 M53)

Coordination is an important problem foreconomics and organization (Ronald H Coase1937 James M March and Herbert A Simon1958 Thomas C Schelling 1960 Kenneth JArrow 1974 Russell W Cooper 1999) Tacitcoordination deals with situations in which eco-nomic actors attempt to match the actions ofothers without knowing what these others willdo and without an agreement about what to do1

This paper addresses the difficulty of tacitcoordination in large groups Considerable ex-perimental evidencemdashmostly from a gameknown as the minimum-effort (or ldquoweak-linkrdquo)coordination gamemdashshows that group size has

a strong effect on the ability of groups to coor-dinate (eg John B Van Huyck et al 1990Weber et al 2001) Large groups of peoplewho cannot speak to one another almost nevercoordinate successfully and repetition alonedoes not solve the problem2

This observation is inconsistent howeverwith the real world where we observe groupsmuch larger than those in experimentsmdashsuch asfirms and communitiesmdashwhere coordinationplays a crucial role but where these largegroups have managed to coordinate success-fully This observation of large efficiently co-ordinated groups outside the laboratory createsa puzzle If ldquolargerdquo laboratory groups cannotcoordinate efficiently how do large communi-ties and firms often do so

This paper demonstrates that the ability oflarge groups to coordinate successfully can becritically affected by the grouprsquos growth pro-cess As previous research shows coordinationis easy in small groups Therefore members ofa small group such as the founding members ofa firm do not face substantial difficulty in co-

Department of Social and Decision Sciences CarnegieMellon University Pittsburgh PA 15213 (e-mailrweberandrewcmuedu) Thanks to Colin CamererDavid Grether Richard McKelvey Tom Palfrey JohnPatty various seminar participants and two anonymousreferees for valuable comments and suggestions Thanks toAlex Markle and Scott Rick for help in conducting theexperiments This research was funded by the Russell SageFoundation Behavioral Economics Small Grant Programand by Carnegie Mellon University

1 Examples of coordination problems include buyers orsellers searching for a market workers with complementaryproduction tasks or consumers purchasing products withnetwork externalities In game-theoretic terms coordinationproblems arise when there are multiple equilibria and play-ers must tacitly resolve which one to play

2 In fact even with communication large-group coordi-nation is still difficult (see for instance Weber et al 2001Ananish Chaudhuri et al 2001) Gary Charness (2000)shows that communication helps in small groups

114

ordinating activity efficiently Once they havedone so they can establish a set of self-reinforcing rules or norms governing what ac-tions are appropriate and these allow the groupto continue to coordinate activity successfullyAs the group grows entrantsrsquo exposure to thesenorms allows these entrants to be aware of theappropriate behavior and creates an expectationfor everyone in the group of what everyone else(including the entrants) will do Thus by coor-dinating efficiently as a small group growingslowly and exposing entrants to the grouprsquosprevious norms a group can become large andefficiently coordinated

The rest of this paper demonstrates the influ-ence of growth on tacit coordination using anexperiment involving the minimum-effort coor-dination game The experiment is motivated bya simple modelmdashdiscussed briefly later in thispapermdashwhich shows why growth should workwhen entrants are exposed to the grouprsquos his-tory In the experiment efficiently coordinatedlarge groups which are impossible to obtainwhen a group starts out at a large size areldquogrownrdquo in the laboratory by starting with smallgroups and adding a few entrants at a time whoare exposed to the grouprsquos history This paperproduces the only laboratory demonstration ofthe regular occurrence of efficient tacit coordi-nation among large groups The experiment alsoreveals however that the ability of entrants toobserve the grouprsquos history is crucialmdashif theyare not aware of what incumbent members didin the past coordination failure results

I The Minimum-Effort Coordination Game

The minimum-effort or weak-link coordina-tion game was first studied experimentally by

Van Huyck et al (1990)3 In the gamemdashwhichis a seven-effort-level version of the stag huntgame (see Crawford 1995)mdashn players choosefrom a set of integers that can be thought of asorderable strategies such as effort or contribu-tion levels Every playerrsquos payoff is a functionof her choice and the minimum choice of all nplayers (thus the term ldquoweak-linkrdquo since everyplayerrsquos payoff is partially determined by thelowest choice in the group) Table 1 presents thepayoff to each player as a function of her choiceand the minimum choice4

When everyone makes the same choice andtherefore receives the same payoff (representedby cells along the diagonal) the outcome is oneof the gamersquos seven pure-strategy Nash equilib-ria The equilibria differ because those corre-sponding to higher choices also yield higherpayoffs Therefore ldquomore efficientrdquo coordina-tion corresponds to all players making higherchoices in equilibrium and the Pareto-optimal (orefficient) equilibrium results when all players se-lect the highest choice 7 and receive $090

Since all symmetric outcomes including theefficient one are equilibria this game does nothave the incentive problem present in the pris-onerrsquos dilemma Nonetheless the efficient equi-librium may not be easy to achieve becauseplayers are faced with strategic uncertainty Ev-eryone may recognize the efficient equilibrium

3 See also Jack Hirshleifer (1983) for an early theoreticaldiscussion of the problem underlying the game

4 The game in Table 1 is the same as Game A in VanHuyck et al (1990) except every payoff is lower by $040Of course real payoffs also differ due to differences inexperimental location and time However experiments us-ing the two sets of payoffs produce similar small versuslarge group results

