Tacit Coordination Games, Strategic Uncertainty, and Coordination ...
American Economic Association
Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure Author(s): John B. Van Huyck, Raymond C. Battalio, Richard O. Beil Source: The American Economic Review, Vol. 80, No. 1 (Mar., 1990), pp. 234-248 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/2006745 Accessed: 07/12/2010 11:25 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=aea. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].
American Economic Association is collaborating with JSTOR to digitize, preserve and extend access to The American Economic Review.
http://www.jstor.org
Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure By JOHN B. VAN HUYCK, RAYMOND C. BATTALIO, AND RICHARD 0. BEIL*
Deductive equilibriummethods-such as Rational Expectations or Bayesian Nash Equilibrium-are powerfultools for analyzing economies that exhibit strategicinterdependence. Typically, deductive equilibrium analysis does not explain the process by which decision makers acquire equilibrium beliefs. The presumption is that actual economies have achieveda steady state. In economies with stable and unique equilibrium points, the influence of inconsistent beliefs and, hence, actions would disappear over time, see Robert Lucas (1987). The power of the equilibriummethod derives from its ability to abstractfrom the complicated dynamic process that inducesequilibrium and to abstractfromthe historicalaccident that initiatedthe process. Unfortunately, deductive equilibrium analysis often fails to determinea unique equilibriumsolutionin manyeconomiesand, hence, often fails to prescribeor predictrational behavior.In economieswith multiple equilibria, the rational decision maker formulatingbeliefs using deductiveequilibrium
*Department of Economics, Texas A&M University, College Station, TX 77843; Department of Economics, Texas A&M University, College Station, TX 77843; and Department of Economics, Auburn University, Auburn, AL 36849, respectively. Mike Baye, John Bryant, Cohn Camerer, Russell Cooper, Vincent Crawford, John Haltiwanger, Pat Kehoe, Tom Saving, Steve Wiggins, Casper de Vries, a referee, and seminar participants at the NBER/FMME 1987 Summer Institute, the 1988 meetings of the Economic Science Association, the University of New Mexico, and Texas A&M University made constructive comments on earlier versions of this paper. Sophon Khanti-Akom, Kirsten Madsen, and Andreas Ortmann provided research assistance. The National Science Foundation, under grants no. SES8420240 and SES-8911032, the Texas A&M University Center for Mineral and Energy Research, and the Lynde and Harry Bradley Foundation provided financial support. A Texas A&M University TEES Fellowship has supported Battalio. 234
concepts is uncertain which equilibrium strategyother decision makerswill use and, when the equilibriaare not interchangeable, this uncertaintywill influence the rational decision-maker'sbehavior. Strategicuncertainty arises even in situationswhereobjectives, feasible strategies, institutions, and equilibriumconventionsarecompletelyspecified and are common knowledge. While multiple equilibriaare common in theoretical analysis, consideration of specific economies suggests that many equilibrium points are implausible and unlikely to be observedin actualeconomies. One response to multipleequilibriais to arguethat some Nash equilibriumpoints are not self-enforcingand, hence, are implausible, because they fail to satisfy one or more of the following refinements:eliminationof individuallyunreasonableactions,sequential rationality, and stability against perturbations of the game-see Elon Kohlburgand Jean-FrancoisMertens(1986) for examples and references.Equilibriumrefinementsdetermine when an outcome that is already expected would be implementedby rational decision makers. In general,many outcomeswill satisfythe conditionsof a givenequilibriumrefinement. The equilibriumselectionliteratureattempts to determine which, if any, self-enforcing equilibriumpoint will be expected.A satisfactory theory of interdependentdecisions must not only identifythe outcomesthat are self-enforcingwhen expectedbut also must identify the expectedoutcomes.Consequently, a theory of equilibriumselection would be a useful complement to the theory of equilibriumpoints. The experimental method provides a tractable and constructiveapproachto the equilibrium selection problem. This paper studies a class of tacit pure coordination games with multiple equilibria,which are
VOL. 80 NO.1
strictly Paretoranked,and it reportsexperiments that provide evidenceon how human subjectsmake decisionsunderconditionsof strategicuncertainty. Game I. A Pure Coordination
To focus the analysisconsiderthe following tacit coordination game, which is a strategic form representation of John Bryant's (1983) Keynesian coordination game. The baseline game is defined as follows: Let el,..., en denote the actions taken by n players.The period gameA is defined by the following payoff function and strategy space for each of n players: (1)
235
VAN HUYCK ETAL.: TACIT COORDINA TION GAMES
a(ei, e1) = a [min(ei, e1)] - bei,
a > b > O, where ei equalsmin(el,..., ei_1,ei+,,..., en). Actions are restrictedto the set of integers from 1 to J. The players have complete information about the payoff function and strategy space and know that the payoff function and strategy space are common knowledge.'
If the players could explicitlycoordinate their actions, the-real or imagined planner'sdecision problemwould be trivial. Given a - b greater than 0, each player
should choose the maximumfeasibleaction, e. Moreover, a negotiated "pregame" agree-
ment to choose e- would be self-enforcin*. Unlike games with incentiveproblems,here
'Apparently, this game is similar to Rousseau's "stag hunt" parable, which he used to motivate his analysis of the social contract, see Crawford (1989, p. 4). In the stag hunt game, each hunter in a group must allocate effort between hunting a stag with the group and hunting rabbits by himself. Let e, denote effort expended on the stag hunt. Since stag hunting in that era required the coordinated effort of all the hunters, the probability of successfully hunting a stag depends on the smallest e. The parameter a in equation (1) reflects the benefits of participating in the stag hunt: eating well should the hunt succeed. Hunting rabbits does not require coordination with the other hunters. The parameter b in equation (1) reflects the opportunity cost of effort allocated to the stag hunt that could have been allocated to rabbit hunting: a meal-however, meager.
the firstbest outcomeis an equilibriumpoint. However,when the playerscannotengagein "pregame"negotiationthey face a nontrivial coordinationproblem. Suppose that the players attempt to use the Nash equilibriumconceptto informtheir strategic behavior in the tacit coordination game A. A player'sbest responseto ei is to choose ei equal to ei. By symmetryit follows that any n-tuple (e,...,
e)
with e E {1,
2,..., e-} satisfies the mutual best response property of a Nash equilibriumpoint. All feasible actions are potentialNash equilibrium outcomes. The Nash concept neither prescribesnor predictsthe outcome of this tacit coordinationgame. (Standardequilibriumrefinementsdo not reducethe set of equilibria.For example,the equilibria are strict-each player has a unique best response-and, hence, trembling-handperfect.) II. Coordination ProblemsandEquilibrium
SelectionPrinciples
The analysis in Section I follows convention and abstractsfrom the equilibriumselection problem. However,a rationalplayer using deductive equilibriumconcepts confronts two nontrivialcoordinationproblems in period game A. First, playersmay fail to correctly forecast the minimum, ei, and, hence, regret their individualchoice, that is, ei = ei. This type of coordinationfailureresults in disequilibrium:outcomesthat do not satisfy the mutual best-responsepropertyof an equilibrium. An equilibriumselectionprincipleidentifies a subset of equilibriumpoints according to some distinctivecharacteristic.An interesting conjectureis that decisionmakersuse some selectionprincipleto identifya specific equilibrium point in situations involving multiple equilibria.This selection principle would solve the problemof coordinatingon a specificequilibriumpoint. Hence, the outcome will satisfy the mutual best-response propertyof an equilibrium. A second coordination problem arises when the equilibriacan be Pareto ranked. In such situations, all players may give a best response,but, nevertheless,implementa
236
THE AMERICAN ECONOMIC REVIEW
Pareto dominated equilibrium, that is, en) # J. Whilenot regrettingtheir individualchoice, they regretthe equilibrium implemented by these individual choices. Consequently,the outcomeresultsin coordination failure. What equilibriumselection principlescould a playeruse to resolvethese two coordinationproblems?2 Deductive selectionprinciplesselect equilibrium points based on the descriptionof the game. Deductiveselectionprinciplespreserve the equilibrium method's desirable propertyof independencefromhistoricalaccidents and from complicateddynamicprocesses. Inductive selection principles select equilibriumpoints based on the history of some pregame.3Hence, inductive selection principles are not independentof accident and process. When multiple equilibriumpoints can be Pareto ranked,it is possible to use concepts of efficiency to select a subset of selfenforcing equilibriumpoints: examples include R. Duncan Luce and HowardRaiffa's (1956, p. 106) concept of joint-admissibility, Tamer Basar and Geert Olsder's(1982, p. 72) concept of admissibility, and John Harsanyiand ReinhardSelten's(1988,p. 81) concept of payoff-dominance.An equilibif it riumpoint is said to be payoff-dominant is not strictlyParetodominatedby any other equilibriumpoint. When unique,considerations of efficiency may induce players to focus on and, hence, select the payoffdominant equilibrium point, see Thomas Schelling(1980, p. 291). In period game A, the equilibriumpoints are strictly Pareto ranked. Each player prefers a larger minimum.The only equilibrium point not Pareto dominated by any other equilibriumpoint is the n-tuple min(el,...,
(e,...,
e):
the payoff-dominant equilibrium
point. Consequently,payoff-dominanceselects the n-tuple(e,..., e-) in gameA. 2The "disequilibrium Keynesians" emphasizethefirst coordinationproblem. The "equilibriumKeynesians" emphasizethe secondcoordinationproblem,see Cooper and John (1988) for examplesand references. 3We use the term inductionin the logical, rather than mathematical,sense of reasoningfrom observed facts-history-to a conclusion.
MARCH 1990
Selecting the unique payoff-dominant equilibriumpoint not only allows playersto coordinateon an equilibriumpoint but also ensures that they will not coordinateon an inefficientone. Payoffdominancesolvesboth the individual and the collective coordination problemsof disequilibriumand coordination failure and, as Harsanyiand Selten suggest, should take precedenceover alternative selectionprinciples. The tacit coordinationgameA providesa severe test of payoffdominance,becausethe minimum rule exacerbatesthe influenceof uncertaintyabout the strategiesof the other n -1 players. Define the cumulativedistribution functionfor a player'sactionas F(e.). In the payoff-dominantequilibrium,F&e) equals 1 and F(e1) equals0 for ej less than e. A well-known theorem is that if el,..., en
are independent and identicallydistributed with common cumulativedistributionfunction F(ej), then the cumulativedistribution function for the minimum, F.,n(e), equals 1- [1 -F(e)]; see A. M. Mood, F. A. Graybill, and D. C. Boes (1974). In the payoff-dominantequilibrium,Fmin(e-) equals 1 and Fmin(e) equals 0 for e less than e-.But
suppose that a player is uncertainthat the n -1 playerswill select the payoff-dominant action, J. Specifically, let F(1) be small but
greater than 0, then as n goes to infinity Fmin(l) goes to 1. Consequently,when the
number of players is large it only takes a remote possibility that an individualplayer will not select the payoff-dominantaction eto motivate defection from the payoffdominantequilibrium. Several deductive selection principles based on the "riskiness"of an equilibrium point have been identifiedand formalized.A maximin action, which is an action (pure strategy)with the largestpayoffin the worst possible outcome, is secure, see John Von Neumann and Oskar Morgenstern(1944, 1972). Given existence, security selects the equilibrium point supported by player's maximin actions. Securitymay select very inefficientequilibriumpointsin nonzerosum games. In period game A, a player can ensurea payoff of a - b by choosing ei equal to 1,
which is the largestpayoffin the worstpossi-
VOL. 80 NO. I
VAN HUYCKETAL: TACIT COORDINATION GAMES
237
TABLE1-EXPERIMENTAL DESIGN
Experiment No.
Date
Size
1 2 3 4 5 6 7
June June June Sept Sept Sept Sept
16 16 14 15 16 16 14
A Payoff A Fullsize
B Payoff B Fullsize
1P,2,... 10 1P,2,..., 10P 1P,2,... 10P 1P,2P,..., 10P 1P,2P".. ,1OP lP,2P,..,10lP 1P,2P,..0. lop
-
11,...,15 11,...115
11P,"...15 11p...,15 11"..., 15 1ip,1... 15
A' Payoff A Fullsize
C Payoff A Size Twoa
-
-
16P,..., 20 16P, ... ,20 16,...,20 16,.,20 20 16, 16,...,22
21, ,27 21,.,27 21,.,25 23,...,25
P- Denotes a period in which subjects made predictions. - - In experiment 4 and 5 pairings were fixed, while in experiments 6 and 7 pairings were random.
