strategic uncertainty, equilibrium selection,consequently, a theory of equilibrium selection would...

STRATEGIC UNCERTAINTY, EQUILIBRIUM SELECTION,AND COORDINATION FAILURE IN

AVERAGE OPINION GAMES*

JOHN B. VAN HUYCKRAYMOND C. BATTALIO

RICHARD O. BEIL

Deductive equilibrium analysis often fails to provide a unique equilibriumsolution in many situations of strategic interdependence. Consequently, a theory ofequilibrium selection would be a useful complement to the theory of equilibriumpoints. A salient equilibrium selection principle would allow decision makers toimplement a mutual best response outcome. This paper uses the experimentalmethod to examine the salience of payoff-dominance, security, and historicalprecedents in related average opinion games. The systematic and, hence, predictablebehavior observed in the experiments suggests that it should be possible toconstruct an accurate theory of equilibrium selection.

Deductive equilibrium methods are powerful tools for analyz-ing economies that exhibit strategic interdependence. Typically,equilibrium analysis does not explain the process by which decisionmakers acquire equilibrium beliefs. The presumption is that actualeconomies have achieved a steady state. In economies with stableand unique equilibrium points, the influence of inconsistent beliefswould disappear over time, see Lucas [1987]. The power of theequilibrium method derives from its ability to abstract from thecomplicated dynamic process that induces equilibrium and toabstract from the historical accident that initiated the process.Unfortunately, deductive equilibrium analysis often fails to deter-mine a unique equilibrium solution in many economies and, hence,often fails to prescribe or predict rational behavior.

In economies with multiple equilibria, the rational decisionmaker formulating beliefs using deductive equilibrium concepts isuncertain which equilibrium strategy other decision makers willuse, and in general, this uncertainty will influence the rationaldecision maker's behavior. Strategic uncertainty arises even insituations where objectives, feasible strategies, and institutions are

*Ann Gillette, Sophon Khanti-Akom, and Kirsten Madsen provided researchassistance. The National Science Foundation (SES-8420240; SES-8911032), theTexas Advanced Technology Program, the Texas Engineering Extension Service,and the Texas A&M University Center for Mineral and Energy Research providedfinancial support,

© 1991 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnology,The Quarterly Journal of Economics, August 1991

886 QUARTERLY JOURNAL OF ECONOMICS

completely specified and are common knowledge.' The deductiveequilibrium method is incomplete. A satisfactory theory of interde-pendent decisions not only must identify the outcomes that areequilibria when expected, but also must explain the process bywhich decision makers acquire equilibrium beliefs. Consequently,a tbeory of equilibrium selection would be a useful complement tothe theory of equilibrium points.

An interesting conjecture is that decision makers may focus onsome selection principle to identify a specific equilibrium point insituations involving multiple equilibria; see Schelling [1980]. Thissalient principle would allow the decision makers to implement anequilibrium. The salience of an equilibrium selection principle isessentially an empirical question.

This paper uses the experimental method to examine thesedience of payoff-dominance, security, and historical precedents inrelated average opinion geimes. The average opinion games studiedexhibit multiple equilibria, which in the baseline experiments werePareto ranked. Hence, deductive equilibrium analysis of theseaverage opinion games is indeterminate. Yet the observed behaviorin the experiments was systematic. The distribution of initialactions in a treatment varied systematically witb considerations ofpayoff-dominance and security. Given an initial median, the me-dian in the remaining periods of a treatment was perfectly predict-able. The systematic and, hence, predictable behavior observed inthe experiments illustrates the importance of equilibrium selectiontheory.

I. AVERAGE OPINION GAMES

Cooper and John [1988] demonstrate that a large number ofsuperficially dissimilar market and nonmarket models with strate-gic complementarities and demand spillovers have a similar strate-gic form representation.^ Rather than developing an extensive formmarket game and then converting it into the strategic form for our

1. Sugden [1989, p. 88] provides a lucid critique of the view that a rationaldecision maker can deduce a unique "rational" strategy from the informationcontained in a complete information description of a game. Strategic uncertjiintyshould not be confused with uncertainty arising from incomplete information aboutother aspects of a decision maker's environment. Kejfnes's [1936, p. 156] discussionof the aversige opinion problem in newspaper beauty contests and in stock marketsis a venerable example of strategic uncertainty.

2, Milgrom and Roberts [1990] demonstrate that extant models of macroeco-nomic coordination failure, bank runs, technolo^ adoption and diffusion, R&Dcompetition, and manufacturing with nonconvexities exhibit strategic complemen-tarities.

AVERAGE OPINION GAMES 887

analysis, we work directly with the strategic form. To focus theanalysis, consider the following average opinion game.

LetXi, . . . ,x^ denote the actions taken by n decision makers,where n is odd, and let M be the median of these actions. The periodgame T is defined by the following payoff function and action spacefor each of ra decision makers indexed by i:

(1) -nix.M) = aM- b[M - x,f + c, a > 0, 6 > 0,

where x^ G [l,2, . . . ,X].A decision maker's payoff is decreasing inthe distance between the decision maker's choice x^, and themedian M, and is incresising in the median M. Assume that thedecision makers have complete information about the payofffunctions and feasible actions, Eind know tbat the payoff functionand feasible actions are common knowledge.