TABLE 1mdashPAYOFFS (IN DOLLARS) FOR MINIMUM-EFFORT GAME

Minimum choice of all players

7 6 5 4 3 2 1Playerrsquos choice 7 090 070 050 030 010 010 030

6 080 060 040 020 000 0205 070 050 030 010 0104 060 040 020 0003 050 030 0102 040 0201 030

115VOL 96 NO 1 WEBER MANAGING GROWTH TO ACHIEVE EFFICIENT COORDINATION

but may be unsure of what others will doTherefore players may choose something otherthan 7 particularly when they think it is likelythat someone else will choose something otherthan 7 Simply being unsure about what otherswill do may lead to choices other than 7

Previous experiments with minimum-effortcoordination games established clear regulari-ties Tacit coordination on the efficient equilib-rium is impossible for large groups Of theseven sessions initially conducted by VanHuyck et al (1990) with groups of size 14 to 16the minimum in all sessions after the third pe-riod was the lowest possible choice For smallgroups (n 2) playing in fixed pairs coordi-nation on the efficient equilibrium was mucheasiermdashit was reached in 12 of 14 (86 percent)of the groups (a result replicated by Colin FCamerer and Marc Knez 2000) Table 2 sum-marizes the distributions of fifth-period minimain several different experiments all using vari-ants of the Van Huyck et al game in whichplayers choose integers from 1 to 7 and choos-ing 7 is efficient

The effect of group size could hardly bestronger Subjects in a group size of 2 are al-most assured to coordinate on the efficient equi-librium while subjects in larger groups (six ormore) are almost assured to converge to theleast efficient outcome in which at least oneplayer chooses 1 Thus there is a strong nega-tive relationship between a grouprsquos size and theability of its members to coordinate efficiently5

II Growing Efficient Coordination

Given the link between coordination in min-imum-effort games and coordination problemsfaced by such real-world groups as firms (egCamerer and Knez 1997 Ashish Nanda 1997Jody H Gittell 2001) the work above suggestsan impossibility to regularly obtaining efficientcoordination in large groups outside the labora-tory This is inconsistent however with theobservation that there exist efficiently coordi-nated real-world groups To see how we mightresolve this apparent inconsistency we need tobegin by recognizing that few large groups startoff at a large size Most groups in fact beginsmall when solving coordination problems iseasier according to the experimental literaturenoted above Once successfully coordinatedthen these groups might be able to remaincoordinated as entrants are addedmdashparticularlyif the entrants are aware of the grouprsquos previoussuccess

Using a simple dynamic model based on Vin-cent P Crawfordrsquos (1995) model of adaptivedynamics in order-statistic coordination games(which include minimum-effort games) Weber(2005) shows how growth can produce largeefficiently coordinated groups (see alsoWeber 2000) The model assumes that playerirsquos discrete action in period t (xit) is determinedby a continuous latent strategy variable (ait) (for

5 Once these groups reached the inefficient outcomethey were not able subsequently to increase the minimum

This result has also been replicated using different versionsof the game (eg Weber et al 2001) A few studies showthat changing incentives improves coordination for groupsof seven to eight players (Siegfried Berninghaus and KarlEhrhart 1998 Gary Bornstein et al 2002)

TABLE 2mdashDISTRIBUTIONS OF FIFTH-PERIOD GROUP MINIMA IN VARIOUS 7-ACTION MINIMUM-EFFORT STUDIES

(1 inefficient 7 efficient)

Minimum choice in fifth periodGroupsize

Number ofgroups Source7 6 5 4 3 2 1

86 3 3 3 0 0 5 2 37 VHBB CK18 4 0 11 15 15 37 3 27 KC CK0 0 0 0 10 10 80 6 10 KC0 0 0 0 0 0 100 8 5 CSS0 0 0 0 0 0 100 9 2 CC0 0 0 0 0 0 100 14ndash16 7 VHBB

Sources Van Huyck et al 1990 (VHBB) Camerer and Knez 2000 (CK) Knez and Camerer 1994 (KC) Gerard P Cachonand Camerer 1996 (CC) Chaudhuri et al 2001 (CSS)

116 THE AMERICAN ECONOMIC REVIEW MARCH 2006

instance by rounding the latent variable to thenearest discrete action) Initial choices are de-termined by a common belief and an idiosyn-cratic stochastic term measuring strategicuncertainty6 When group size is fixed subse-quent choices are determined by a simple linearadjustment from the playerrsquos own previouschoice toward the previous minimum as well asby a mean zero error term representing strategicuncertainty7 The variance of the error termdecreases with experience reflecting the reso-lution of strategic uncertainty8

To include the possibility of growing groupsthe model defines a ldquogrowth pathrdquo as a sequenceof weakly increasing sets of players The set ofplayers in any period t includes three possiblekinds of players incumbents (those who playedthe game in period t 1) informed entrants(those who did not play the game in period t 1 but observed the entire sequence of groupminima) and uninformed entrants (those whodid not play the game in period t 1 and didnot observe any of the previous minima) Thecomposition of the group in any period is com-mon knowledge

Incumbent playersrsquo dynamic adjustment oc-curs in the same manner as in the model withoutgrowth (as in footnote 7) Uninformed entrantsrsquochoices are determined by a draw from theinitial distribution of choices (as in footnote 6)Informed entrants also generate a draw from theinitial distribution but then perform one itera-tion of the dynamic adjustment process in thedirection of the minimum in the last period9