Notice that the principlesof efficiencyand security can be defined independentlyof equilibrium.The experimentsin this paper, (1,...,1). Since payoff-dominance and security select differentequilibriumpoints in this which are designedto study the conflictbetacit coordinationgame (equilibriumpoints tween efficiency and security, are not dewith the highest and lowest payoffs,respec- signed to study how repeatedplay of period tively), an importantand tractableempirical game A influences the set of equilibrium questionis which, if any, deductiveselection points for A(T). principleorganizesthe experimentaldata. Having t periods of experiencein A(T) It is often possibleto applymorethanone provides a player with observed facts, in deductive selection principle to a game. addition to the descriptionof the game,that Hence, subjects may choose disequilibrium can be used to reasonabout the equilibrium selection problem in the continuationgame outcomes unless they behaveas if thereis a hierarchyof selection principles.When de- A(T - t). This experience may influence the ductive selectionprinciplesfail to coordinate outcome of the continuationgame A(T- t) beliefs and actions, inductiveselectionprin- by focusing expectationson a specificequiciples based on repeated interaction may librium point. For example, one adaptive hypothesis is that players will give a best allow players to learn to coordinate. response to the minimumobserved in the Consider a finitely repeatedgame A(T), which involves the n playersplayingperiod previous period. This adaptive behavior game A for T periods.The payoff-dominant would immediatelyconvergeto an equilibequilibrium of A(T) is just the repeated rium in A(T-1). The selected equilibrium implementation of the payoff-dominant involves all players choosing the period 1 minimum for the T- 1 periods of the conequilibriumof period game A, because the tinuationgame A(T -1). first-best outcome for period game A is
ble outcome. Consequently,in this tacit coordinationgame, securityselects the n-tuple
(e,..., e). Similarly, the secure equilibrium
of A(T) is just the repeatedimplementation of the secureequilibriumof periodgameA.4 4Crawford (1989)emphasizesthat the secureequilibrium is also the only equilibriumthat is evolutionary stable.In repeatedplay,playersusingadaptivebehavior may be led to implementthe secureequilibrium.Hence, while the experimentsreportedin this paper discriminate sharplybetween strategicstabilityand evolutionary stability,they do not discriminatebetweenlearning to use securityand certainkinds of adaptivebehavior.
III. Experimental Design
Table 1 outlines the design of the seven experimentsreportedin this paper. The instructionswere read aloud to ensurethat the descriptionof the game was commoninformation,if not, commonknowledge.'No pre5The originalworkingpaper,Van Huyck, Battalio, and Beil (1987), which includesthe actualinstructions,
MARCH 1990
THE AMERICAN ECONOMIC REVIEW
238
PAYOFFTABLE A
Smallest Value of X Chosen
Your Choice of X
7 6 5 4 3 2 1
7
6
5
4
3
2
1
1.30 -
1.10 1.20 -
0.90 1.00 1.10 -
0.70 0.80 0.90 1.00 -
0.50 0.60 0.70 0.80 0.90 -
0.30 0.40 0.50 0.60 0.70 0.80 -
0.10 0.20 0.30 0.40 0.50 0.60 0.70
-
-
PAYOFFTABLE B
Smallest Value of X Chosen Your Choice of X X
7 6 5 4 3 2 1
7
6
5
4
3
2
1
1.30 -
1.20 1.20 -
1.10 1.10 1.10 -
1.00 1.00 1.00 1.00 -
0.90 0.90 0.90 0.90 0.90 -
0.80 0.80 0.80 0.80 0.80 0.80 -
0.70 0.70 0.70 0.70 0.70 0.70 0.70
play negotiation was allowed. After each repetitionof the periodgame, the minimum action was publicly announcedand the subjects calculated their earningsfor that period. The only commonhistoricaldata available to the subjectswas the minimum. During the courseof an experimentsome design parameterswerealteredresultingin a sequence of treatmentslabeled A, B, A', and C. Instructionsfor continuationtreatments were given to the subjectsafterearlier treatmentshad been completed.The feasible actions in all treatmentsof all experiments were the integers 1 through 7: hence, e equaled7. In treatmentA and A', the followingvalues were assignedto the parametersin equation (1): parametera was set equal to $0.20, parameterb was set equal to $0.10, and a
payoff tables, questionnaire,extra instructions and recordsheet used in the experimentsand a moreextensive analysis of the experimentalresults, is available from the authorsupon request.
-
-
-
constant of $0.60 was added to ensurethat all payoffs were strictly positive. Consequently, the payoff-dominantequilibrium, (7, ... , 7), paid $1.30 while the secure equilibrium, (1,...,1), paid $0.70 per subject per
period.6Subjectsweregiventhis information in the form of a payoff table, see payoff Table A. In treatmentA, the periodgameA was repeatedten times.The numberof players, n, varied between 14 and 16 subjects. (In treatment C, the numberof players, n, was reduced to two.) TreatmentA' designates the resumptionof these conditionsafter treatmentB. In treatment B, parameterb in equation (1) was set equal to zero,see payoffTable B. This gives the subjectsa dominatingstrategy (play 7 regardlessof the minimum),which eliminates the coordination problem. The numberof players, n, remainedthe same as in treatmentA. 6For the remainderof this paper, an equilibrium denotes a mutual best-responseoutcome in the period game.
VOL. 80 NO. 1
VAN HUYCK ETAL: TACIT COORDINATION GAMES
239
Occasionally,subjectswere asked to pre- riod one was never greaterthan 4. Hence, dict the actions of all the subjects in the the largest payoff in period one was $1.00 treatment.7 For each prediction in the and some payoffs were $0.10. (The payoffSeptemberexperiments,a subjectwas paid dominant equilibriumwould have paid ev$0.70 less 0.02 times the sum of the absolute eryone $1.30). All of these outcomes are value of the differencebetween the actual inefficient.The subjectswere unable to use and predictedactions.(The rule used in the any deductive selectionprincipleto coordiJune experimentswas less sensitive to pre- nate on an equilibriumpoint. diction errors).At the end of the experiment, Only 10 percentof the subjectspredicted the subjectswere told the actualdistribution an equilibriumoutcome in period one. Inof actionsand werepaid. stead, most subjects(95 of 106) predicteda The subjectswere undergraduatestudents disequilibriumoutcome.8Moreover,the subattending Texas A&M Universityand were jects' predictionswere dispersed:one third recruited form sophomoreand junior ecoof the subjectspredictedat least one 1 and nomics courses.A total of 107 studentspar- one 7-a rangeof 6-and the averagerange ticipated in the seven experiments.After of the predictionswas 4.0. The subjects'disreading the instructions,but before the ex- persedpredictionssuggestthat they expected periment began, the students filled out a other subjectsto respondto the payoff table questionnaireto determinethat they under- differently than they did. These data are stood how to read the payofftable for treat- inconsistent with any theoryof equilibrium ment A, that is, map actions into money selection that assumesthat, becausea player payoffs. The instructionswould have been will derive his prior probabilitydistribution re-read if needed, but all 107 students re- over other players' pure strategies strictly from the parametersof the game,all players sponded correctly. will have the samepriorprobabilitydistribuIV. Experimental Results tion. Instead, some subjectsmade optimistic predictions and some subjects made pesTable 2 reports the experimentalresults simisticpredictions. for treatmentA. The data in periodone are While the subjectswere unable to coordiparticularlyinterestingbecause the subjects nate beliefs and actions,in almost all cases can only use deductiveselectionprinciplesto their individualpredictionsand actionswere inform their behavior. consistent. Of the 107 subjects,106 subjects In periodone, the payoff-dominant action, predicted that at least one other subject 7, was chosen by 31 percentof the subjects would choose an actionequalto or less than (33 of 107) and the secure action, 1, was their choice. (Only one subjectpredictedhe chosen by 2 percent of the subjects (2 of would determinethe minimum.)Most sub107). Neither deductive selection principle jects mapped predictionsinto actions in a succeeds in organizingmuch of the data, reasonable way. Those subjects who made althoughpayoff-dominanceis more success- pessimisticpredictionsabout what the other ful than security.The popularityof actions4 subjects would do chose small values for and 5-chosen by 18 and 34 subjects,re- their action and subjects who made optispectively-is consistentwith many subjects mistic predictionsaboutwhat the other subhaving nearly diffusepriorbeliefs about the jects would do chose large values for their outcome of the periodgame. action.9 The initial play of all seven experiments exhibitboth individualand collectivecoordination failure.The minimumaction for pe8One subjectis excludeddue to predictingonly 15 choices. 7In two earlier pilot experimentspredictionswere not made in any period. The substantiveresultswere the same as those reportedhere.
9Ananomalyis that,of the 95 subjectswhopredicted a minimumless than 7, 87 subjectschose an action greaterthan the minimumthey predicted.Van Huyck, Battalio, and Beil (1987) provide an expected value model to explainthis anomaly.