If the decision makers could explicitly coordinate their actions,the—real or imagined—planner's decision problem would be triv-ial. Efficiency requires that each decision maker choose the largestfeasible actionX, that is, tbe unique efficient outcome is(X, . . ., X).Moreover, a negotiated "pregame" aigreement to choose {X, . . . ,X)would be self-enforcing.

However, when the decision makers cannot engage in pregamenegotiation, they face a nontrivial coordination problem: an aver-age opinion problem. In game F, decision meiker i's best response isto set Xi equal to i's forecast of tbe median action. Tbe principle ofmutually expected rationality implies that, when forecasting themedian, decision maker i expects decision maker j to set Xj equal toj's forecast of the median action. Hence, decision maker i's bestresponse becomes set JC; equal to i's forecast of the median of theforecasts of the median. Again, the principle of mutually expectedrationEility applies, and decision maker i confronts an infiniteregress of forecasts of the median of the forecasts of the median ofthe forecasts of t h e . . . .̂

Suppose that the decision makers attempt to use the Nashequilibrium concept to inform their strategic behavior in the tacitaverage opinion game F. Formally, an ra-tuple of feasible actions{x'l X*) constitutes a Nasb equilibrium point, if

(2) TTix^M*) < T:{X;M*)

for all Xi £ (1,2, . . . ,X} and for all i. Since a decision maker's

3, Frydman and Phelps [1983, introduction] provide a related discussion of theaverage opinion problem in Rational Expectations Equilibria,


unique best response to M is to choose x, equal to M, by symmetry itfollows that any n-tuple {x, . . . ,x) with x E. {1,2, . . . , X) is a Nashequilibrium point.

All feasible actions can be rationalized as part of some Nashequilibrium. The Nash equilibrium concept neitber prescribes norpredicts the outcome of this tacit coordination game. The deductiveequilibrium analysis is indeterminate. A conventional response tothis indeterminacy is to argue that some Nash equilibria are notself-enforcing. However, all of tbe pure strategy equilibria arestrict, that is, each decision maker has a unique best response to M.Hence, the usual refinements do not reduce the set of equilibriumpoints; see Van Damme [1987, p. 20]. Moreover, the indeterminacyof the equilibrium analysis and the resulting strategic uncertaintyundermines the presumption that the outcome of F will satisfy themutual best response condition (2).

II. EQUILIBRIUM SELECTION PRINCIPLES

An equilibrium selection principle identifies a subset of equilib-rium points according to some distinctive characteristic of thegame's description or of the decision makers' experiences. Deduc-tive selection principles select equilibrium points based on thegame's description. Inductive selection principles select equilib-rium points based on the decision makers' experiences.

An interesting conjecture is that decision makers may focus onsome selection principle to identify a specific equilibrium point insituations involving multiple equilibria. Hence, the outcome ofsituations involving strategic uncertainty may, nevertheless, sat-isfy the mutual best response condition (2). A salient principleselects an equilibrium point based on its conspicuous uniqueness insome respect. The salience of an equilibrium selection principle isessentially an empirical question.

A. Deductive Selection Principles

When multiple equilibrium points can be Pareto ranked, it ispossible to use the concepts of efficiency to select a subset ofequilibrium points: examples include Luce and Raiffa's [1957, p.106] concept of joint-admissibility, Basar and Olsder's [1982, p. 72]concept of admissibility, and Harsansn and Selten's [1988, p. 81]concept of payoff-dominance. An equilibrium point is said to bepayoff-dominant if it is not strictly Pareto dominated by any other


equilibrium point.'' Considerations of efficiency may induce deci-sion mEikers to focus on and, hence, select a payoff-dominantequilibrium point if it is unique; see Schelling [1980, p. 291].

When equilibria can be Pareto ranked, it is important todistinguish between disequilibrium outcomes—outcomes tbat donot satisfy the mutual best response condition—and coordinationfailure—an inefficient equilibrium outcome.^ Wben a unique payoff-dominant equilibrium point is salient, tbis not only allows decisionmakers to coordinate on an equilibrium point, but also insures tbattbey will not coordinate on an inefficient one. Hence, payoff-dominance solves both the disequilibrium and coordination failureproblems. For example, in game F, payoff-dominance selects {X,.. .,X), wbicb implies not only tbat X; = M for all i, but also tbat M = X.