One final and important difference betweenthe models without and with growth is that inthe latter the change in the variance of the errorterm from period t 1 to period t depends onthe composition of the group in period t Spe-

cifically if all of the players are aware of thefull history of minima (ie the group consistsonly of incumbents and informed entrants) thenthe variance decreases from the previous period(as in footnote 8) If however there are anyuninformed entrants in period t the variance ofthe error term is higher than it was in period t 1 for all players This increase in variance cap-tures the increased strategic uncertainty pro-duced by the entry of players unaware of thegrouprsquos history

Weber (2005) shows the following result forthe model above for any two groups such thatone group maintains a constant size and theother group starts small and grows to that sizeby adding only informed entrants both choicesand minima in any period will be stochasticallyhigher in the grown group than in the group ofconstant size1011 The intuition behind the re-sult is simple In the first period the smallergroup obtains a (stochastically) higher mini-mum In subsequent periods three conditionsensure that the grown group maintains this ad-vantage First the linear adjustment processnever moves downward ldquofasterrdquo for players inthe grown group than for players in the fixed-size group Second as long as entrants are in-formed the variance on the term measuringstrategic uncertainty is never greater in thegrown group than in the fixed-size group Andthird the number of players is never greater inthe grown group than in the fixed-size groupThese last two conditions ensure that the sto-chastic component of choice never decreasesthe minimum in the grown group faster than inthe fixed-size group12

Importantly the result above does not hold if

6 Specifically ai1 0 i1 where 0 represents acommon belief and i1 is distributed normally with meanzero and variance 1

2 0 Thus playersrsquo initial choices areiid draws from a common distribution that is independentof group size

7 Player irsquos latent strategy in period t is given by ait (1 )xit1 yt1 it where yt1 is the minimum inperiod t 1 and 0 1 The it are distributednormally with mean zero and variance t

28 Specifically t

2 kt12 where 0 k 1

9 That is an informed entrantrsquos choice is given by ait (1 )xi1 yt1 it

10 The term ldquostochastically higherrdquo refers to an upwardshift in the probability distribution of the variable in themanner of first-order stochastic dominance (cf Crawford1995)

11 Note that this does not imply ldquolock inrdquo on equilibriaabove the least efficient However all that is necessary toallow such ldquolock inrdquo is that the variance of the it decreaseto zero (ie as when k 0 in footnote 8)

12 A second result shows that for any two groups thatboth grow only by adding informed entrants if one groupgrows more slowly than the other (meaning that it starts offsmaller is never larger and never adds more people in aperiod) the choices and minima will be stochasticallyhigher in the more slowly grown group


groups grow by adding uninformed entrantsThis is because the second condition above nolonger holds The increase in strategic uncer-tainty due to the introduction of uninformedentrants in a period increases the variance ofchoices and stochastically decreases the mini-mum in that period meaning that the group thatstarts small (with a higher minimum) can loseits advantage when uninformed entrants areadded

The result above implies a clear prescriptionfor alleviating large-group coordination failureBy starting at a small size and adding onlyentrants who are aware of the grouprsquos historyone should be able to produce a group of anyfinite size with higher minima on average thangroups that start off at the large size The abilityof new entrants to observe the grouprsquos historyhowever is critical since it underlies the de-crease in strategic uncertainty This implies thatgrowth without such ldquoexposurerdquo of new en-trants should not be successful in producinglarge groups with higher minima The experi-ment in the next section directly tests the effec-tiveness of growth with and without suchexposure

III Growing Efficient Coordinationin the Laboratory

This experiment tests the hypothesis thatgrowth coupled with exposure of entrants to thegrouprsquos history produces groups with higherminima than control groups that start off at alarge size The experiment also tests the hypoth-esis that growth without exposure of entrants tothe grouprsquos history will not produce higher min-ima The results are clear while none of thecontrol groups or ldquono-historyrdquo groups managesto coordinate on a minimum above one severalgrown groups manage to coordinate at higherlevels of efficiency including on the highestpossible minimum

A Experimental Design

This experiment explored whether a slowsteady growth rate determined by the experi-menter could create large groups coordinatedmore efficiently than fixed-size large groups Ineach session of the experiment a group of 12

students (at Stanford University the Universityof California at Santa Cruz or Carnegie MellonUniversity) were assembled in one room as-signed participant numbers from 1 through 12and presented the game in Table 113 Instruc-tions were read aloud and before starting thegame participants answered several questionsto ensure their comprehension of the instruc-tions and the game At the end of the experi-ment subjects were anonymously paid theirearnings in cash

There were three kinds of sessions In controlsessions no mention was made of growth and agroup of 12 subjects played the game for 12periods In two kinds of growth sessions thegroup of subjects playing the game grew in sizefrom 2 to 12

Two growth treatments tested the effect ofentrantsrsquo awareness of the grouprsquos history Inthe history condition entrants observed thegrouprsquos history (the minima in all previous pe-riods) and this was common knowledge In theno-history condition entrants were not aware ofany of the previous minima

Each of the growth sessions consisted of 22periods In the first several periods only partic-ipants 1 and 2 played the game while the othersubjects sat quietly (history condition) or werein another room (no-history condition) Partici-pants were told that they would receive a fixedpositive ldquofairrdquo amount for periods in whichthey were not playing the game but that theexact amount would not be revealed until theend of the experiment14 In each period partic-ipating subjects recorded a number from 1 to 7(indicating the contribution time for their sec-tion of the report) on a piece of paper and

13 The game was framed in the context of a reportcompletion as in Weber et al (2001) Subjects were alsogiven Table 1 with an explanation of the payoffs Weber etal (2001) found no difference in behavior with and withoutsuch context