240
THE AMERICAN ECONOMIC REVIEW
MARCH 1990
TABLE2-EXPERIMENTAL RESULTSFOR TREATMENTA
Period 1
2
3
4
5
6
7
8
9
10
No. of 7's No. of 6's No. of 5's No.of4's No. of 3's No. of 2's No.of l's
8 3 2 1 1 1 0
1 2 3 6 2 2 0
1 1 2 5 5 2 0
0 0 1 4 5 4 2
0 0 0 1 4 8 3
0 0 0 1 1 7 7
0 0 1 1 1 8 5
0 0 0 0 1 6 9
0 0 0 0 0 4 12
1 0 0 0 1 1 13
Minimum Experiment 2 No. of 7's No. of 6's No. of 5's No. of 4's No. of 3's No. of 2's No. of l's
2
2
2
1
1
1
1
1
1
1
4 1 3 4 1 3 0
0 0 3 6 4 2 1
1 1 2 2 2 6 2
0 0 1 3 5 5 2
0 0 0 3 0 5 8
0 1 0 0 1 9 5
0 0 1 0 1 3 11
0 0 1 0 0 4 11
0 0 0 0 1 3 12
1 0 1 0 0 1 13
2
1
1
1
1
1
1
1
1
1
4 2 5 3 0 0 0
4 0 6 3 0 1 0
1 2 1 2 7 1 0
0 0 1 1 6 4 2
1 0 1 2 0 5 5
1 0 0 1 2 3 7
1 0 0 0 3 6 4
0 0 0 0 0 3 11
0 0 0 0 0 2 12
2 0 0 1 0 2 9
Minimum Experiment 4 No. of 7's No. of 6's No.of5's No. of 4's No. of 3's No. of 2's No.of l's
4
2
2
1
1
1
1
1
1
1
6 0 8 1 0 0 0
0 6 5 1 2 1 0
1 2 5 4 3 0 0
1 0 5 6 2 0 1
0 0 0 7 4 2 2
0 1 1 1 3 3 6
1 0 0 2 2 7 3
0 0 0 1 2 4 8
0 0 0 1 1 2 11
0 0 0 0 0 2 13
Minimum
4
2
3
1
1
1
1
1
1
1
Experiment1
Minimum Experiment 3 No. of 7's No. of 6's No.of 5's No. of 4's No.of 3's No. of 2's No. of l's
An interestingquestionis whetherthe subjects' predictions correspondto the actual distribution of actions more closely than predictions based on payoff-dominanceor security. Using the numberof actions correctly predictedas a statistic,the data reveal that 95 percentof the subjectspredictedthe actions of the other n -1 subjects more accurately than did payoff-dominance. This statistic is used to measureprediction accuracybecause the subjectspayoff were a linear transformationof the predictionaccuracy score. The differenceof the mean prediction accuracy score was always positive and in most cases significantlydifferentfrom
zero at the 1 percentlevel: a result that is robust to non-parametricstatistical procedures. Obviously,securitydoes even worse. Subjects predicted the observed heterogenous responseto the descriptionof the game and the resultingcoordinationfailurein period one. Repeated play of the period game allows the subjectsto use inductiveselectionprinciples or to learn to use deductiveselection principles. Hence, repeated play makes it more likely that subjects will be able to obtain mutualbest-responseoutcomesin the continuation game. Repeating the period game does cause actions to converge to a
VOL. 80 NO. I
VAN HUYCK ETAL: TACIT COORDINA TION GAMES
241
TABLE2-EXPERIMENTAL RESULTSFOR TREATMENT A, Continued
Period
Experiment 5 No. of 7's No. of 6's No.of5's No. of 4's No.of3's No. of 2's No. of l's Minimum Experiment 6 No. of 7's No. of 6's No. of 5's No. of 4's No.of3's No.of2's No. of l's Minimum Experiment 7 No. of 7's No. of 6's No. of 5's No. of 4's No.of3's No. of 2's No. of l's Minimum
1
2
3
4
5
6
7
8
9
10
2 1 9 3 1 0 0
2 3 3 4 2 2 0
3 1 0 6 2 2 2
1 0 4 2 4 3 2
1 0 1 1 6 4 3
1 0 0 2 0 6 7
1 0 2 0 0 5 8
0 0 0 2 0 2 12
0 0 0 1 0 5 10
0 0 0 1 1 3 11
3
2
1
1
1
1
1
1
1
1
5 2 5 2 1 0 1
3 0 1 3 5 2 2
1 0 0 4 4 4 3
1 0 0 0 2 5 8
1 1 0 0 2 3 9
1 0 1 0 2 3 9
2 0 0 0 1 6 7
2 0 0 0 0 4 10
2 0 0 0 2 5 7
3 0 0 0 0 5 8
1
1
1
1
1
1
1
1
1
1
4 1 2 4 1 1 1 1
3 0 3 0 3 3 2 1
1 0 0 1 2 2 8 1
1 0 0 2 1 2 8 1
1 0 0 1 1 4 7 1
1 0 0 0 0 4 9 1
1 0 0 0 0 4 9 1
1 0 0 0 0 4 9 1
1 0 0 0 0 5 8 1
1 0 0 0 0 3 10 1
stable outcome, see Table2. But ratherthan converging to the payoff-dominantequilibrium or to the initial outcome of the treatment, the most inefficientoutcome obtains in all seven experiments. The change in a subject'saction between period one and period two providesinsight into the subjects'dynamicbehavior.Of the eleven subjects who determinedthe minimum in period one the averagechange in action betweenperiodone and two was 0.73:t seven subjects increasedtheir action, three did not change, and one decreasedhis action. In every experimentsomeonewho had not determinedthe minimumin period one determines the minimum in period two. Moreover, in experimentsone throughfive the intersection of the set of subjectswho determinethe minimumin period one with the set of subjectswho determinethe minimum in period two is empty. Since a subject's payoff is increasing in ei when he
(she) uniquelydeterminesthe minimum,this adaptivebehaviorcan be rationalized. However, a subject'spayoff is decreasing in ei when he (she) played above the minimum and those subjectsthat played above the minimumreducedtheirchoice of action. The observed mean reductionis increasing in the differencebetween a subject'saction and the reported minimum,and the mean reductionis smallerthan this difference.The observed correlation between the current choice of these optimisticsubjects and the minimumreportedin period 1 suggeststhat behavior in the continuationgame A(9) is not independentof the historyleadingup to continuation game A(9). However,only 14 of 107 subjects give a best-responseto the period one minimumin periodtwo. Some subjectsplay below the minimumof the preceding period. This observed"overshooting" cannot be reconciledwith adaptive theories that predict the currentaction
242
THE AMERICAN ECONOMIC REVIEW
will be a convex combinationof last periods action and last periodsoutcome.Apparently, some subjects learn how "risky" it is to choose an action other than the secure action, 1, underthe minimumrule and learnto use security to inform their behaviorin the continuationgame. Although it failed to predict the initial outcome, security predicts the stable outcome of period game A. By period ten 72 percentof the subjects(77 out of 107) adopt their secure action, 1, and the minimumfor all seven experimentswas a 1. The observed coordinationfailureappearsto resultfrom a few subjects concludingit is too "risky"to choose an action other than the secure action and from most subjectsfocusingon the minimum reportedin earlierperiod games. The minimumrule interactingwith this dynamic behaviorcausesthis treatmentto converge to the most inefficientoutcome. In treatment B, parameterb of equation (1) was set equal to zero. Becausea player's action is no longer penalized, the payoffdominant action, 7, is a best responseto all feasible minimums.Action 7 is a dominating strategy. Hence, treatmentB tests equilibrium refinementsbasedon the eliminationof individually unreasonableactions. For example, a simple dominanceargumenteliminates all of the equilibriumpoints except one: (7,.. ., 7). Any strategic uncertainty would cause an individuallyrationalplayer to choose the payoff-dominantaction,7. Table 3 reports the experimentalresults for treatmentB and treatmentA'. In period eleven, the payoff-dominantaction, 7, was chosen by 84 percentof the subjects(76 of 91). However,the minimumin periodeleven was never more than 4 and in experiments four, five, six, and sevenit was a 1.10 Of course, a subjectthat adopts action 7 need not worry about what actions other
10At least one subject did not understand how the payoff table had changed. Subject 3 in experiment 5, who plays a 1 in every period of the B treatment, predicts that all 16 players will choose 1 but only he does so. When the actual distribution was revealed, subject 3 appeared genuinely amazed and confessed that he had not understood how the payoffs had changed.
MARCH 1990
subjects take and, apparently,most subjects did not. This propertyof dominatingstrategies resulted in the B treatmentexhibiting different dynamics than the A treatment. Like the A treatment, those players who determine the minimum increase their action, but, unlike the A treatment,those players who were above the minimumdo not decreasetheir action.11This dynamicbehavior converges to the efficientoutcome-the payoff-dominantequilibrium-in four of the six experiments.By periodfifteen,96 percent of the subjects chose the payoff-dominant action, 7. Even in the experimentsthat obtainedthe efficient outcome, the B treatmentwas not sufficientto induce the groupsto implement the payoff-dominantequilibriumin treatment A': parameter b equals $0.10 once again. Returningto the originalpayoff table in period sixteen, 25 percentof the subjects chose the payoff-dominantaction,7.12 However, 37 percent chose the secure action, 1. Period sixteen predictionswere peakedwith subjects choosing a 7 predictingmost subjects would choose 7 and subjectschoosinga 1 predictingmost subjectswould choose 1. This bi-modal distributionof actions and predictionssuggeststhat play priorto period sixteen influenced subjects'behavior.However, the subjectsexhibit a heterogenousresponse to this history. Security predicts the stable outcome of treatmentA'. In treatmentA', the minimum in all periods of all six experimentswas 1. By period twenty, 84 percentof the subjects chose the secure action, 1, and 94 percent chose an action less than or equal to 2. (Experimentstwo and four even satisfy the
"1The two exceptions were due to subject 3 in experiment five, see fn. 10, and subject 12 in experiment six. Subject 12 predicts that he will uniquely determine the minimum in period 11, verifies this in period 12 by choosing a 3, and then chooses a one for the remainder of the B treatment. Perhaps, subject 12 became vindictive. He had chosen a 7 in period 1. 12The large fluctuations in behavior resulting from changes in the parameter b -between treatment A and B and between treatment B and A'-suggest that subjects are influenced by the description of the game. In our view, these data are inconsistent with backwardlooking theories of adaptive behavior.
VOL. 80 NO. 1
TABLE 3-EXPERIMENTAL
Experiment 2 No.of 7's No. of 6's No. of 5's No.of 4's No. of 3's No. of 2's No.of l's Minimum Experiment 3 No. of 7's No. of 6's No. of 5's No. of 4's No. of 3's No. of 2's No. of l's Minimum Experiment 4 No. of 7's No. of 6's No.of5's No. of 4's No. of 3's No. of 2's No.of l's Minimum Experiment 5 No. of 7's No. of 6's No.of5's No. of 4's No. of 3's No. of 2's No.ofl's Minimum Experiment 6 No.of 7's No. of 6's No.of S's No. of 4's No. of 3's No. of 2's No. of l's
243
VAN HUYCK ETAL.: TACIT COORDINATION GAMES RESULTS FOR TREATMENT
Treatment B 14 13
B AND
TREATMENT
A'
Treatment A' 19 18
15
16
17
16 0 0 0 0 0 0
16 0 0 0 0 0 0
8 0 1 1 1 3 2
2 0 0 2 1 3 8
0 0 0 0 1 4 11
0 0 0 0 1 2 13
7*
7*
7*
1
1
1
1
13 0 0 0 1 0 0
12 1 1 0 0 0 0
13 1 0 0 0 0 0
14 0 0 0 0 0 0
6 1 0 1 0 2 4
2 0 2 0 0 4 6
2 0 1 0 0 2 9
1 0 0 0 0 3 10
1 0 0 1 0 0 12
4
3
5
6
7*
1
1
1
1
1
12 0 1 0 0 0 2
13 0 0 1 1 0 0
14 0 0 1 0 0 0
14 0 1 0 0 0 0
15 0 0 0 0 0 0
3 0 0 2 2 2 6
1 0 0 0 0 1 13
0 0 0 0 0 2 13
0 0 0 0 0 0 15
0 0 0 0 0 0 15
1
3
4
5
7*
1
1
1
1*
13 0 1 1 0 0 1
13 0 1 1 0 0 1
15 0 0 0 0 0 1
15 0 0 0 0 0 1
15 0 0 0 0 0 1
1 0 0 0 1 3 11
0 0 0 0 1 4 11
0 0 0 0 0 2 14
0 0 O 0 0 2 14
0 0 0 0 0 3 13
1
1
1
1
1
1
1
1
1
1
13 0 0 1 0 1 1
13 1 1 0 1 0 0
12 1 1 1 0 0 1
12 1 0 1 1 0 1
13 0 1 0 0 1 1
2 0 0 1 1 5 7
2 0 0 0 0 6 8
2 0 0 0 0 7 7
2 0 0 0 0 6 8
2 0 0 0 0 5 9
11
12
13 1 0 1 1 0 0
15 0 1 0 0 0 0
16 0 0 0 0 0 0
3
5
13 0 0 1 0 0 0
20 0 0 0 0 0 0 16 1*
1*
Minimum Experiment 7 No. of 7's No. of 6's No.of S's No.of 4's No.of 3's No. of 2's No.of l's
1
3
1
1
1
1
1
1
1
1
12 0 0 1 0 0 1
14 0 0 0 0 0 0
13 1 0 0 0 0 0
13 0 0 0 1 0 0
14 0 0 0 0 0 0
3 0 1 2 2 2 4
4 0 0 0 0 4 6
2 0 0 0 0 2 10
2 0 0 0 0 2 10
2 0 0 0 0 1 11
Minimum
1
7*
6
3
7*
1
1
1
1
1
*
Denotes a mutual best-response outcome.