However, payoff-dominance may not be salient in many strate-gic situations because it does not take account of out-of-equilibrium payoffs. Several selection principles based on the"riskiness" of an equilibrium point have been identified andformalized: examples include Von Neumann and Morgenstern's[1972] concept of maximin and Harsanjd and Selten's [1988]concept of risk-dominance. A secure action is an action wbosesmallest payoff is at least as large as the smallest payoff to anyother feasible action. Security selects equilibrium points imple-mented by secure actions. Security, in contrast to payoff-dominance, may select very inefficient equilibrium points in non-zero-sum games.*

Consider a specific representation of game F, wbich illustratessecurity and is used in the experiments reported below. PayoffTable F sets parameter X equal to 7, parameter a equal to 0.1,parameter b equal to 0.05, and parameter c equal to 0.6 in equation(1). Tbe cells along tbe diagonal give tbe payoffs corresponding to

4, Luce and Raiffa's [1957] concept makes comparisons with feasible butdisequilibrium outcomes, Basar and Olsder's [1982] concept does not require thatall decision makers be strictly better off. In our experiments the payoff-dominantequilibrium is also tbe best feasible outcome.

5, Tbe recent literature on macroeconomic coordination games (see Bryant[1983] and Cooper and John [1988] for examples and references) emphasizes thepossibility of coordination failure; while the older literature emphasizes disequilib-rium.

6, Security, like payoff-dominance, has the practical advantEige that it onlyrequires ordinal preferences that are increasing in the score of the observable game.Hence, it is consistent with the hypothesis that, all else equal, people preferoutcomes with higher money payoffs. The experimentalist confronts a moreambitious task when attempting to induce Von Neumann and Morgensternpreferences over the score of" em observable game, which is required for mixedstrategy equilibria and risk-dominance.


PAYOFF TABLE r

YourchoiceofX

7654321

7

1,301,251.100.850.500.05

-0.50

6

1,151.201.151.000,750.40

-0.05

MediEin value of X chosen

5

0.901,051,101,050.900.650.30

4

0,550,800.951.000.950.800.55

3

0.100.450.700,850,900,850.70

2

-0,450.000.350,600,750,800,75

1

-1.10-0,55-0,10

0.250.500,650,70

the seven strict equilibrium points. Hence, payoffs range from 1.30in the payoff-dominant equilibrium (7, . . . , 7), to 0.70 in the leastefficient equilibrium (1, . . . , 1). The secure equilibrium is(3, . . . , 3), which pays 0.90 in equilibrium and insures a payoff ofat least 0.50. In game F both payoff-dominance and security select aunique equilibrium point, and hence, both are potentially salient.

In game F there is a tension between efficiency and security.This tension may undermine the salience of both selection princi-ples unless it is common knowledge which selection principle takespriority in average opinion games, which seems unlikely. A plausi-ble conjecture is that payoff-dominance is more likely to be salientif it does not conflict with security and, conversely, that security ismore likely to be salient if it does not conflict with payoff-dominance.

Consider a game that has the same equilibrium actions andpayoffs as F, but that has a zero payoff to all disequilibriumoutcomes. The period game fl is defined by Payoff Table O. In gameil, unlike game F, all actions are equally secure because they allinsure a payoff of zero.' Hence, security cannot be a salientequilibrium selection principle for game H, but payoff-dominanceuniquely selects (7, . . . , 7) and, hence, is potentially salient.

Alternatively, consider the game obtained by setting parame-ter a equal to zero and parameter c equal to 0.7 in equation (1).Specifically, the period game <& is defined by the Payoff Table O.Game $ has the same set of equilibrium points as game F, but the

7, There exists a mixed strategy that insures an expected payoff greater thanzero.


YourchoiceofX

7654321

7

1,30000000

6

0

PAYOFF TABLE ft

Median value ofX chosen

5

01,20 0

00000

1,100000

4

000

1.00000

3

0000

0,9000

2

00000

0,800

1

000000

0.70

payoffs associated with the equilibrium points are no longerincreasing in the meditin. In game $, unlike game F, all strictequilibria are included in the set of payoff-dominant equilibria.Hence, payoff-dominance cannot be a salient equilibrium selectionprinciple. Security selects (4, . . . , 4), which insures a payoff of0.25. Hence, security is a potentially salient equilibrium selectionprinciple for game <&.

Our discussion of deductive selection principles has focused onthe simple principles of efficiency and security that are directlyapplicable to the average opinion games F, fl, and $. Our experimen-tal research attempts to determine how people actually use thestrategic details of their environment to solve equilibrium selectionproblems.

B. Inductive Selection Principles

If decision makers fail to coordinate on an equilibrium,repeated interaction may allow decision makers to learn to coordi-

YourchoiceofX

7654321

7

0.700.650,500,25

-0.10-0.55-1.10

6

0.650.700.650.500,25

-0.10-0.55

PAYOFF TABLE <D

Median value of X chosen

5

0,500,650.700.650,500,25

-0.10

4

0.250,500,650,700.650.500.25

3

-0,100.250,500,650.700.650,50

2

-0,55-0,10

0,250,500.650.700,65

1

-1.10-0.55-0.10

0.250.500,650,70


nate. Consider a finitely repeated game G{T), which involves the ndecision makers plasang one of the average opinion games, either F,n, or 4>, for T periods. Having t periods of experience in G{T)provides a decision maker with observed facts, in addition to thedescription of the game, that can be used to reason about theequilibrium selection problem in the continuation game G{T - t).This experience may influence the outcome of the continuationgame GiT — t) by focusing expectations on a specific equilibriumpoint.