14 This was done because of the concern that announcingthe per-period amount might create a focal point that couldinfluence choices Subjects were told that they were notinformed of the amount because ldquowe do not want this toinfluence your choicesrdquo To ensure that they believed theamount was fixedmdashand not dependent on what happened inthe experimentmdashthe experimenter placed a large envelopeat the front of the room and told subjects that the amountwas written on a sheet inside the envelope This sheet wasshown to subjects at the end of the experiment


handed it to the experimenter15 The experi-menter then calculated the minimum wrote thisnumber on the board asked subjects to calculatetheir payoffs and checked to make sure thatthey had done so correctly16 Before proceedingto the next round the experimenter erased out-come information from the board

At various preannounced and commonlyknown periods other participants joined thegroup of those actively playing the game Forexample in one of the sessions a third partici-pant (3) was added in period 7 joining the firsttwo participants (1 and 2) who continued toparticipate In the no-history condition the en-trant was brought in from the other room Par-ticipant numbers of those playing the game in aperiod were written on the board at the begin-ning of the period In each session a schedule ofsuch additions (a ldquogrowth pathrdquo) was handed toall subjects at the beginning of the experimentSubjects all knew the predetermined growthpath In all growth paths 12 subjects were par-ticipating by the last few periods

B Results Control Sessions

Five control sessions (n 60) were con-ducted using undergraduates at Stanford Uni-versity California Institute of Technology andCarnegie Mellon University The results are re-ported in Figure 1 which presents the minimumchoice across all 12 periods for each session17

In addition the two thicker lines present the

15 To prevent players from knowing which others wereparticipating in the history condition all players handed inslips of paper (nonparticipants checked a box indicatingthey were not playing) In the no-history condition subjectsknew the participant number of subjects 3 to 12 since theyentered the room once they were added (though participantsalready in the group were seated facing away from the doorso that they could not look at or make eye contact with theentrant) The history and control sessions (with anonymity)were designed and conducted before deciding to run theno-history condition (where anonymity is difficult) at thesuggestion of two anonymous reviewers It would have beenpossible to maintain anonymity in the no-history sessionsby for instance using a code to report the grouprsquos history orreporting history via private messages but both these pro-cedures eliminate common knowledge of outcomes amongthose in the group

16 In all conditions the action chosen by any participantwas anonymousmdashexcept for choices by participants 1 and 2in the no-history condition where a subject choosing ahigher number could infer both the identity and the choiceof the other subject Since the results for two-person groupsare similar in both the history and no-history conditions thisappears not to have influenced behavior

17 The dataset is available at httpwwwe-aerorgdatamar06_data_20020652zip

FIGURE 1 PERIOD MINIMA IN CONTROL SESSIONS


average of the session minima and the averagechoice across all sessions

Overall the results replicate previous find-ings on large groups playing the minimum-effort coordination game The minimum fell to1 in all five control groups Both the averagechoice and the average of the minima consis-tently decreased and ended up at or near 1 bythe final periods

C Results Growth Sessions

Twelve sessions with predetermined growthpaths (n 144) were conducted at Stanford Uni-versity (sessions 1 to 4) University of CaliforniaSanta Cruz (sessions 5 to 7) and Carnegie MellonUniversity (sessions 8 to 12) Of these nine ses-sions (1 to 9) were in the history condition whilethree (10 to 12) were in the no-history condition

Figure 2 shows the three growth paths used inthe experiment The growth paths were de-signed first to allow the establishment of suc-cessful coordination in a two-person group (byallowing several periods before adding moreparticipants) and to then add players in a slowand regular manner Thus with one exception(the last two players added in growth path 3)the growth paths add only one player at a time

Figures 3A through 3C present the minimumchoices for sessions 1 through 9 (history) while

Figure 3D presents the minimum choices forsessions 10 through 12 (no history) Each figurealso presents the corresponding growth pathThe marker-less solid line in each figure showsthe growth path (measured on the left verticalaxis) while each of the remaining lines presentsthe minima in a particular session (measured onthe right vertical axis)

Table 3 presents summary statistics of theminima in each session for varying rangesof group size The second through sixth col-umns contain the average and median groupminima for the periods in which the group sizewas within a particular range (for instance thesecond column corresponds to a group size of2) The final column presents the minimum in thefirst period in which that group reached a sizeof 12

An examination of behavior in individualsessions reveals several behavioral regularitiesFirst the small groups were able to coordinateefficiently in both history and no-history ses-sions In all but one of the sessions the group ofsize 2 coordinated on a minimum of 7 for atleast two consecutive periods prior to growingThe remaining group (session 7) coordinated ona high level of efficiency (minimum 6)

A second observation is that growth did notalways work even with history In three historysessions (1 3 and 6) the minimum by the first

FIGURE 2 PREDETERMINED GROWTH PATHS


period in which the group played as a 12-persongroup was 1 Therefore growth did not alwaysproduce efficiently coordinated groups evenwith growth paths designed to do so

An equally strong regularity however is thatgrowth often produced large groups coordinatedat high levels of efficiency In three sessions (4

8 and 9) the minimum remained at 7 through-out the growth process and these groups ob-tained a minimum of 7 in at least two periods ata group size of 12 (recall that control 12-persongroups never manage to coordinate on a mini-mum of 7 in any period) In another session (2)the minimum remained at 5 throughout the

FIGURE 3A GROWTH PATH 1 AND PERIOD MINIMA FOR SESSIONS 1 AND 2 (HISTORY)