244
THE AMERICAN ECONOMIC REVIEW
mutual best-responsepropertyof an equilibrium by period twenty.) Obtainingthe efficient outcome in treatmentB failed to reverse the observedcoordinationfailure.Like the A treatment, the most inefficientoutcome obtained. V. ExperimentalResultsfor TreatmentC: GroupSize Two
TreatmentC was addedto the September experiments to determineif subjects were influencedby groupsize whenchoosingtheir actions. In theory,any uncertaintyabout the actions of an individualplayerin the gameis exacerbated by the minimum rule as the number of players increases,see Section II. The C treatmentreducesgroup size to two. Table 4 reports the experimentalresults for the C treatmentof experimentsfour and five, which permanentlypairedsubjectswith an unknown partner.In period twenty-one, 42 percent of the subjectsplay theirpayoffdominant action, 7, and 74 percent of the subjectsincreasetheiraction.This resultoccurred even though the minimum for the precedingfive periodshad been a 1 and all 31 subjects had played either a 1 (28 subjects) or a 2 (3 subjects)in period twenty. Clearly,eitherthe subjectsthoughtthat their partner in treatment C would change his (her) action in response to reduced group size or the subjectsexpectedalternativedynamicsin repeatedplay. The subjectsin experimentsfour and five used an adaptive behaviorin the C treatment similarto the adaptivebehaviorexhibited in the A treatment.Subjectsthat played the minimumincreasedeffortby an average of +2.0 and subjectsthat played above the minimum rediced effort by an averageof - 1.9. However,unlikethe A treatment,there was no ".overshooting" to the secureaction, 1. Occasionally, both subjects simultaneously chose the payoff-dominantaction, 7V13 13Recall that subjects only observe the minimum and their own action. Hence, it is not possible to unilaterally "signal" a willingness to implement the payoff-dominant equilibrium, that is, subjects could not use Os-
MARCH 1990
This dynamic behaviorconvergedto the efficient outcome-the payoff-dominantequilibrium-in 12 of 14 trails.Hence,even with an extremely negative history payoff-dominance predicts the stable outcome of the tacit coordinationgamewith fixedpairs. Experiments six and seven randomly paired subjects with an unknownpartner.'4 Hence, experiments six and seven test whetherthe resultsobtainedin Experiments four and five were due to subjectsrepeating the period game with the same opponent.In the first period of these experiments,37 percent of the subjectschose the payoff-dominant action, 7, and 73 percentof the subjects increasedtheir choice of action,see Table 5. Moreover, the subjects' dynamic behavior was similarto that found in the fixedpair C treatment.While the resultsfor the random pair C treatment are influencedby group size, no stable outcomeobtains. The C treatmentconfirmsthat there are two consequencesof the minimumrule.First, groupsize interactingwith the minimumrule alters the subjects' initial choice of action. Second, group size interactingwith the minimum rule alters the convergenceof the subjects' dynamicbehaviorin disequilibrium.
borne's (1987) refinement of a "convincing deviation" to inform their behavior, see also van Damme (1987). 14Experiments by Cooper, DeJong, Forsythe, and Ross (1987) report that after eleven repetitions randomly paired groups of size two almost always obtain the payoff-dominant equilibrium. However, Cooper et al. also report coordination failure when subjects can choose from a strategy space that includes certain kinds of dominated cooperative strategies. Their game illustrates an interesting distinction between Luce and Raiffa's "solution in the strict sense," which depends on joint-admissibility and Harsanyi and Selten's solution, which depends on payoff-dominance. Because the first-best outcome requires using a strictly dominated strategy-as in the prisoner's dilemma game-and because joint-admissibility admits efficiency comparisons with disequilibrium outcomes, the Cooper et al. game with dominated cooperative strategies has no "solution in the strict sense" of Luce and Raiffa. Because the first-best outcome is an equilibrium in period game A, joint-admissibility-appropriately defined for n person games-and payoff-dominance select the same equilibrium point in period game A.
VOL. 80 NO. 1
VAN HUYCK ETAL.: TACIT COORDINATION GAMES
245
TABLE4-EXPERIMENTAL RESULTSFOR TREATMENTC: FIXED PAIRINGS
Period 21
22
23
24
25
26
7 7
7 7
7 7
7 7
7 7
7 7
7 7
Minimum Pair 2 Subject 2 Subject 15
7*
7*
7*
7*
7*
7*
7*
7 1
2 7
7 3
7 6
7 7
7 7
7 7
Minimum Pair 3 Subject 3 Subject 14
1
2
7
7
7
7
7
1 1
1 1
1 7
1 1
1 1
1 1
1 7
Minimum Pair 4 Subject 4 Subject 13
1*
1*
1
1*
1*
1*
1
1 7
7 2
7 5
7 7
7 7
7 7
7 7
Minimum Pair 5 Subject 5 Subject 12
1
2
5
7*
7*
7*
7*
1 1
7 4
4 7
7 7
7 7
7 7
7 7
Minimum Pair 6 Subject6 Subject 11 Minimum Pair 7 Subject 7 Subject 10
1
4
4
7*
7*
7*
7*
5 7 5
7 7 7*
7 7 7*
7 7 7*
7 7 7*
7 7 7*
7 7 7*
1 5
7 3
6 6
7 7
7 7
7 7
7 7
Minimum Pair 8 Subject 8 Subject 9
1
3
6*
7*
7*
7*
7*
7 3
6 5
6 7
7 7
7 7
7 7
7 7
3
5
6
7*
7*
7*
7*
7 2
7 3
4 6
5 6
6 7
6 7
7 7
Minimum Pair 2 Subject 3 Subject 14
2
3
4
5
6
6
7*
5 7
7 7
7 7
7 7
7 7
7 7
7 7
Minimum Pair 3 Subject 4 Subject 13
5
7*
7*
7*
7*
7*
7*
1 7
1 1
1 1
1 3
4 1
4 1
1 2
Minimum Pair 4 Subject 5 Subject 12 Minimum
1
1*
1*
1
1
1
1
5 7 5
7 7 7*
7 7 7*
7 7 7*
7 7 7*
7 7 7*
7 7 7*
Experiment 5 Pair 1 Subject 1 Subject l6
Minimum Experiment 6 Pair 1 Subject 2 Subject 15
27
246
THE AMERICAN ECONOMIC REVIEW TABLE4-FIXED
MARCH 1990
PAIRINGS,Continued
Period 21
22
23
24
25
26
4 4
5 5
7 7
7 7
7 7
7 7
7 7
4*
5*
7*
7*
7*
7*
7*
SubjectlO
5 5
7 7
7 7
7 7
7 7
7 7
7 7
Minimum
5*
7*
7*
7*
7*
7*
7*
Pair5 Subject6 Subject11 Minimum Pair6 Subject7
*
27
Denotes a mutualbest-responseoutcome.
TABLE 5-DISTRIBUTION OF ACTIONSFOR TREATMENTC: RANDOMPAIRINGS
Period 21
22
23
24
25
5 0 2 3 1 1 4
5 1 5 1 1 1 2
4 3 3 1 1 2 2
10 0 3 1 0 2 0
8 0 4 1 0 2 1
-
-
6 1 0 2 2 0 3
5 0 3 1 0 0 5
5 1 0 4 0 1 3
Experiment6 No. of 7's No. of 6's No. of 5's No. of 4's No. of 3's No. of 2's No. of l's Experiment 7 No. of 7's No. of 6's No. of 5's No. of 4's No. of 3's No. of 2's No. of l's
VI. Treatment A withMonitoring As a refereepoints out, a reasonableconjecture is that revealingthe distributionof actions each period-in additionto the minimum-might influencethe reporteddynamics. For example, subjects could signal a willingness to coordinate on the payoffdominant equilibriumand optimistic subjects might delay reducing their action if they knew the minimumwas determinedby just one subject.Two experiments,each using payoff Table A and 16 naive subjects, were conductedin whichthe entiredistribution of actionswas recordedon a blackboard
at the end of each periodand was left there for the entire experiment. The initial distributionof actions and the dynamicsof the two monitoringexperiments were similar to those reported above, see Table 6. If anything, the convergenceof actions to the secure action, 1, was more rapid under the monitoring treatment.In fact, a mutual best-responseoutcome was obtained in one experiment:without monitoring mutual best-responseoutcomes are not observedfor period game A until treatment A'. Apparently,monitoringhelps solve the individualcoordinationproblem-more subjects give a best-responsesooner-but
VOL. 80 NO. 1
247
VAN HUYCK ETAL.: TACIT COORDINA TION GAMES TABLE6-DISTRIBUTION OF ACTIONSFOR TREATMENT A WITH MONITORING
Period
Experiment 8 No.of7's No. of 6's No. of 5's No. of 4's No. of 3's No. of 2's No. of l's Minimum Experiment 9 No. of 7's No. of 6's No. of 5's No. of 4's No. of 3's No. of 2's No. of l's Minimum
1P
2P
3
4
5
6
7
8
9
lop
4 1 4 5 1 0 1
0 1 0 4 4 2 5
0 0 1 2 1 1 11
0 0 0 1 0 2 13
2 0 0 0 0 3 11
0 0 1 0 2 2 11
0 0 0 1 0 1 14
1 0 0 0 0 1 14
1 0 0 0 0 2 13
1 0 1 0 0 1 13
1
1
1
1
1
1
1
1
1
1
6 1 2 4 1 0 2
2 2 2 2 5 1 2
0 0 1 3 3 0 9
1 0 0 1 1 5 8
0 0 0 0 0 4 12
0 0 0 0 0 1 15
0 0 0 0 0 0 16
0 0 1 0 0 0 15
0 0 0 0 1 1 14
0 0 0 1 0 2 13
1
1
1
1
1
1
1
1
1
1*
P- Denotes a period in which subjects made predictions. * - Denotes a mutual best-response outcome.
not the collectivecoordinationproblem-the minimumwas a 1 in all ten periodsof both experiments. VII. ConcludingComments
These experimentsprovide an interesting example of coordinationfailure. The minimum was never above four in period one and all seven experimentsconvergedto a minimum of one within four periods. Since the payoff-dominantequilibriumwouldhave paid all subjects $19.50 in the A and A' treatments-excluding predictions-and the average earnings were only $8.80, the observed behavior cost the average subject $10.70 in lost earnings. This inefficientoutcomeis not due to conflictingobjectivesas in "prisoner'sdilemma" games or to asymmetricinformationas in "moralhazard"games.Rather,coordination failure results from strategic uncertainty: some subjectsconcludethat it is too "risky" to choose the payoff-dominantaction and most subjects focus on outcomesin earlier period games.The minimumrule interacting with this dynamicbehaviorcausesthe A and
A' treatmentsto convergeto the most inef-
ficientoutcome. Deductive methodsimply that all feasible actions are consistentwith some equilibrium point in this experimental coordination game. However, the experimentalresults suggest that the first-bestoutcome,which is the payoff-dominantequilibrium,is an extremely unlikely outcome either initially or in repeatedplay. Instead,the resultssuggest that the initial outcomewill not be an equilibriumpoint and only the secure-but very inefficient-equilibrium describes behavior that actual subjectsare likely to coordinate on in repeatedplay of periodgame A when the numberof playersis not small. REFERENCES Basar, Tamer and Olsder, Geert J., Dynamic Noncooperative Game Theory, New York:
AcademicPress,1982. Bryant,John, "A Simple Rational Expectations Keynes-Type Model," Quarterly Journal of Economics, August 1983, 98, no. 3, 525-28.
248
THE AMERICAN ECONOMIC REVIEW
MARCH 1990
Cooper, Russell, DeJong, Douglas V., Forsythe, Robertand Ross, ThomasW., "Selection Cn-
Luce, R. DuncanandRaiffa,Howard,Games and
teria in CoordinationGames: Some Experimental Results," unpublishedmanuscript, October1987.
Mood, A. M., Graybill,F. A. and Boes, D. C., Introduction to the Theoryof Statistics, 3rd
and John, Andrew,"Coordinating Co-
ordinationFailuresin KeynesianModels," Quarterly Journal of Economics, August
1988, 103, no. 3, 441-64. Crawford,Vincent P., "An 'Evolutionary' In-
terpretationof Van Huyck, Battalio,-and Beil's ExperimentalResults on Coordination," unpublished manuscript,January 1989. Harsanyi,John C., andSelten,Reinhard,A General Theory of Equilibrium Selection in
Games,Cambridge:MIT Press,1988. " On Kohlberg,Elon and Mertens,Jean-Francois,
the Strategic Stability of Equilibria,"
Decisions,New York:Wiley& Sons, 1957. ed., New York: McGraw-Hill,1974. Osbome, Martin J., "Signaling, Forward In-
duction, and Stability in Finitely Repeated Games," unpublishedmanuscript, November1987. Schelling,ThomasC., The Strategy of Conflict,
Cambridge: Harvard University Press, 1980. vanDamme,Eric,"StableEquilibriaand Forward Induction," UniversitiitBonn Discussion PaperNo. 128, August1987. Van Huyck, John B., Battalio,RaymondC. and Beil, RichardO., "Keynesian Coordination
Games, StrategicUncertainty,and Coordination Failure," unpublished manuscript, October1987.
Econometrica, 54, no. 5, September 1986. Lucas, RobertE., Jr., "AdaptiveBehaviorand Economic Theory," in Rational Choice, R.
Von Neumann, John and Morgenstern,Oskar, Theory of Games and Economic Behavior,
Hogarth and M. Reder, eds. Chicago: Universityof ChicagoPress,1987.