In the continuation game G(T — t) decision makers can useprecedent to solve their coordination problem. Selecting an equilib-rium based on precedent requires decision makers to focus on somesalient analogy to a past instance of the present equilibriumselection problem and to expect others to focus on the sameanalogy. Hence, precedent requires decision makers to have someshared experience.

There are too many ways to use precedent to enumerate themall. However, a plausible conjecture in average opinion games isthat the historical median provides a salient precedent for thepresent equilibrium selection problem. However, the plausibility ofthe conjecture depends on how similar the past and presentequilibrium selection problems are. It is useful to distinguishbetween two forms of precedent, which we denote as strongprecedent and weak precedent for clarity.*

A historical outcome of the period game G may provide astrong precedent for G{T — t) when it is observed by all thedecision makers. Suppose that decision makers observe {M,3̂ 2> • • • .M,], then decision makers can use the strong precedent establishedby a historical median or some statistic of historical medians toinform their strategic behavior in the continuation game G{T - t).

A historical outcome of a related average opinion game G'^(T),some pregame, may provide a weak precedent for GiT) when it isobserved by all the decision makers. (For example, let G'̂ (T) equalIKT), and let GiT) equal F(T).) Suppose that decision makersobserve [M\M2 M'',], then decision makers can use the weakprecedent established by a historical median or some statistic of

8. Lewis [1969] contrasts precedent, which is hased on shared experience, andconvention, which is based on common knowledge of how memhers of a populationsolve the present equilibrium selection problem and common knowledge that allrelevant decision makers belong to that population. We assume that the equilibriumselection problem in an abstract game is sufficiently novel that subjects cannot useconventions in the experiments reported below.


historical medians in a related average opinion game to informtheir strategic behavior in the present game GiT). This paperreports evidence on the salience of strong and weak precedent inrepeated average opinion games.

It is possible, even likely, that decision makers will focus ondifferent inductive or deductive selection principles. Hence, thedecision makers must form complex beliefs about the averageopinion of the salience of alternative selection principles. If attempt-ing to reason about the equilibrium selection problem is toocomplex, some decision makers may adopt simple trial and errorlearning rules. Many learning rules coevolve to an equilibriumpoint in average opinion geunes; see Crawford [1991] for an"evolutionary" interpretation of our results.

III. EXPERIMENTAL DESIGN

A laboratory environment capturing the essential aspects ofthe equilibrium selection problem in a many-person decentralizedeconomy must include two features. First, the environment mustnot assume away the problem by allowing an tirbiter—or any otherindividual—to make common knowledge pregame assignments.*Second, the environment must allow repeated interaction amongthe decision makers so that they have a chance to learn tocoordinate. The repeated average opinion games described aboveare well suited for an experimental study of equilibrium selection.

The gEimes were described to the subjects using payoff tables F,n, and $ discussed above, where the payoffs denote dollars.Treatment Gamma denotes repeated play of F, Treatment Omegadenotes repeated play of H, and Treatment Phi denotes repeatedplay of $. All initial treatments last ten periods. Nine subjectsparticipated in each of the twelve experiments reported in the text.

In treatment Gammadm subjects participated simultaneouslyin two games: one set n equal to nine, and the other set n equal totwenty-seven. See Appendix A for a discussion of the dual marketdesign and experimental results.

The instructions were read aloud to insure that the descriptionof the game was common information. Appendix C contains theinstructions used in Treatment Gamma. No preplay negotiation

9. See Van Huyck, Gillette, and Battalio [forthcoming] for experiments on theability of an arbiter to select equilibrium points and for references to the "cheaptalk" literature.


TABLE IEXPERIMENTAL DESIGN MATRIX

iljXp

Nm.

123456789

101112

eriment

Date

3/1/883/1/883/1/883/3/88°3/3/88°3/3/88°3/31/883/31/883/31/883/30/883/30/884/5/88

111111

GammaTable T

*,2*,2,. . . ,*,2,. . . ,*,2,. . . ,*,2,. . . ,*,2,. . . ,

———

—

—

101010101010

Treatment

OmegaTable n

11*,. .11*,. .11*,. .11*,. .11*,. .11*,. .1*,2,.1*,2, .1*,2,.

. ,18

. ,18

.,18

. ,15

. ,15

. ,15.,10.,10.,10

PhiTable *

—

—

1*,2,. . . , 101*,2,. . . , 101*,2 10

_

Table r

19*,2019*,2019*,20

11*, . . . , 1511*,. . . , 1511*,. . . , 1511*, . . . , 1 511*, . . . , 1511* 15

Notes. Nm. = number, 'denotes a period in which subjects made predictions. " = dual market treatment.

was allowed. After each repetition of the period game, the medianaction was publicly announced, and the subjects calculated theirearnings for that period. The only common historical data availableto the subjects were the reported medians.