FIGURE 3B GROWTH PATH 2 AND PERIOD MINIMA FOR SESSIONS 3 AND 4 (HISTORY)


growth process (falling only in the last peri-od)18 while in another session (5) it did not fall

below 3 as the group grew In five out of ninesessions in the history condition groups played

18 In several sessions (2 4 5 and 8) the minimum in oneof the final two periods fell to a lower value than that atwhich the group was coordinated previously This ldquoend ofexperiment effectrdquo frequently occurs in these games andresults from a few subjects changing their choices in thefinal periods (perhaps to do better that others or perhaps

because they believe that others will do so) While thisphenomenon is interesting this paper is not principallyconcerned with what occurs in the final periods (aftergrowth is completed) but rather with coordination duringand immediately after growth

FIGURE 3C GROWTH PATH 3 AND PERIOD MINIMA FOR SESSIONS 5 THROUGH 9 (HISTORY)

FIGURE 3D GROWTH PATH 3 AND PERIOD MINIMA FOR SESSIONS 10 THROUGH 12 (NO HISTORY)


a minimum of three or higher for at least twoperiods at a size of 12 and in three of those theycoordinated on the efficient equilibrium

In addition in all the sessions that ended upat a minimum of 1 the minimum was higher atleast through a group size of 9 This higher levelof efficiency for groups of up to size 9 is sur-prising in light of the fact that the minimum wasalways 1 for the large groups (8 or larger) inTable 219 Thus there is clear support for thehypothesis that starting with a two-persongroup which can reliably reach efficiency andthen adding informed players at a slow rateenables more efficient coordination than start-ing with a large group

An equally important result is that growthdoes not work in the three no-history sessions(10 to 12) In all of these sessions the minimumfell below 7 by the time the group reached a sizeof 4 and the minimum equaled 1 by the time thegroup reached a size of 8 Moreover this failure

was never driven entirely by only one sub-jectmdashin every session at least 6 of the 10entrants chose a number below 7 in their firstplay20

To test directly whether growth with historyresults in higher minima we need to comparechoices in the control and no-history sessionswith those in the history sessions Table 4 com-pares the distributions of subject choices in thefive control sessions the three no-history ses-sions and the nine history sessions in the fourthperiod in which participants played at a groupsize of 1221

19 While Table 2 reports the fifth-period minima thefirst-period minima in previous experiments were not ashigh as in the sessions reported here and there was never aminimum of 7 Note that in the control sessions the mini-mum was never above 4

20 Overall there appears to be a relationship betweenchanges in group size and decreases in the minimum Of the34 total decreases 24 (71 percent) coincided with an in-crease in group size 6 (18 percent) occurred after the groupreached a size of 12 (see footnote 18) 3 (9 percent) occurredbefore the growth process began and one (3 percent) oc-curred during a ldquopauserdquo in growth These percentages do notdiffer significantly between the history and no-historysessions

21 A reasonable comparison is to compare the controlgroups in period t with the t-th period in which the growngroups played as groups of size 12 The analysis here setst 4 because this is the greatest value of t for which thereare data in all the growth sessions

TABLE 3mdashAVERAGE MINIMA (MEDIANS) BY SESSION FOR RANGES OF GROUP SIZE

Growth path 1 Group size (number of periods at that size)First n 12

minimum2 (6) 3 (2) 4ndash6 (4) 7ndash11 (5) 12 (5)

Session 1 (h) 70 (7) 60 (6) 45 (45) 20 (2) 10 (1) 1Session 2 (h) 63 (65) 55 (55) 53 (5) 50 (5) 42 (5) 5

Growth path 2 Group size (number of periods at that size)

2 (5) 3 (4) 4ndash6 (4) 7ndash11 (5) 12 (4)



2 (5) 3 (4) 4ndash6 (4) 7ndash10 (4) 12 (5)

Session 5 (h) 66 (7) 70 (7) 70 (7) 33 (3) 26 (3) 3Session 6 (h) 70 (7) 70 (7) 70 (7) 35 (35) 10 (1) 1Session 7 (h) 60 (6) 60 (6) 48 (6) 40 (4) 20 (1) 4Session 8 (h) 70 (7) 70 (7) 70 (7) 70 (7) 58 (7) 7Session 9 (h) 70 (7) 70 (7) 70 (7) 70 (7) 70 (7) 7

Session 10 (nh) 58 (6) 70 (7) 38 (4) 18 (1) 10 (1) 1Session 11 (nh) 70 (7) 70 (7) 28 (25) 10 (1) 10 (1) 1Session 12 (nh) 56 (6) 10 (1) 10 (1) 10 (1) 10 (1) 1


The frequency of subjects choosing 1 washigh in all three conditions (13 of 60 in thecontrol 35 of 36 in no history 33 of 108 inhistory) but was clearly highest in the no-history condition While the proportion washigher for history sessions than for the controlthis is not surprising since in three of the ninehistory sessions the minimum was 1 even beforethe group reached a size of 12 In these groupsthere were more previous periods for subjectsrsquochoices to converge toward the inefficient equi-librium than in the control22

Just as interesting however is the differencein the frequency with which subjects choose 7In the control and no-history sessions only 3 of60 (5 percent) and 0 of 36 (0 percent) respec-tively did so However 32 of 108 subjects (30percent) in the history sessions chose 7 There-fore the number of subjects playing the strategycorresponding to the efficient equilibrium wasmuch higher in the grown groups than in thecontrol sessions The distributions of choices inthe history condition and the two other condi-tions are significantly different in a one-tailedKolmogorov-Smirnov test (COH D 0294p 0001 NHOH D 0667 p 0001) Acomparison of the mean choice in the historycondition (389) with the mean choices in the

control (318) and no-history (102) conditionsusing a one-tailed t-test (with unequal vari-ances) also produces significant differences(COH t16178 199 p 002 NHOHt10988 1194 p 0001)23