Princeton: Princeton University Press, 1972.
Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure Author(s): John B. Van Huyck, Raymond C. Battalio, Richard O. Beil Source: The American Economic Review, Vol. 80, No. 1 (Mar., 1990), pp. 234-248 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/2006745 Accessed: 07/12/2010 11:25 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=aea. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].
American Economic Association is collaborating with JSTOR to digitize, preserve and extend access to The American Economic Review.
http://www.jstor.org
Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure By JOHN B. VAN HUYCK, RAYMOND C. BATTALIO, AND RICHARD 0. BEIL*
Deductive equilibriummethods-such as Rational Expectations or Bayesian Nash Equilibrium-are powerfultools for analyzing economies that exhibit strategicinterdependence. Typically, deductive equilibrium analysis does not explain the process by which decision makers acquire equilibrium beliefs. The presumption is that actual economies have achieveda steady state. In economies with stable and unique equilibrium points, the influence of inconsistent beliefs and, hence, actions would disappear over time, see Robert Lucas (1987). The power of the equilibriummethod derives from its ability to abstractfrom the complicated dynamic process that inducesequilibrium and to abstractfromthe historicalaccident that initiatedthe process. Unfortunately, deductive equilibrium analysis often fails to determinea unique equilibriumsolutionin manyeconomiesand, hence, often fails to prescribeor predictrational behavior.In economieswith multiple equilibria, the rational decision maker formulatingbeliefs using deductiveequilibrium
*Department of Economics, Texas A&M University, College Station, TX 77843; Department of Economics, Texas A&M University, College Station, TX 77843; and Department of Economics, Auburn University, Auburn, AL 36849, respectively. Mike Baye, John Bryant, Cohn Camerer, Russell Cooper, Vincent Crawford, John Haltiwanger, Pat Kehoe, Tom Saving, Steve Wiggins, Casper de Vries, a referee, and seminar participants at the NBER/FMME 1987 Summer Institute, the 1988 meetings of the Economic Science Association, the University of New Mexico, and Texas A&M University made constructive comments on earlier versions of this paper. Sophon Khanti-Akom, Kirsten Madsen, and Andreas Ortmann provided research assistance. The National Science Foundation, under grants no. SES8420240 and SES-8911032, the Texas A&M University Center for Mineral and Energy Research, and the Lynde and Harry Bradley Foundation provided financial support. A Texas A&M University TEES Fellowship has supported Battalio. 234
concepts is uncertain which equilibrium strategyother decision makerswill use and, when the equilibriaare not interchangeable, this uncertaintywill influence the rational decision-maker'sbehavior. Strategicuncertainty arises even in situationswhereobjectives, feasible strategies, institutions, and equilibriumconventionsarecompletelyspecified and are common knowledge. While multiple equilibriaare common in theoretical analysis, consideration of specific economies suggests that many equilibrium points are implausible and unlikely to be observedin actualeconomies. One response to multipleequilibriais to arguethat some Nash equilibriumpoints are not self-enforcingand, hence, are implausible, because they fail to satisfy one or more of the following refinements:eliminationof individuallyunreasonableactions,sequential rationality, and stability against perturbations of the game-see Elon Kohlburgand Jean-FrancoisMertens(1986) for examples and references.Equilibriumrefinementsdetermine when an outcome that is already expected would be implementedby rational decision makers. In general,many outcomeswill satisfythe conditionsof a givenequilibriumrefinement. The equilibriumselectionliteratureattempts to determine which, if any, self-enforcing equilibriumpoint will be expected.A satisfactory theory of interdependentdecisions must not only identifythe outcomesthat are self-enforcingwhen expectedbut also must identify the expectedoutcomes.Consequently, a theory of equilibriumselection would be a useful complement to the theory of equilibriumpoints. The experimental method provides a tractable and constructiveapproachto the equilibrium selection problem. This paper studies a class of tacit pure coordination games with multiple equilibria,which are
VOL. 80 NO.1
strictly Paretoranked,and it reportsexperiments that provide evidenceon how human subjectsmake decisionsunderconditionsof strategicuncertainty. Game I. A Pure Coordination
To focus the analysisconsiderthe following tacit coordination game, which is a strategic form representation of John Bryant's (1983) Keynesian coordination game. The baseline game is defined as follows: Let el,..., en denote the actions taken by n players.The period gameA is defined by the following payoff function and strategy space for each of n players: (1)
235
VAN HUYCK ETAL.: TACIT COORDINA TION GAMES
a(ei, e1) = a [min(ei, e1)] - bei,
a > b > O, where ei equalsmin(el,..., ei_1,ei+,,..., en). Actions are restrictedto the set of integers from 1 to J. The players have complete information about the payoff function and strategy space and know that the payoff function and strategy space are common knowledge.'
If the players could explicitlycoordinate their actions, the-real or imagined planner'sdecision problemwould be trivial. Given a - b greater than 0, each player
should choose the maximumfeasibleaction, e. Moreover, a negotiated "pregame" agree-
ment to choose e- would be self-enforcin*. Unlike games with incentiveproblems,here
'Apparently, this game is similar to Rousseau's "stag hunt" parable, which he used to motivate his analysis of the social contract, see Crawford (1989, p. 4). In the stag hunt game, each hunter in a group must allocate effort between hunting a stag with the group and hunting rabbits by himself. Let e, denote effort expended on the stag hunt. Since stag hunting in that era required the coordinated effort of all the hunters, the probability of successfully hunting a stag depends on the smallest e. The parameter a in equation (1) reflects the benefits of participating in the stag hunt: eating well should the hunt succeed. Hunting rabbits does not require coordination with the other hunters. The parameter b in equation (1) reflects the opportunity cost of effort allocated to the stag hunt that could have been allocated to rabbit hunting: a meal-however, meager.
the firstbest outcomeis an equilibriumpoint. However,when the playerscannotengagein "pregame"negotiationthey face a nontrivial coordinationproblem. Suppose that the players attempt to use the Nash equilibriumconceptto informtheir strategic behavior in the tacit coordination game A. A player'sbest responseto ei is to choose ei equal to ei. By symmetryit follows that any n-tuple (e,...,
e)
with e E {1,
2,..., e-} satisfies the mutual best response property of a Nash equilibriumpoint. All feasible actions are potentialNash equilibrium outcomes. The Nash concept neither prescribesnor predictsthe outcome of this tacit coordinationgame. (Standardequilibriumrefinementsdo not reducethe set of equilibria.For example,the equilibria are strict-each player has a unique best response-and, hence, trembling-handperfect.) II. Coordination ProblemsandEquilibrium
SelectionPrinciples
The analysis in Section I follows convention and abstractsfrom the equilibriumselection problem. However,a rationalplayer using deductive equilibriumconcepts confronts two nontrivialcoordinationproblems in period game A. First, playersmay fail to correctly forecast the minimum, ei, and, hence, regret their individualchoice, that is, ei = ei. This type of coordinationfailureresults in disequilibrium:outcomesthat do not satisfy the mutual best-responsepropertyof an equilibrium. An equilibriumselectionprincipleidentifies a subset of equilibriumpoints according to some distinctivecharacteristic.An interesting conjectureis that decisionmakersuse some selectionprincipleto identifya specific equilibrium point in situations involving multiple equilibria.This selection principle would solve the problemof coordinatingon a specificequilibriumpoint. Hence, the outcome will satisfy the mutual best-response propertyof an equilibrium. A second coordination problem arises when the equilibriacan be Pareto ranked. In such situations, all players may give a best response,but, nevertheless,implementa
236
THE AMERICAN ECONOMIC REVIEW
Pareto dominated equilibrium, that is, en) # J. Whilenot regrettingtheir individualchoice, they regretthe equilibrium implemented by these individual choices. Consequently,the outcomeresultsin coordination failure. What equilibriumselection principlescould a playeruse to resolvethese two coordinationproblems?2 Deductive selectionprinciplesselect equilibrium points based on the descriptionof the game. Deductiveselectionprinciplespreserve the equilibrium method's desirable propertyof independencefromhistoricalaccidents and from complicateddynamicprocesses. Inductive selection principles select equilibriumpoints based on the history of some pregame.3Hence, inductive selection principles are not independentof accident and process. When multiple equilibriumpoints can be Pareto ranked,it is possible to use concepts of efficiency to select a subset of selfenforcing equilibriumpoints: examples include R. Duncan Luce and HowardRaiffa's (1956, p. 106) concept of joint-admissibility, Tamer Basar and Geert Olsder's(1982, p. 72) concept of admissibility, and John Harsanyiand ReinhardSelten's(1988,p. 81) concept of payoff-dominance.An equilibif it riumpoint is said to be payoff-dominant is not strictlyParetodominatedby any other equilibriumpoint. When unique,considerations of efficiency may induce players to focus on and, hence, select the payoffdominant equilibrium point, see Thomas Schelling(1980, p. 291). In period game A, the equilibriumpoints are strictly Pareto ranked. Each player prefers a larger minimum.The only equilibrium point not Pareto dominated by any other equilibriumpoint is the n-tuple min(el,...,
(e,...,
e):
the payoff-dominant equilibrium
point. Consequently,payoff-dominanceselects the n-tuple(e,..., e-) in gameA. 2The "disequilibrium Keynesians" emphasizethefirst coordinationproblem. The "equilibriumKeynesians" emphasizethe secondcoordinationproblem,see Cooper and John (1988) for examplesand references. 3We use the term inductionin the logical, rather than mathematical,sense of reasoningfrom observed facts-history-to a conclusion.
MARCH 1990
Selecting the unique payoff-dominant equilibriumpoint not only allows playersto coordinateon an equilibriumpoint but also ensures that they will not coordinateon an inefficientone. Payoffdominancesolvesboth the individual and the collective coordination problemsof disequilibriumand coordination failure and, as Harsanyiand Selten suggest, should take precedenceover alternative selectionprinciples. The tacit coordinationgameA providesa severe test of payoffdominance,becausethe minimum rule exacerbatesthe influenceof uncertaintyabout the strategiesof the other n -1 players. Define the cumulativedistribution functionfor a player'sactionas F(e.). In the payoff-dominantequilibrium,F&e) equals 1 and F(e1) equals0 for ej less than e. A well-known theorem is that if el,..., en
are independent and identicallydistributed with common cumulativedistributionfunction F(ej), then the cumulativedistribution function for the minimum, F.,n(e), equals 1- [1 -F(e)]; see A. M. Mood, F. A. Graybill, and D. C. Boes (1974). In the payoff-dominantequilibrium,Fmin(e-) equals 1 and Fmin(e) equals 0 for e less than e-.But
suppose that a player is uncertainthat the n -1 playerswill select the payoff-dominant action, J. Specifically, let F(1) be small but
greater than 0, then as n goes to infinity Fmin(l) goes to 1. Consequently,when the
number of players is large it only takes a remote possibility that an individualplayer will not select the payoff-dominantaction eto motivate defection from the payoffdominantequilibrium. Several deductive selection principles based on the "riskiness"of an equilibrium point have been identifiedand formalized.A maximin action, which is an action (pure strategy)with the largestpayoffin the worst possible outcome, is secure, see John Von Neumann and Oskar Morgenstern(1944, 1972). Given existence, security selects the equilibrium point supported by player's maximin actions. Securitymay select very inefficientequilibriumpointsin nonzerosum games. In period game A, a player can ensurea payoff of a - b by choosing ei equal to 1,
which is the largestpayoffin the worstpossi-
VOL. 80 NO. I
VAN HUYCKETAL: TACIT COORDINATION GAMES
237
TABLE1-EXPERIMENTAL DESIGN
Experiment No.
Date
Size
1 2 3 4 5 6 7
June June June Sept Sept Sept Sept
16 16 14 15 16 16 14
A Payoff A Fullsize
B Payoff B Fullsize
1P,2,... 10 1P,2,..., 10P 1P,2,... 10P 1P,2P,..., 10P 1P,2P".. ,1OP lP,2P,..,10lP 1P,2P,..0. lop
-
11,...,15 11,...115
11P,"...15 11p...,15 11"..., 15 1ip,1... 15
A' Payoff A Fullsize
C Payoff A Size Twoa
-
-
16P,..., 20 16P, ... ,20 16,...,20 16,.,20 20 16, 16,...,22
21, ,27 21,.,27 21,.,25 23,...,25
P- Denotes a period in which subjects made predictions. - - In experiment 4 and 5 pairings were fixed, while in experiments 6 and 7 pairings were random.