After ten periods of a treatment, the subjects were switchedinto a continuation treatment. Instructions for continuation treat-ments were given to the subjects after earlier treatments had beencompleted. Table I outlines the design of the twelve experimentsreported in this paper.

At the beginning of each treatment, the subjects predicted thedistribution of actions in that period.'" For each prediction asubject was paid $0.50 less $0.02 times the sum of the absolutevalue of the difference between the actual and predicted actions. Atthe end of the experiment, the subjects were told the actualdistribution of actions and were paid for their predictions.

The subjects were sophomore and junior economics studentsattending Texas A&M University. A total of 108 students partici-pated in the twelve experiments. After reading the instructions,but before the experiment began, the students filled out a question-naire to determine that they understood how to read the payoff

10. In two earlier pilot experiments predictions were not made in any period.The substantive results were the same as those reported bere.


TABLE IIDISTRIBUTION OF CHOICES IN PERIOD 1

Action

7654321

Total

Gamma

Nm.

5388300

27

(Pr.)

(18)(11)(30)(30)(11)

(0)(0)

(100)

Gammadm

Nm.

317

11600

27

(Pr.)

(11)(4)

(26)(41)(18)

(0)(0)

(100)

Treatment

Combined(baseline)

Nm.

84

1519800

54

(Pr.)

(15)(7)

(28)(35)(15)

(0)(0)

(100)

Omega

Nm.

14193000

27

(Pr.)

(52)(4)

(33)(11)

(0)(0)(0)

(100)

Nm

239

11200

27

Phi

. (Pr.)

(7.5)(11)(33)(41)(7.5)

(0)(0)

(100)

Notes. Nm. = number of subjects. Pr. = percent of subjects.

tahle for the treatment—that is, how to map actions into moneypayoffs—and how to calculate the median of nine numhers. All ofthe suhjects read the payoff tahle correctly. In a few experiments asuhject failed to calculate the median correctly. On those occasionsthe instructions were read again.

The experiments take less than two hours to conduct. Conse-quently, the suhjects could earn significantly more than theminimum wage. For example, in experiments 1-3 if all suhjectschoose 7 in each period, each suhject would earn $27.

rv. EXPERIMENTAL RESULTS: THE SALIENCE OFPAYOFF-DOMINANCE AND SECURITY

This section reports the period 1 results for the twelveexperiments. The data in period 1 are particularly interestinghecause the suhjects can only use deductive selection principles toinform their hehavior. Hence, the period 1 data provide a direct testof the salience of payoflF-dominance and security. Recall that in

11. The text groups the choice data from treatments Gamma and Gammadm.Nonparametric tests failed to reject the hypothesis that the sample distribution ofchoices was drawn from the same population distribution. The chi-square statisticwas 1.7 with a probability value of 0.19.

12. Since the secure mixed strategy assigns a probability of 48 percent toactions 1,2, Eind 3, the fact that these actions are never observed is inconsistent withthe hypothesis that subjects were using the secure mixed strategy.


period game F payoff-dominance selects (7,. . . , 7), and securityselects (3, . . . , 3); in period game fl payofF-dominance selects(7, . . . , 7), and security does not apply; and in period game $payoff-dominance does not apply and security selects (4, . . . , 4).

Neither payoff-dominance nor security is salient in periodgame T. In the basehne treatments—Gamma and Gammadm,action 7 was chosen hy only 15 percent (8 out of 54) of the suhjectsand action 3 was also chosen hy 15 percent of the suhjects; seeTahle II." However, suhjects' behavior was not diffuse. Ratherthan play either the payoff-dominant or the secure action, 70percent (38 out of 54) of the suhjects chose an action hetween 7 and3. In the six haseline experiments, the median action was 4 in threeexperiments and 5 in three experiments.

Payoff-dominance predicts the modal response to period gamen. In Treatment Omega, action 7 was chosen hy 52 percent (14 outof 27) of the suhjects; see Tahle II. Actions 4, 5, and 6 were chosenhy the remaining suhjects.'^ In the three Omega treatments themedian action was 7 in two experiments and 5 in one experiment.Using nonparametric procedures, the difference hetween the distri-bution of actions in Treatment Omega and the distribution ofactions in the haseline treatment is statistically significant at the 1percent level. This contrast hetween period games F and Cl isconsistent with the conjecture that eliminating considerations ofsecurity in F increases the salience of payoff-dominance.

Security predicts the modal response to period game <P. Thesecure action 4 was chosen hy 41 percent (11 out of 27) of thesubjects. Hence, there is some support for the proposition thateliminating considerations of payoff-dominance in F increases thesalience of security.