The tests above rely on the assumption thateach choice is an independent observationwhich is doubtful since subjects in a session areinfluenced by a shared history Therefore thelevel of significance reported by the statistics isexaggerated One way to address this problem isto treat each session as the independent unit ofanalysis and examine only the minima Theseminima are reported in the final row of Table4 Note that the fourth-period minimum in allbut one of the control sessions and no-historysessions was 1 While the minimum in four ofthe history sessions was also 1 the minimumwas greater than 1 in the remaining five ses-sions Moreover the history sessions producedminima of at least 5 three times which neveroccurred in the control sessions A one-tailedMann-Whitney U test of the minima revealssignificant differences between control and his-tory sessions (z 133 p 01) and no-historyand history sessions (z 155 p 007) andno difference between the control and no-history sessions (z 078) While the p-valuesin the tests above are not highly significant thistest is extremely conservative since it treatseach group of 12 subjects as just oneobservation

IV Conclusion

The experiment above shows that growth andexposure of entrants to history can alleviate theproblem of large group coordination failureThe history sessions regularly produce indepen-dent 12-person groups coordinated at higherminima than 1 (including minima of 7) a resultthat has never been found in previous experi-

22 This aspect of the comparison works against the hy-pothesis that growth works since choices in unsuccessfullygrown groups are usually almost all 1 even prior to the firstperiod as a 12-person group

23 The difference between the control and no-historyconditions is also significant in two-tailed versions of thesame tests (t6109 1028 p 0001 D 0756 p 0001)This is largely due to the fact that by the fourth period at asize of 12 groups in the no-history condition are almostperfectly coordinated on the minimum choice of 1

TABLE 4mdashDISTRIBUTIONS OF SUBJECT CHOICES IN FOURTH

PERIOD AS 12-PERSON GROUPS

Control

Growthand

history

Growthand nohistory

Choice 7 3 (5) 32 (30) 0 (0)6 0 (0) 3 (3) 0 (0)5 6 (10) 13 (12) 0 (0)4 22 (37) 8 (7) 0 (0)3 7 (12) 10 (9) 0 (0)2 9 (15) 9 (8) 1 (3)1 13 (22) 33 (31) 35 (97)

Total 60 108 36

Minima 1 1 11 4

1 1 11 3 45 7 7

1 1 1


ments on minimum-effort coordination games24

Growth however does not always work andeven groups that were grown slowly with expo-sure of entrants to history did not always remaincoordinated efficiently

Further evidence of successful growth can beseen in Weber (2005) which reports an exper-iment in which a subject in the role of ldquoman-agerrdquo decided how to grow the group in eachsession In all four sessions managers initiallygrew the group too quickly resulting in a dropin the minimum25 However two of the fourmanagers then started over and succeeded ingrowing their groups more slowly resulting ingroups of size 12 coordinated on high minima(6 and 7) The results of these two sessions alsosuggest that a better approach to growth than therigid growth paths used in this paper might beone in which a manager waits until some set ofconditions is met before growing The fact thatboth of these sessions produced successfulgrowth points to the potential value of suchendogenous rules

Another interesting result of the experimentreported here was unintended but merits men-tion In one session (session 1) subjects ap-peared to form norms about how to react notonly to the previous minimum but also to whathappened in previous experiences with growthIn this session the group appeared to develop aprecedent that ldquowhen the group grows the min-imum falls by 1rdquo In fact in later periods severalgroup members anticipated this change in theminimum and adjusted their behavior accord-ingly26 This result suggests that players mayform ldquohigher-orderrdquo precedents based not juston levels of previous play (eg ldquoexpect the

previous minimum to be the minimum againrdquo)but also on the relation between levels of pre-vious play and group sizes or transitions Thatis previous experience with growth may affecthow group members subsequently jointly reactto a similar event

The main result in this papermdashthat slowgrowth coupled with the exposure of entrants toa grouprsquos history might be one reason why weobserve large successfully coordinated groupsin the real worldmdashcan be extended to provideprescriptions for more efficient coordination ingroups and firms outside the laboratory Forinstance the success of growth is linked to theexposure of entrants to the grouprsquos historywhich can be thought of as a form of the train-ing or acculturation that new members of anorganization frequently undergo (see NoelTichy 2001)

Of course the results also reveal that evenslow growth and acculturation of new entrantscan fail Moreover the results in this paper donot provide a prescription for how a group thatis already large and coordinated on an ineffi-cient equilibrium might turn things around Nodoubt in practice there are other ways to im-prove coordination in large groups27 Nonethe-less the results address the important questionof how groups can reach a large size whileremaining efficiently coordinated

REFERENCES

Arrow Kenneth J The limits of organizationNew York W W Norton amp Company 1974

Berninghaus Siegfried and Ehrhart Karl ldquoTimeHorizon and Equilibrium Selection in TacitCoordination Gamesrdquo Journal of EconomicBehavior and Organization 1998 37(2) pp231ndash48

Bornstein Gary Gneezy Uri and Nagel Rose-marie ldquoThe Effect of Intergroup Competitionon Group Coordination An ExperimentalStudyrdquo Games and Economic Behavior2002 41(1) pp 1ndash25