Notice that the principlesof efficiencyand security can be defined independentlyof equilibrium.The experimentsin this paper, (1,...,1). Since payoff-dominance and security select differentequilibriumpoints in this which are designedto study the conflictbetacit coordinationgame (equilibriumpoints tween efficiency and security, are not dewith the highest and lowest payoffs,respec- signed to study how repeatedplay of period tively), an importantand tractableempirical game A influences the set of equilibrium questionis which, if any, deductiveselection points for A(T). principleorganizesthe experimentaldata. Having t periods of experiencein A(T) It is often possibleto applymorethanone provides a player with observed facts, in deductive selection principle to a game. addition to the descriptionof the game,that Hence, subjects may choose disequilibrium can be used to reasonabout the equilibrium selection problem in the continuationgame outcomes unless they behaveas if thereis a hierarchyof selection principles.When de- A(T - t). This experience may influence the ductive selectionprinciplesfail to coordinate outcome of the continuationgame A(T- t) beliefs and actions, inductiveselectionprin- by focusing expectationson a specificequiciples based on repeated interaction may librium point. For example, one adaptive hypothesis is that players will give a best allow players to learn to coordinate. response to the minimumobserved in the Consider a finitely repeatedgame A(T), which involves the n playersplayingperiod previous period. This adaptive behavior game A for T periods.The payoff-dominant would immediatelyconvergeto an equilibequilibrium of A(T) is just the repeated rium in A(T-1). The selected equilibrium implementation of the payoff-dominant involves all players choosing the period 1 minimum for the T- 1 periods of the conequilibriumof period game A, because the tinuationgame A(T -1). first-best outcome for period game A is
ble outcome. Consequently,in this tacit coordinationgame, securityselects the n-tuple
(e,..., e). Similarly, the secure equilibrium
of A(T) is just the repeatedimplementation of the secureequilibriumof periodgameA.4 4Crawford (1989)emphasizesthat the secureequilibrium is also the only equilibriumthat is evolutionary stable.In repeatedplay,playersusingadaptivebehavior may be led to implementthe secureequilibrium.Hence, while the experimentsreportedin this paper discriminate sharplybetween strategicstabilityand evolutionary stability,they do not discriminatebetweenlearning to use securityand certainkinds of adaptivebehavior.
III. Experimental Design
Table 1 outlines the design of the seven experimentsreportedin this paper. The instructionswere read aloud to ensurethat the descriptionof the game was commoninformation,if not, commonknowledge.'No pre5The originalworkingpaper,Van Huyck, Battalio, and Beil (1987), which includesthe actualinstructions,
MARCH 1990
THE AMERICAN ECONOMIC REVIEW
238
PAYOFFTABLE A
Smallest Value of X Chosen
Your Choice of X
7 6 5 4 3 2 1
7
6
5
4
3
2
1
1.30 -
1.10 1.20 -
0.90 1.00 1.10 -
0.70 0.80 0.90 1.00 -
0.50 0.60 0.70 0.80 0.90 -
0.30 0.40 0.50 0.60 0.70 0.80 -
0.10 0.20 0.30 0.40 0.50 0.60 0.70
-
-
PAYOFFTABLE B
Smallest Value of X Chosen Your Choice of X X
7 6 5 4 3 2 1
7
6
5
4
3
2
1
1.30 -
1.20 1.20 -
1.10 1.10 1.10 -
1.00 1.00 1.00 1.00 -
0.90 0.90 0.90 0.90 0.90 -
0.80 0.80 0.80 0.80 0.80 0.80 -
0.70 0.70 0.70 0.70 0.70 0.70 0.70
play negotiation was allowed. After each repetitionof the periodgame, the minimum action was publicly announcedand the subjects calculated their earningsfor that period. The only commonhistoricaldata available to the subjectswas the minimum. During the courseof an experimentsome design parameterswerealteredresultingin a sequence of treatmentslabeled A, B, A', and C. Instructionsfor continuationtreatments were given to the subjectsafterearlier treatmentshad been completed.The feasible actions in all treatmentsof all experiments were the integers 1 through 7: hence, e equaled7. In treatmentA and A', the followingvalues were assignedto the parametersin equation (1): parametera was set equal to $0.20, parameterb was set equal to $0.10, and a
payoff tables, questionnaire,extra instructions and recordsheet used in the experimentsand a moreextensive analysis of the experimentalresults, is available from the authorsupon request.
-
-
-
constant of $0.60 was added to ensurethat all payoffs were strictly positive. Consequently, the payoff-dominantequilibrium, (7, ... , 7), paid $1.30 while the secure equilibrium, (1,...,1), paid $0.70 per subject per
period.6Subjectsweregiventhis information in the form of a payoff table, see payoff Table A. In treatmentA, the periodgameA was repeatedten times.The numberof players, n, varied between 14 and 16 subjects. (In treatment C, the numberof players, n, was reduced to two.) TreatmentA' designates the resumptionof these conditionsafter treatmentB. In treatment B, parameterb in equation (1) was set equal to zero,see payoffTable B. This gives the subjectsa dominatingstrategy (play 7 regardlessof the minimum),which eliminates the coordination problem. The numberof players, n, remainedthe same as in treatmentA. 6For the remainderof this paper, an equilibrium denotes a mutual best-responseoutcome in the period game.
VOL. 80 NO. 1
VAN HUYCK ETAL: TACIT COORDINATION GAMES
239
Occasionally,subjectswere asked to pre- riod one was never greaterthan 4. Hence, dict the actions of all the subjects in the the largest payoff in period one was $1.00 treatment.7 For each prediction in the and some payoffs were $0.10. (The payoffSeptemberexperiments,a subjectwas paid dominant equilibriumwould have paid ev$0.70 less 0.02 times the sum of the absolute eryone $1.30). All of these outcomes are value of the differencebetween the actual inefficient.The subjectswere unable to use and predictedactions.(The rule used in the any deductive selectionprincipleto coordiJune experimentswas less sensitive to pre- nate on an equilibriumpoint. diction errors).At the end of the experiment, Only 10 percentof the subjectspredicted the subjectswere told the actualdistribution an equilibriumoutcome in period one. Inof actionsand werepaid. stead, most subjects(95 of 106) predicteda The subjectswere undergraduatestudents disequilibriumoutcome.8Moreover,the subattending Texas A&M Universityand were jects' predictionswere dispersed:one third recruited form sophomoreand junior ecoof the subjectspredictedat least one 1 and nomics courses.A total of 107 studentspar- one 7-a rangeof 6-and the averagerange ticipated in the seven experiments.After of the predictionswas 4.0. The subjects'disreading the instructions,but before the ex- persedpredictionssuggestthat they expected periment began, the students filled out a other subjectsto respondto the payoff table questionnaireto determinethat they under- differently than they did. These data are stood how to read the payofftable for treat- inconsistent with any theoryof equilibrium ment A, that is, map actions into money selection that assumesthat, becausea player payoffs. The instructionswould have been will derive his prior probabilitydistribution re-read if needed, but all 107 students re- over other players' pure strategies strictly from the parametersof the game,all players sponded correctly. will have the samepriorprobabilitydistribuIV. Experimental Results tion. Instead, some subjectsmade optimistic predictions and some subjects made pesTable 2 reports the experimentalresults simisticpredictions. for treatmentA. The data in periodone are While the subjectswere unable to coordiparticularlyinterestingbecause the subjects nate beliefs and actions,in almost all cases can only use deductiveselectionprinciplesto their individualpredictionsand actionswere inform their behavior. consistent. Of the 107 subjects,106 subjects In periodone, the payoff-dominant action, predicted that at least one other subject 7, was chosen by 31 percentof the subjects would choose an actionequalto or less than (33 of 107) and the secure action, 1, was their choice. (Only one subjectpredictedhe chosen by 2 percent of the subjects (2 of would determinethe minimum.)Most sub107). Neither deductive selection principle jects mapped predictionsinto actions in a succeeds in organizingmuch of the data, reasonable way. Those subjects who made althoughpayoff-dominanceis more success- pessimisticpredictionsabout what the other ful than security.The popularityof actions4 subjects would do chose small values for and 5-chosen by 18 and 34 subjects,re- their action and subjects who made optispectively-is consistentwith many subjects mistic predictionsaboutwhat the other subhaving nearly diffusepriorbeliefs about the jects would do chose large values for their outcome of the periodgame. action.9 The initial play of all seven experiments exhibitboth individualand collectivecoordination failure.The minimumaction for pe8One subjectis excludeddue to predictingonly 15 choices. 7In two earlier pilot experimentspredictionswere not made in any period. The substantiveresultswere the same as those reportedhere.
9Ananomalyis that,of the 95 subjectswhopredicted a minimumless than 7, 87 subjectschose an action greaterthan the minimumthey predicted.Van Huyck, Battalio, and Beil (1987) provide an expected value model to explainthis anomaly.