However, seven times as many suhjects played ahove thesecure action 4 as played helow it. This is curious given that, unlikeF and H, Payoff Table 4> has the same values in the upper leftcorner of the tahle as in the lower right corner. Hence, suhjectsmust he responding to description-specific details of the payofftahle. (We placed the payoff-dominant equilihrium in the upper leftcorner of F and fi for this reason; that is, we helieved that we werehiasingthe experiments in favor of payoff-dominance.) The medianaction was a 4 in one experiment and a 5 in two experimentsheginning with Treatment Phi.

Suhjects' predictions of individual hehavior were dispersed.Ninty-one percent (49 out of 54) of the suhjects predicted aheterogeneous response to the Payoff Tahle F. Similar results


obtain for treatments Omega and Phi. Only 18 percent (5 out of 27)of the subjects predicted that everyone would chose the payoff-dominant action, 7, in response to Payoff Table il. Only 4 percent(1 out of 27) of the subjects predicted that everyone would chosethe secure action 4 in response to Payoff Table O. Hence, 85percent (46 out of 54) of the subjects predicted a heterogeneousresponse to payoff tables ft and <&.

The subjects' dispersed predictions suggest that they expectsome of the other subjects to respond to the payoff table differentlythan they did. The data are inconsistent with any theory ofequilibrium selection that assumes that, because a decision makerwill derive his prior probability distribution over other decisionmakers' pure strategies strictly from the parameters of the game,all decision makers will have the seime prior probability distribu-tion. Instead, most subjects predict that some participants will bemore optimistic or pessimistic than themselves.

The subjects' predictions and actions are consistent." Thosewho made optimistic predictions chose large values, and those whomade pessimistic predictions chose small values. The inefficientdisequilibrium outcomes observed in period 1 appears to resultfrom the heterogeneous response of subjects to the description ofthe game and from the subjects' expectation of a heterogeneousresponse to the description of the game.

V. EXPERIMENTAL RESULTS: THE SALIENCE OF STRONGPRECEDENTS

Repeated interaction may allow subjects to learn to coordinate.At the end of each period, the median was reported to the subjects.Hence, the experimentsd design allowed subjects to use theirshared experience in past period games, the historical median, toinform their behavior in the current period game, which providesevidence on the salience of strong precedents.

The strong precedent of the initial median is salient in theseaverage opinion games. Table III reports the median for periods 1through 10 in treatments Gamma, Gammadm, Omega, and Phi.While the period 1 median differed across experiments and treat-

13. The number of subjects that gave a best response to the median of theirpredictions was as follows: 21 out of 27 subjects in Treatment Gamma, 13 out of 27subjects in Treatment Gammadm, 22 out of 27 subjects in Treatment Omega, £ind22 out of 27 in Treatment Phi.


TABLE IIIMEDIAN CHOICE FOR THE FIRST TEN PERIODS OF ALL EXPERIMENTS

Treatment

GammaExp. 1Exp. 2Exp. 3

GammadmExp. 4Exp. 5Exp. 6

OmegaExp. 7Exp. 8Exp. 9

PhiExp. 10Exp. 11Exp. 12

1

455

45

5

55

2

455

45

5

55

3

455

45

5

55

4

455

4*5

5

5*5*

5

455

5

5*

5*5*

Period

6

455

5

5*

5*5*

7

4*55

4*5

5*

5*5*

8

455

5*

5*

5*5*

9

4*55

5*

5*

5*5*

10

4*55*

5*

5*

5*5*

Notes. Exp. - experiment. * = indicates a mutual best response outcome.

ments, the median in subsequent periods was always equal to theperiod 1 median. This stability was observed whether the initialoutcome was efficient or inefficient and whether the treatment wasGamma, Gammadm, Omega, or Phi. In eleven out of twelveexperiments, the period 10 outcome satisfied the mutual bestresponse condition (2). Tbe equilibrium selected always generateda median equal to the period 1 median. Hence, in these treatmentssubjects select an equilibrium that is determined by tbe historicalaccident of tbe initial median."

ExEimination of individual bebavior in treatments Omega, Phi,and Gammadm confirms the salience of the initial median. Tbenumber of subjects selecting a period 2 action equal to tbe period 1median- was 18 out of 27 in Treatment Omega, 17 out of 27 intreatment Pbi, 18 out of 27 in Treatment Gammadm, and 8 out of

14. The dynamics of these average opinion games are remarkably differentfrom the tacit coordination game studied in Vsin Huyck, Battalio, and Beil [1990],which used larger teams and a minimum rule—^-rather them a median rule—toaggregate choices. Under a minimum rule, the initial outcome plays no role inselecting the equilibrium. Instead, the process always converges to the mostinefficient equilibrium point.


27 in Treatment Gamma. The data on individual behavior inTreatment Gamma reveal systematic adaptive hehavior.

A suhject's payoff is decreasing in x^ when he (she) playedahove the median, and those suhjects that played ahove the medianshow a strong tendency to reduce their action. The ohserved meanreduction in x^ is increasing in the difference hetween a suhject'speriod 1 action and the period 1 median, hut the mean reduction issignificantly smaller than this difference. Conversely, a suhject'spayoff is increasing in X; when he (she) played helow the median,and those suhjects that played helow the median show a tendencyto increase their action. The ohserved mean increase in :c, isincreasing in the difference hetween the suhject's period 1 actionand the period 1 median, hut the mean increase is significantlysmaller than this difference. Consequently, these average opiniongames quickly converge to an outcome that satisfies the mutualhest response property of a Nash equilihrium.