Brandts Jordi and Cooper David J ldquoA Change

24 Another way to view this result is as a demonstrationof transfer across games from n-person games to similarn 1ndashperson games (cf Camerer and Knez 2000 DavidCooper and John Kagel 2004)

25 This indicates that the rate of growth might affect thesuccess of growth In addition it also indicates that the needfor slow growth might not be immediately apparent topeople deciding how fast groups should grow

26 This phenomenon is even more striking in one of theendogenously grown sessions in Weber (2005) in which thegroup developed a norm that ldquothe minimum falls every timethe group grows but in every period in which the groupdoes not grow the minimum rises by exactly 1rdquo All 12subjects acted in precise concordance with the rule in laterperiods

27 For instance management might temporarily modifypayoffs to change the game to one in which efficient coor-dination is easier (see Jordi Brandts and Cooper forthcom-ing)


Would Do You Good An ExperimentalStudy of How to Overcome CoordinationFailure in Organizationsrdquo American Eco-nomic Review (forthcoming)

Cachon Gerard P and Camerer Colin F ldquoLoss-Avoidance and Forward Induction in Exper-imental Coordination Gamesrdquo QuarterlyJournal of Economics 1996 111(1) pp165ndash94

Camerer Colin F and Knez Marc ldquoCoordina-tion in Organizations A Game-TheoreticPerspectiverdquo in Zur Shapiro ed Organiza-tional decision making Cambridge Cam-bridge University Press 1997 pp 158ndash88

Camerer Colin F and Knez Marc ldquoIncreasingCooperation in Social Dilemmas by Estab-lishing a Precedent of Efficiency in Coordi-nation Gamesrdquo Organizational Behavior andHuman Decision Processes 2000 82(2) pp194ndash216

Charness Gary ldquoSelf-Serving Cheap Talk ATest of Aumannrsquos Conjecturerdquo Games andEconomic Behavior 2000 33(2) pp 177ndash94

Chaudhuri Ananish Schotter Andrew and So-pher Barry ldquoTalking Ourselves to Effi-ciency Coordination in Inter-GenerationalMinimum Games with Private Almost Com-mon and Common Knowledge of AdvicerdquoNew York University C V Starr Center forApplied Economics Working Papers No 01-11 2001

Coase Ronald H ldquoThe Nature of the FirmrdquoEconometrica 1937 4(16) pp 386ndash405

Cooper David and Kagel John ldquoLearning andTransfer in Signaling Gamesrdquo UnpublishedPaper 2004

Cooper Russell W Coordination gamesComplementarities and macroeconomicsCambridge Cambridge University Press1999

Crawford Vincent P ldquoAdaptive Dynamics inCoordination Gamesrdquo Econometrica 199563(1) pp 103ndash43

Gittell Jody H ldquoSupervisory Span Relational

Coordination and Flight Departure Perfor-mance A Reassessment of PostbureaucracyTheoryrdquo Organization Science 2001 12(4)pp 468ndash83

Hirshleifer Jack ldquoFrom Weakest-Link to Best-Shot The Voluntary Provision of PublicGoodsrdquo Public Choice 1983 41(3) pp371ndash86

Knez Marc and Camerer Colin F ldquoCreatingExpectational Assets in the Laboratory Co-ordination in Weakest-Link Gamesrdquo Strate-gic Management Journal 1994 15(8) pp101ndash19

March James G and Simon Herbert A Organi-zations New York John Wiley amp Sons1958

Nanda Ashish ldquoImplementing OrganizationalChangerdquo in Miguel Pina e Cunha and CarlosAlves Marques eds Organizational changein a changing context Lisbon ISPA 1999pp 489ndash522

Schelling Thomas C The strategy of conflictCambridge MA Harvard University Press1960

Tichy Noel ldquoNo Ordinary Boot Camprdquo Har-vard Business Review 2001 79(4) pp 63ndash70

Van Huyck John B Battalio Raymond C andBeil Richard O ldquoTacit Coordination GamesStrategic Uncertainty and Coordination Fail-urerdquo American Economic Review 199080(1) pp 234ndash48

Weber Roberto A ldquoInterdependence in Organi-zations and Laboratory Groupsrdquo PhD dissCalifornia Institute of Technology 2000

Weber Roberto A ldquoManaging Growth toAchieve Efficient Coordination Theoryand Experimental Evidencerdquo UnpublishedPaper

Weber Roberto A Camerer Colin F Rottenst-reich Yuval and Knez Marc ldquoThe Illusion ofLeadership Misattribution of Cause in Coor-dination Gamesrdquo Organization Science2001 12(5) pp 582ndash98


ordinating activity efficiently Once they havedone so they can establish a set of self-reinforcing rules or norms governing what ac-tions are appropriate and these allow the groupto continue to coordinate activity successfullyAs the group grows entrantsrsquo exposure to thesenorms allows these entrants to be aware of theappropriate behavior and creates an expectationfor everyone in the group of what everyone else(including the entrants) will do Thus by coor-dinating efficiently as a small group growingslowly and exposing entrants to the grouprsquosprevious norms a group can become large andefficiently coordinated