240
THE AMERICAN ECONOMIC REVIEW
MARCH 1990
TABLE2-EXPERIMENTAL RESULTSFOR TREATMENTA
Period 1
2
3
4
5
6
7
8
9
10
No. of 7's No. of 6's No. of 5's No.of4's No. of 3's No. of 2's No.of l's
8 3 2 1 1 1 0
1 2 3 6 2 2 0
1 1 2 5 5 2 0
0 0 1 4 5 4 2
0 0 0 1 4 8 3
0 0 0 1 1 7 7
0 0 1 1 1 8 5
0 0 0 0 1 6 9
0 0 0 0 0 4 12
1 0 0 0 1 1 13
Minimum Experiment 2 No. of 7's No. of 6's No. of 5's No. of 4's No. of 3's No. of 2's No. of l's
2
2
2
1
1
1
1
1
1
1
4 1 3 4 1 3 0
0 0 3 6 4 2 1
1 1 2 2 2 6 2
0 0 1 3 5 5 2
0 0 0 3 0 5 8
0 1 0 0 1 9 5
0 0 1 0 1 3 11
0 0 1 0 0 4 11
0 0 0 0 1 3 12
1 0 1 0 0 1 13
2
1
1
1
1
1
1
1
1
1
4 2 5 3 0 0 0
4 0 6 3 0 1 0
1 2 1 2 7 1 0
0 0 1 1 6 4 2
1 0 1 2 0 5 5
1 0 0 1 2 3 7
1 0 0 0 3 6 4
0 0 0 0 0 3 11
0 0 0 0 0 2 12
2 0 0 1 0 2 9
Minimum Experiment 4 No. of 7's No. of 6's No.of5's No. of 4's No. of 3's No. of 2's No.of l's
4
2
2
1
1
1
1
1
1
1
6 0 8 1 0 0 0
0 6 5 1 2 1 0
1 2 5 4 3 0 0
1 0 5 6 2 0 1
0 0 0 7 4 2 2
0 1 1 1 3 3 6
1 0 0 2 2 7 3
0 0 0 1 2 4 8
0 0 0 1 1 2 11
0 0 0 0 0 2 13
Minimum
4
2
3
1
1
1
1
1
1
1
Experiment1
Minimum Experiment 3 No. of 7's No. of 6's No.of 5's No. of 4's No.of 3's No. of 2's No. of l's
An interestingquestionis whetherthe subjects' predictions correspondto the actual distribution of actions more closely than predictions based on payoff-dominanceor security. Using the numberof actions correctly predictedas a statistic,the data reveal that 95 percentof the subjectspredictedthe actions of the other n -1 subjects more accurately than did payoff-dominance. This statistic is used to measureprediction accuracybecause the subjectspayoff were a linear transformationof the predictionaccuracy score. The differenceof the mean prediction accuracy score was always positive and in most cases significantlydifferentfrom
zero at the 1 percentlevel: a result that is robust to non-parametricstatistical procedures. Obviously,securitydoes even worse. Subjects predicted the observed heterogenous responseto the descriptionof the game and the resultingcoordinationfailurein period one. Repeated play of the period game allows the subjectsto use inductiveselectionprinciples or to learn to use deductiveselection principles. Hence, repeated play makes it more likely that subjects will be able to obtain mutualbest-responseoutcomesin the continuation game. Repeating the period game does cause actions to converge to a
VOL. 80 NO. I
VAN HUYCK ETAL: TACIT COORDINA TION GAMES
241
TABLE2-EXPERIMENTAL RESULTSFOR TREATMENT A, Continued
Period
Experiment 5 No. of 7's No. of 6's No.of5's No. of 4's No.of3's No. of 2's No. of l's Minimum Experiment 6 No. of 7's No. of 6's No. of 5's No. of 4's No.of3's No.of2's No. of l's Minimum Experiment 7 No. of 7's No. of 6's No. of 5's No. of 4's No.of3's No. of 2's No. of l's Minimum
1
2
3
4
5
6
7
8
9
10
2 1 9 3 1 0 0
2 3 3 4 2 2 0
3 1 0 6 2 2 2
1 0 4 2 4 3 2
1 0 1 1 6 4 3
1 0 0 2 0 6 7
1 0 2 0 0 5 8
0 0 0 2 0 2 12
0 0 0 1 0 5 10
0 0 0 1 1 3 11
3
2
1
1
1
1
1
1
1
1
5 2 5 2 1 0 1
3 0 1 3 5 2 2
1 0 0 4 4 4 3
1 0 0 0 2 5 8
1 1 0 0 2 3 9
1 0 1 0 2 3 9
2 0 0 0 1 6 7
2 0 0 0 0 4 10
2 0 0 0 2 5 7
3 0 0 0 0 5 8
1
1
1
1
1
1
1
1
1
1
4 1 2 4 1 1 1 1
3 0 3 0 3 3 2 1
1 0 0 1 2 2 8 1
1 0 0 2 1 2 8 1
1 0 0 1 1 4 7 1
1 0 0 0 0 4 9 1
1 0 0 0 0 4 9 1
1 0 0 0 0 4 9 1
1 0 0 0 0 5 8 1
1 0 0 0 0 3 10 1
stable outcome, see Table2. But ratherthan converging to the payoff-dominantequilibrium or to the initial outcome of the treatment, the most inefficientoutcome obtains in all seven experiments. The change in a subject'saction between period one and period two providesinsight into the subjects'dynamicbehavior.Of the eleven subjects who determinedthe minimum in period one the averagechange in action betweenperiodone and two was 0.73:t seven subjects increasedtheir action, three did not change, and one decreasedhis action. In every experimentsomeonewho had not determinedthe minimumin period one determines the minimum in period two. Moreover, in experimentsone throughfive the intersection of the set of subjectswho determinethe minimumin period one with the set of subjectswho determinethe minimum in period two is empty. Since a subject's payoff is increasing in ei when he
(she) uniquelydeterminesthe minimum,this adaptivebehaviorcan be rationalized. However, a subject'spayoff is decreasing in ei when he (she) played above the minimum and those subjectsthat played above the minimumreducedtheirchoice of action. The observed mean reductionis increasing in the differencebetween a subject'saction and the reported minimum,and the mean reductionis smallerthan this difference.The observed correlation between the current choice of these optimisticsubjects and the minimumreportedin period 1 suggeststhat behavior in the continuationgame A(9) is not independentof the historyleadingup to continuation game A(9). However,only 14 of 107 subjects give a best-responseto the period one minimumin periodtwo. Some subjectsplay below the minimumof the preceding period. This observed"overshooting" cannot be reconciledwith adaptive theories that predict the currentaction
242
THE AMERICAN ECONOMIC REVIEW
will be a convex combinationof last periods action and last periodsoutcome.Apparently, some subjects learn how "risky" it is to choose an action other than the secure action, 1, underthe minimumrule and learnto use security to inform their behaviorin the continuationgame. Although it failed to predict the initial outcome, security predicts the stable outcome of period game A. By period ten 72 percentof the subjects(77 out of 107) adopt their secure action, 1, and the minimumfor all seven experimentswas a 1. The observed coordinationfailureappearsto resultfrom a few subjects concludingit is too "risky"to choose an action other than the secure action and from most subjectsfocusingon the minimum reportedin earlierperiod games. The minimumrule interactingwith this dynamic behaviorcausesthis treatmentto converge to the most inefficientoutcome. In treatment B, parameterb of equation (1) was set equal to zero. Becausea player's action is no longer penalized, the payoffdominant action, 7, is a best responseto all feasible minimums.Action 7 is a dominating strategy. Hence, treatmentB tests equilibrium refinementsbasedon the eliminationof individually unreasonableactions. For example, a simple dominanceargumenteliminates all of the equilibriumpoints except one: (7,.. ., 7). Any strategic uncertainty would cause an individuallyrationalplayer to choose the payoff-dominantaction,7. Table 3 reports the experimentalresults for treatmentB and treatmentA'. In period eleven, the payoff-dominantaction, 7, was chosen by 84 percentof the subjects(76 of 91). However,the minimumin periodeleven was never more than 4 and in experiments four, five, six, and sevenit was a 1.10 Of course, a subjectthat adopts action 7 need not worry about what actions other
10At least one subject did not understand how the payoff table had changed. Subject 3 in experiment 5, who plays a 1 in every period of the B treatment, predicts that all 16 players will choose 1 but only he does so. When the actual distribution was revealed, subject 3 appeared genuinely amazed and confessed that he had not understood how the payoffs had changed.
MARCH 1990
subjects take and, apparently,most subjects did not. This propertyof dominatingstrategies resulted in the B treatmentexhibiting different dynamics than the A treatment. Like the A treatment, those players who determine the minimum increase their action, but, unlike the A treatment,those players who were above the minimumdo not decreasetheir action.11This dynamicbehavior converges to the efficientoutcome-the payoff-dominantequilibrium-in four of the six experiments.By periodfifteen,96 percent of the subjects chose the payoff-dominant action, 7. Even in the experimentsthat obtainedthe efficient outcome, the B treatmentwas not sufficientto induce the groupsto implement the payoff-dominantequilibriumin treatment A': parameter b equals $0.10 once again. Returningto the originalpayoff table in period sixteen, 25 percentof the subjects chose the payoff-dominantaction,7.12 However, 37 percent chose the secure action, 1. Period sixteen predictionswere peakedwith subjects choosing a 7 predictingmost subjects would choose 7 and subjectschoosinga 1 predictingmost subjectswould choose 1. This bi-modal distributionof actions and predictionssuggeststhat play priorto period sixteen influenced subjects'behavior.However, the subjectsexhibit a heterogenousresponse to this history. Security predicts the stable outcome of treatmentA'. In treatmentA', the minimum in all periods of all six experimentswas 1. By period twenty, 84 percentof the subjects chose the secure action, 1, and 94 percent chose an action less than or equal to 2. (Experimentstwo and four even satisfy the
"1The two exceptions were due to subject 3 in experiment five, see fn. 10, and subject 12 in experiment six. Subject 12 predicts that he will uniquely determine the minimum in period 11, verifies this in period 12 by choosing a 3, and then chooses a one for the remainder of the B treatment. Perhaps, subject 12 became vindictive. He had chosen a 7 in period 1. 12The large fluctuations in behavior resulting from changes in the parameter b -between treatment A and B and between treatment B and A'-suggest that subjects are influenced by the description of the game. In our view, these data are inconsistent with backwardlooking theories of adaptive behavior.
VOL. 80 NO. 1
TABLE 3-EXPERIMENTAL
Experiment 2 No.of 7's No. of 6's No. of 5's No.of 4's No. of 3's No. of 2's No.of l's Minimum Experiment 3 No. of 7's No. of 6's No. of 5's No. of 4's No. of 3's No. of 2's No. of l's Minimum Experiment 4 No. of 7's No. of 6's No.of5's No. of 4's No. of 3's No. of 2's No.of l's Minimum Experiment 5 No. of 7's No. of 6's No.of5's No. of 4's No. of 3's No. of 2's No.ofl's Minimum Experiment 6 No.of 7's No. of 6's No.of S's No. of 4's No. of 3's No. of 2's No. of l's
243
VAN HUYCK ETAL.: TACIT COORDINATION GAMES RESULTS FOR TREATMENT
Treatment B 14 13
B AND
TREATMENT
A'
Treatment A' 19 18
15
16
17
16 0 0 0 0 0 0
16 0 0 0 0 0 0
8 0 1 1 1 3 2
2 0 0 2 1 3 8
0 0 0 0 1 4 11
0 0 0 0 1 2 13
7*
7*
7*
1
1
1
1
13 0 0 0 1 0 0
12 1 1 0 0 0 0
13 1 0 0 0 0 0
14 0 0 0 0 0 0
6 1 0 1 0 2 4
2 0 2 0 0 4 6
2 0 1 0 0 2 9
1 0 0 0 0 3 10
1 0 0 1 0 0 12
4
3
5
6
7*
1
1
1
1
1
12 0 1 0 0 0 2
13 0 0 1 1 0 0
14 0 0 1 0 0 0
14 0 1 0 0 0 0
15 0 0 0 0 0 0
3 0 0 2 2 2 6
1 0 0 0 0 1 13
0 0 0 0 0 2 13
0 0 0 0 0 0 15
0 0 0 0 0 0 15
1
3
4
5
7*
1
1
1
1*
13 0 1 1 0 0 1
13 0 1 1 0 0 1
15 0 0 0 0 0 1
15 0 0 0 0 0 1
15 0 0 0 0 0 1
1 0 0 0 1 3 11
0 0 0 0 1 4 11
0 0 0 0 0 2 14
0 0 O 0 0 2 14
0 0 0 0 0 3 13
1
1
1
1
1
1
1
1
1
1
13 0 0 1 0 1 1
13 1 1 0 1 0 0
12 1 1 1 0 0 1
12 1 0 1 1 0 1
13 0 1 0 0 1 1
2 0 0 1 1 5 7
2 0 0 0 0 6 8
2 0 0 0 0 7 7
2 0 0 0 0 6 8
2 0 0 0 0 5 9
11
12
13 1 0 1 1 0 0
15 0 1 0 0 0 0
16 0 0 0 0 0 0
3
5
13 0 0 1 0 0 0
20 0 0 0 0 0 0 16 1*
1*
Minimum Experiment 7 No. of 7's No. of 6's No.of S's No.of 4's No.of 3's No. of 2's No.of l's
1
3
1
1
1
1
1
1
1
1
12 0 0 1 0 0 1
14 0 0 0 0 0 0
13 1 0 0 0 0 0
13 0 0 0 1 0 0
14 0 0 0 0 0 0
3 0 1 2 2 2 4
4 0 0 0 0 4 6
2 0 0 0 0 2 10
2 0 0 0 0 2 10
2 0 0 0 0 1 11
Minimum
1
7*
6
3
7*
1
1
1
1
1
*
Denotes a mutual best-response outcome.