The speed of convergence to equilihrium appears to differacross treatments. Treatments Omega and Phi all converge withinfive periods, while only one of six haseline treatments convergeswithin five periods. In the haseline treatments, suhjects appear to"explore" the consequences of choosing an action one ahove orhelow last period's median. Inspection of Payoff Tahle F revealsthat choosing an action one ahove or helow the median cost onlyfive cents. However, the "exploring" hehavior ohserved in thehaseline treatments would he very costly in Treatment Omega—emy error results in zero earnings—and pointless in TreatmentPhi—all strict equilihrium points are contained in the set ofpayoff-dominant equilihrium points.

In the haseline treatments suhjects never coordinated on thepayoff-dominant equilihrium. Payoff-dominance was not a salientselection principle. Hence, these treatments provide striking exam-ples in which strategic uncertainty leads to coordination failure.The ohserved coordination failure results from suhjects' heteroge-neous response to strategic uncertainty, the adaptive hehaviorsuhjects exhihit in disequilihrium, the median rule, and thesalience of the historical median.'^

The strong precedent of the initial median M^ was salient to

15. Van Huyck, Battalio, and BeU [1990] and Cooper, DeJong, Forsythe, andRoss [1990] also report experimental games exhibiting coordination failure. In bothVan Huyck et al. and Cooper et al., the secure hut inefficient equilihrium obtains.(Cooper et al. results also depend on the presence of a strictly dominated"cooperative" strategy.) These results contrast with the coordination failure of thetext, which does not depend on suhjects adopting a secure action.


enough subjects that the historical accident of the initial medianMj selected the equilibrium in all of the average opinion treat-ments. Even though equilibrium analysis of G(9) is indeterminate,the median in periods 2 through 10 is perfectly predictable giventhe initial median M^ in these average opinion games.

VI. EXPERIMENTAL RESULTS: THE SALIENCE OF WEAK PRECEDENTS

This section reports the results of the continuation treatmentsin the twelve experiments, which test the salience of the weakprecedent established by a historical equilibrium in a related game.The subjects received new instructions and a new payoff table inperiod 11. Hence, the subjects were seeing PayoflF Table F or fl forthe first time.

We discuss the influence of weak precedents on gammacontinuation treatments first. In experiments 10, 11, and 12, theequilibria obtained in pretreatment Phi produced a median M\o ofeither 4 or 5, while the period 11 median in continuation Treat-ment Gamma was either 6 or 7; see Table IV. Only 5, out of 27subjects chose an action in period 11 equal to the period 10 median.The other 22 subjects chose an action greater than the period 10median.

The weak precedent of the historical median in a related gamewas not salient in the Phi-Gamma experiments. While not salient,the historical median does appear to anchor subjects' behavior inthe continuation treatment; that is, they appear to use it as a lowerbound on the median from which improvements can safely besought. Also, the presentation of a new table in and of itself mayhelp them coordinate their attempt to move to a better equilibrium.

The three experiments using the sequence Omega-Gamma aremore difficult to interpret, because in experiments 7 and 9 thepayoff-dominant equilibrium obtained in treatment Omega. Hence,experiments 7 and 9 cannot distinguish between the salience of thepayoff-dominant equilibrium point and the salience of the weakprecedent of M\o. However, an inefficient equilibrium, M^Q = 5,obtained in experiment 8; see Table IV. In period 11 of the Gammatreatment of experiment 8, only two out of nine subjects chose anaction equal to 5. The other seven subjects chose an action greaterthan 5. As in the Phi-Gamma experiments, the historical medianwas not salient. Instead, most subjects used it to einchor theirbeliefs and coordinate on the payoff-dominant equilibrium.

A comparison between the distribution of actions in period 1


TABLE IVMEDIAN CHOICE FOR THE CONTINUATION TREATMENTS OF ALL EXPERIMENTS

Exp. 1Exp. 2Exp, 3

Exp, 4Exp, 5Exp, 6

Exp, 7Exp, 8Exp, 9

Exp. 10Exp. 11Exp. 12

4*55*

^ " " " " •

4*4*5*

7*5*7*

M™

4*5*5*

11

5

75

7*

677

12

5

7*5*

7*

67*7*

13

5*

Omega

7*5*

Gamma

7*

Gamma

67*7*

14

Period

15

Omega

5*

7*5*

7*

67*7*

5*

7*5*

7*

67*7*

16

7*

5*

—

—

17

5*

Z

—

z

18

5*

—

—

Z

19 20

Gamma

7

5*

—

—

—

7*

5*

—

—

—

Notes. M (̂, = period 10 median in related game r. Exp, = experiment. * = indicates a mutual best responseoutcome. — - not applicable.