The rest of this paper demonstrates the influ-ence of growth on tacit coordination using anexperiment involving the minimum-effort coor-dination game The experiment is motivated bya simple modelmdashdiscussed briefly later in thispapermdashwhich shows why growth should workwhen entrants are exposed to the grouprsquos his-tory In the experiment efficiently coordinatedlarge groups which are impossible to obtainwhen a group starts out at a large size areldquogrownrdquo in the laboratory by starting with smallgroups and adding a few entrants at a time whoare exposed to the grouprsquos history This paperproduces the only laboratory demonstration ofthe regular occurrence of efficient tacit coordi-nation among large groups The experiment alsoreveals however that the ability of entrants toobserve the grouprsquos history is crucialmdashif theyare not aware of what incumbent members didin the past coordination failure results

I The Minimum-Effort Coordination Game

The minimum-effort or weak-link coordina-tion game was first studied experimentally by

Van Huyck et al (1990)3 In the gamemdashwhichis a seven-effort-level version of the stag huntgame (see Crawford 1995)mdashn players choosefrom a set of integers that can be thought of asorderable strategies such as effort or contribu-tion levels Every playerrsquos payoff is a functionof her choice and the minimum choice of all nplayers (thus the term ldquoweak-linkrdquo since everyplayerrsquos payoff is partially determined by thelowest choice in the group) Table 1 presents thepayoff to each player as a function of her choiceand the minimum choice4

When everyone makes the same choice andtherefore receives the same payoff (representedby cells along the diagonal) the outcome is oneof the gamersquos seven pure-strategy Nash equilib-ria The equilibria differ because those corre-sponding to higher choices also yield higherpayoffs Therefore ldquomore efficientrdquo coordina-tion corresponds to all players making higherchoices in equilibrium and the Pareto-optimal (orefficient) equilibrium results when all players se-lect the highest choice 7 and receive $090

Since all symmetric outcomes including theefficient one are equilibria this game does nothave the incentive problem present in the pris-onerrsquos dilemma Nonetheless the efficient equi-librium may not be easy to achieve becauseplayers are faced with strategic uncertainty Ev-eryone may recognize the efficient equilibrium

3 See also Jack Hirshleifer (1983) for an early theoreticaldiscussion of the problem underlying the game

4 The game in Table 1 is the same as Game A in VanHuyck et al (1990) except every payoff is lower by $040Of course real payoffs also differ due to differences inexperimental location and time However experiments us-ing the two sets of payoffs produce similar small versuslarge group results

TABLE 1mdashPAYOFFS (IN DOLLARS) FOR MINIMUM-EFFORT GAME

Minimum choice of all players

7 6 5 4 3 2 1Playerrsquos choice 7 090 070 050 030 010 010 030

6 080 060 040 020 000 0205 070 050 030 010 0104 060 040 020 0003 050 030 0102 040 0201 030



























28 Specifically t

2 kt12 where 0 k 1



































































2 (5) 3 (4) 4ndash6 (4) 7ndash11 (5) 12 (4)



2 (5) 3 (4) 4ndash6 (4) 7ndash10 (4) 12 (5)








IV Conclusion






Control

Growthand

history

Growthand nohistory

Choice 7 3 (5) 32 (30) 0 (0)6 0 (0) 3 (3) 0 (0)5 6 (10) 13 (12) 0 (0)4 22 (37) 8 (7) 0 (0)3 7 (12) 10 (9) 0 (0)2 9 (15) 9 (8) 1 (3)1 13 (22) 33 (31) 35 (97)

Total 60 108 36

Minima 1 1 11 4

1 1 11 3 45 7 7

1 1 1









REFERENCES


























































28 Specifically t

2 kt12 where 0 k 1



































































2 (5) 3 (4) 4ndash6 (4) 7ndash11 (5) 12 (4)



2 (5) 3 (4) 4ndash6 (4) 7ndash10 (4) 12 (5)








IV Conclusion






Control

Growthand

history

Growthand nohistory

Choice 7 3 (5) 32 (30) 0 (0)6 0 (0) 3 (3) 0 (0)5 6 (10) 13 (12) 0 (0)4 22 (37) 8 (7) 0 (0)3 7 (12) 10 (9) 0 (0)2 9 (15) 9 (8) 1 (3)1 13 (22) 33 (31) 35 (97)

Total 60 108 36

Minima 1 1 11 4

1 1 11 3 45 7 7

1 1 1









REFERENCES






























































































2 (5) 3 (4) 4ndash6 (4) 7ndash11 (5) 12 (4)



2 (5) 3 (4) 4ndash6 (4) 7ndash10 (4) 12 (5)








IV Conclusion






Control

Growthand

history

Growthand nohistory

Choice 7 3 (5) 32 (30) 0 (0)6 0 (0) 3 (3) 0 (0)5 6 (10) 13 (12) 0 (0)4 22 (37) 8 (7) 0 (0)3 7 (12) 10 (9) 0 (0)2 9 (15) 9 (8) 1 (3)1 13 (22) 33 (31) 35 (97)

Total 60 108 36

Minima 1 1 11 4

1 1 11 3 45 7 7

1 1 1









REFERENCES





































IV Conclusion






Control

Growthand

history

Growthand nohistory

Choice 7 3 (5) 32 (30) 0 (0)6 0 (0) 3 (3) 0 (0)5 6 (10) 13 (12) 0 (0)4 22 (37) 8 (7) 0 (0)3 7 (12) 10 (9) 0 (0)2 9 (15) 9 (8) 1 (3)1 13 (22) 33 (31) 35 (97)

Total 60 108 36

Minima 1 1 11 4

1 1 11 3 45 7 7

1 1 1









REFERENCES








































REFERENCES

































Managing Growth to Achieve EfÞcient Coordination in Large...

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Transcript of Managing Growth to Achieve EfÞcient Coordination in Large...