244
THE AMERICAN ECONOMIC REVIEW
mutual best-responsepropertyof an equilibrium by period twenty.) Obtainingthe efficient outcome in treatmentB failed to reverse the observedcoordinationfailure.Like the A treatment, the most inefficientoutcome obtained. V. ExperimentalResultsfor TreatmentC: GroupSize Two
TreatmentC was addedto the September experiments to determineif subjects were influencedby groupsize whenchoosingtheir actions. In theory,any uncertaintyabout the actions of an individualplayerin the gameis exacerbated by the minimum rule as the number of players increases,see Section II. The C treatmentreducesgroup size to two. Table 4 reports the experimentalresults for the C treatmentof experimentsfour and five, which permanentlypairedsubjectswith an unknown partner.In period twenty-one, 42 percent of the subjectsplay theirpayoffdominant action, 7, and 74 percent of the subjectsincreasetheiraction.This resultoccurred even though the minimum for the precedingfive periodshad been a 1 and all 31 subjects had played either a 1 (28 subjects) or a 2 (3 subjects)in period twenty. Clearly,eitherthe subjectsthoughtthat their partner in treatment C would change his (her) action in response to reduced group size or the subjectsexpectedalternativedynamicsin repeatedplay. The subjectsin experimentsfour and five used an adaptive behaviorin the C treatment similarto the adaptivebehaviorexhibited in the A treatment.Subjectsthat played the minimumincreasedeffortby an average of +2.0 and subjectsthat played above the minimum rediced effort by an averageof - 1.9. However,unlikethe A treatment,there was no ".overshooting" to the secureaction, 1. Occasionally, both subjects simultaneously chose the payoff-dominantaction, 7V13 13Recall that subjects only observe the minimum and their own action. Hence, it is not possible to unilaterally "signal" a willingness to implement the payoff-dominant equilibrium, that is, subjects could not use Os-
MARCH 1990
This dynamic behaviorconvergedto the efficient outcome-the payoff-dominantequilibrium-in 12 of 14 trails.Hence,even with an extremely negative history payoff-dominance predicts the stable outcome of the tacit coordinationgamewith fixedpairs. Experiments six and seven randomly paired subjects with an unknownpartner.'4 Hence, experiments six and seven test whetherthe resultsobtainedin Experiments four and five were due to subjectsrepeating the period game with the same opponent.In the first period of these experiments,37 percent of the subjectschose the payoff-dominant action, 7, and 73 percentof the subjects increasedtheir choice of action,see Table 5. Moreover, the subjects' dynamic behavior was similarto that found in the fixedpair C treatment.While the resultsfor the random pair C treatment are influencedby group size, no stable outcomeobtains. The C treatmentconfirmsthat there are two consequencesof the minimumrule.First, groupsize interactingwith the minimumrule alters the subjects' initial choice of action. Second, group size interactingwith the minimum rule alters the convergenceof the subjects' dynamicbehaviorin disequilibrium.
borne's (1987) refinement of a "convincing deviation" to inform their behavior, see also van Damme (1987). 14Experiments by Cooper, DeJong, Forsythe, and Ross (1987) report that after eleven repetitions randomly paired groups of size two almost always obtain the payoff-dominant equilibrium. However, Cooper et al. also report coordination failure when subjects can choose from a strategy space that includes certain kinds of dominated cooperative strategies. Their game illustrates an interesting distinction between Luce and Raiffa's "solution in the strict sense," which depends on joint-admissibility and Harsanyi and Selten's solution, which depends on payoff-dominance. Because the first-best outcome requires using a strictly dominated strategy-as in the prisoner's dilemma game-and because joint-admissibility admits efficiency comparisons with disequilibrium outcomes, the Cooper et al. game with dominated cooperative strategies has no "solution in the strict sense" of Luce and Raiffa. Because the first-best outcome is an equilibrium in period game A, joint-admissibility-appropriately defined for n person games-and payoff-dominance select the same equilibrium point in period game A.
VOL. 80 NO. 1
VAN HUYCK ETAL.: TACIT COORDINATION GAMES
245
TABLE4-EXPERIMENTAL RESULTSFOR TREATMENTC: FIXED PAIRINGS
Period 21
22
23
24
25
26
7 7
7 7
7 7
7 7
7 7
7 7
7 7
Minimum Pair 2 Subject 2 Subject 15
7*
7*
7*
7*
7*
7*
7*
7 1
2 7
7 3
7 6
7 7
7 7
7 7
Minimum Pair 3 Subject 3 Subject 14
1
2
7
7
7
7
7
1 1
1 1
1 7
1 1
1 1
1 1
1 7
Minimum Pair 4 Subject 4 Subject 13
1*
1*
1
1*
1*
1*
1
1 7
7 2
7 5
7 7
7 7
7 7
7 7
Minimum Pair 5 Subject 5 Subject 12
1
2
5
7*
7*
7*
7*
1 1
7 4
4 7
7 7
7 7
7 7
7 7
Minimum Pair 6 Subject6 Subject 11 Minimum Pair 7 Subject 7 Subject 10
1
4
4
7*
7*
7*
7*
5 7 5
7 7 7*
7 7 7*
7 7 7*
7 7 7*
7 7 7*
7 7 7*
1 5
7 3
6 6
7 7
7 7
7 7
7 7
Minimum Pair 8 Subject 8 Subject 9
1
3
6*
7*
7*
7*
7*
7 3
6 5
6 7
7 7
7 7
7 7
7 7
3
5
6
7*
7*
7*
7*
7 2
7 3
4 6
5 6
6 7
6 7
7 7
Minimum Pair 2 Subject 3 Subject 14
2
3
4
5
6
6
7*
5 7
7 7
7 7
7 7
7 7
7 7
7 7
Minimum Pair 3 Subject 4 Subject 13
5
7*
7*
7*
7*
7*
7*
1 7
1 1
1 1
1 3
4 1
4 1
1 2
Minimum Pair 4 Subject 5 Subject 12 Minimum
1
1*
1*
1
1
1
1
5 7 5
7 7 7*
7 7 7*
7 7 7*
7 7 7*
7 7 7*
7 7 7*
Experiment 5 Pair 1 Subject 1 Subject l6
Minimum Experiment 6 Pair 1 Subject 2 Subject 15
27
246
THE AMERICAN ECONOMIC REVIEW TABLE4-FIXED
MARCH 1990
PAIRINGS,Continued
Period 21
22
23
24
25
26
4 4
5 5
7 7
7 7
7 7
7 7
7 7
4*
5*
7*
7*
7*
7*
7*
SubjectlO
5 5
7 7
7 7
7 7
7 7
7 7
7 7
Minimum
5*
7*
7*
7*
7*
7*
7*
Pair5 Subject6 Subject11 Minimum Pair6 Subject7
*
27
Denotes a mutualbest-responseoutcome.
TABLE 5-DISTRIBUTION OF ACTIONSFOR TREATMENTC: RANDOMPAIRINGS
Period 21
22
23
24
25
5 0 2 3 1 1 4
5 1 5 1 1 1 2
4 3 3 1 1 2 2
10 0 3 1 0 2 0
8 0 4 1 0 2 1
-
-
6 1 0 2 2 0 3
5 0 3 1 0 0 5
5 1 0 4 0 1 3
Experiment6 No. of 7's No. of 6's No. of 5's No. of 4's No. of 3's No. of 2's No. of l's Experiment 7 No. of 7's No. of 6's No. of 5's No. of 4's No. of 3's No. of 2's No. of l's
VI. Treatment A withMonitoring As a refereepoints out, a reasonableconjecture is that revealingthe distributionof actions each period-in additionto the minimum-might influencethe reporteddynamics. For example, subjects could signal a willingness to coordinate on the payoffdominant equilibriumand optimistic subjects might delay reducing their action if they knew the minimumwas determinedby just one subject.Two experiments,each using payoff Table A and 16 naive subjects, were conductedin whichthe entiredistribution of actionswas recordedon a blackboard
at the end of each periodand was left there for the entire experiment. The initial distributionof actions and the dynamicsof the two monitoringexperiments were similar to those reported above, see Table 6. If anything, the convergenceof actions to the secure action, 1, was more rapid under the monitoring treatment.In fact, a mutual best-responseoutcome was obtained in one experiment:without monitoring mutual best-responseoutcomes are not observedfor period game A until treatment A'. Apparently,monitoringhelps solve the individualcoordinationproblem-more subjects give a best-responsesooner-but
VOL. 80 NO. 1
247
VAN HUYCK ETAL.: TACIT COORDINA TION GAMES TABLE6-DISTRIBUTION OF ACTIONSFOR TREATMENT A WITH MONITORING
Period
Experiment 8 No.of7's No. of 6's No. of 5's No. of 4's No. of 3's No. of 2's No. of l's Minimum Experiment 9 No. of 7's No. of 6's No. of 5's No. of 4's No. of 3's No. of 2's No. of l's Minimum
1P
2P
3
4
5
6
7
8
9
lop
4 1 4 5 1 0 1
0 1 0 4 4 2 5
0 0 1 2 1 1 11
0 0 0 1 0 2 13
2 0 0 0 0 3 11
0 0 1 0 2 2 11
0 0 0 1 0 1 14
1 0 0 0 0 1 14
1 0 0 0 0 2 13
1 0 1 0 0 1 13
1
1
1
1
1
1
1
1
1
1
6 1 2 4 1 0 2
2 2 2 2 5 1 2
0 0 1 3 3 0 9
1 0 0 1 1 5 8
0 0 0 0 0 4 12
0 0 0 0 0 1 15
0 0 0 0 0 0 16
0 0 1 0 0 0 15
0 0 0 0 1 1 14
0 0 0 1 0 2 13
1
1
1
1
1
1
1
1
1
1*
P- Denotes a period in which subjects made predictions. * - Denotes a mutual best-response outcome.
not the collectivecoordinationproblem-the minimumwas a 1 in all ten periodsof both experiments. VII. ConcludingComments
These experimentsprovide an interesting example of coordinationfailure. The minimum was never above four in period one and all seven experimentsconvergedto a minimum of one within four periods. Since the payoff-dominantequilibriumwouldhave paid all subjects $19.50 in the A and A' treatments-excluding predictions-and the average earnings were only $8.80, the observed behavior cost the average subject $10.70 in lost earnings. This inefficientoutcomeis not due to conflictingobjectivesas in "prisoner'sdilemma" games or to asymmetricinformationas in "moralhazard"games.Rather,coordination failure results from strategic uncertainty: some subjectsconcludethat it is too "risky" to choose the payoff-dominantaction and most subjects focus on outcomesin earlier period games.The minimumrule interacting with this dynamicbehaviorcausesthe A and
A' treatmentsto convergeto the most inef-
ficientoutcome. Deductive methodsimply that all feasible actions are consistentwith some equilibrium point in this experimental coordination game. However, the experimentalresults suggest that the first-bestoutcome,which is the payoff-dominantequilibrium,is an extremely unlikely outcome either initially or in repeatedplay. Instead,the resultssuggest that the initial outcomewill not be an equilibriumpoint and only the secure-but very inefficient-equilibrium describes behavior that actual subjectsare likely to coordinate on in repeatedplay of periodgame A when the numberof playersis not small. REFERENCES Basar, Tamer and Olsder, Geert J., Dynamic Noncooperative Game Theory, New York:
AcademicPress,1982. Bryant,John, "A Simple Rational Expectations Keynes-Type Model," Quarterly Journal of Economics, August 1983, 98, no. 3, 525-28.
248
THE AMERICAN ECONOMIC REVIEW
MARCH 1990
Cooper, Russell, DeJong, Douglas V., Forsythe, Robertand Ross, ThomasW., "Selection Cn-
Luce, R. DuncanandRaiffa,Howard,Games and
teria in CoordinationGames: Some Experimental Results," unpublishedmanuscript, October1987.
Mood, A. M., Graybill,F. A. and Boes, D. C., Introduction to the Theoryof Statistics, 3rd
and John, Andrew,"Coordinating Co-
ordinationFailuresin KeynesianModels," Quarterly Journal of Economics, August
1988, 103, no. 3, 441-64. Crawford,Vincent P., "An 'Evolutionary' In-
terpretationof Van Huyck, Battalio,-and Beil's ExperimentalResults on Coordination," unpublished manuscript,January 1989. Harsanyi,John C., andSelten,Reinhard,A General Theory of Equilibrium Selection in
Games,Cambridge:MIT Press,1988. " On Kohlberg,Elon and Mertens,Jean-Francois,
the Strategic Stability of Equilibria,"
Decisions,New York:Wiley& Sons, 1957. ed., New York: McGraw-Hill,1974. Osbome, Martin J., "Signaling, Forward In-
duction, and Stability in Finitely Repeated Games," unpublishedmanuscript, November1987. Schelling,ThomasC., The Strategy of Conflict,
Cambridge: Harvard University Press, 1980. vanDamme,Eric,"StableEquilibriaand Forward Induction," UniversitiitBonn Discussion PaperNo. 128, August1987. Van Huyck, John B., Battalio,RaymondC. and Beil, RichardO., "Keynesian Coordination
Games, StrategicUncertainty,and Coordination Failure," unpublished manuscript, October1987.
Econometrica, 54, no. 5, September 1986. Lucas, RobertE., Jr., "AdaptiveBehaviorand Economic Theory," in Rational Choice, R.
Von Neumann, John and Morgenstern,Oskar, Theory of Games and Economic Behavior,
Hogarth and M. Reder, eds. Chicago: Universityof ChicagoPress,1987.
Princeton: Princeton University Press, 1972.