and period 11 for Treatment Gamma reveals the dramatic infiu-ence experience has on subject behavior; compare Tables II and V.'*Only 15 percent of the inexperienced suhjects chose the payoff-dominant action 7 in period 1, while 74 percent of the experiencedsuhjects chose the payoff-dominant action 7 when first exposed toPayoff Tahle F." Chi-square tests reject the hypothesis that thedistrihution of actions for gamma in period 1 and period 11 are thesame at the 1 percent level of statisticsd significeince. Paradoxically,experienced suhjects, who could use weak precedents, focus on the

16. Prediction accuracy in period 11 is similar to the prediction accuracy inperiod 1, An analysis of prediction accuracy can be found in our June 1988 workingpaper,

17, The distribution of actions in Gamma is only marginally influenced by thepretreatment of Omega rather than Phi. Chi-square tests fail to reject thehypothesis that the sample distributions were drawn from the same population.


TABLE VDISTRIBUTION OF INDIVIDUAL CHOICES IN PERIOD 11

Action

7654321

Total

Omega(followingbaseline)

Nm,

322

146000

54

(Pr.)

(59)(4)

(26)(11)(0)(0)(0)

(100)

Treatment

Gamma(followingOmega)

Nm.

23220000

27

(Pr,)

(85)(7.5)(7.5)

(0)(0)(0)(0)

(100)

Gamma(following

Phi)

Nm,

17433000

27

(Pr,)

(63)(15)(11)(11)(0)(0)(0)

(100)

Gamma(combined)

Nm,

40653000

54

(Pr.)

(74)(11)(9)(6)(0)(0)(0)

(100)

Notes. Nm. = numberof subjects. Pr. = percent of subjects.

payofF-dominant equilibrium much more strongly than do inexperi-enced subjects, who could not.

In experiments 1, 2, and 3, we used an A-B-A design, specifi-cally Gamma-Omega-Gamma, to determine whether behavior wasreversible. As Table IV reveals, the median in period 18 equals themedian in period 10 in only one of the three experiments. Given theextreme path dependence observed in the experiments, we aban-doned the A-B-A designs for the remaining experiments in theproject.

Experienced subjects when first exposed to Payoff Table il,following Payoff Table F, do not focus on the equilibrium closest tothe historical equilibrium of the "pregame" in four out of sixexperiments; see Table IV. Only 18 out of 54 subjects chose a period11 action equal to the period 11 median. Thirty-five subjects chosean action greater than the period 10 median. Most subjects ignoredthe weak precedent of a historical equilibrium in a related gameand focused on the payoflF-dominant equilibrium instead. A Chi-square test fails to reject the hj^othesis that the distribution ofactions for Treatment Omega in periods 1 and 11 are drawn fromthe same population distribution; compare Table II and V.

The outcome of the continuation treatment in period 11, likethe initial treatment, is extremely stable in these average opinion


games. Table IV reports the median for periods 11 through the endof the experiment. While the outcome of the continuation treat-ment in period 11 differed across experiments and treatments, themedian in subsequent periods was always equal to the period 11median.

In eleven out of twelve experiments, the period 15 outcomesatisfied the mutual best response condition (2). The equilibriumselected always generated a median equal to the period 11 median.In the continuation treatment, convergence to a mutual bestresponse outcome is extremely rapid, occurring in one or twoperiods. In nine out of twelve experiments the period 12 outcomesatisfied the mutual best response condition (2). Like the initialtreatment the historical accident of M^ determines which equilib-rium point obtains in the continuation treatment. The median inperiods 12 through 15 is perfectly predictable given M^ in theseaverage opinion games.

VII. SUMMARY

In the baseline average opinion game, which has a uniquepayoff-dominant equilibrium and a unique secure equilibrium,neither payoff-dominance nor security was a salient equilibriumselection principle. Instead, the modal response was between thepayoff-dominant and the secure action. Repeated interaction pro-duced simple dynamics that converged to the inefficient equilib-rium selected by the historical accident of the initial median. Thebaseline experiments provide a striking example of how strategicuncertainty can lead to coordination failure.

Treatment Omega reduced the strategic uncertainty confront-ing subjects by eliminating considerations of security. Payoff-dominance accurately predicts the modal response to period gamen. Treatment Phi reduced the strategic uncertainty confrontingsubjects by eliminating considerations of payoff-dominance. Secu-rity accurately predicts the modal response to period game <1>. Aswith the baseline treatment, the historical accident of the initialmediein selected the equilibrium outcome implemented in all threeOmega treatments and all three Phi treatments.

Continuation treatments found little evidence for the salienceof weak precedents. Instead, experience in related games increasedthe salience of payoff-dominance. However, like the initial treat-ment, the historical accident of the period 11 median determines


which equilibrium point obtains in the continuation treatment.Hence, all treatments provide evidence that the initial median in atreatment provides a salient precedent.

Deductive equilibrium analysis of these average opinion gainesis indeterminate. Yet the observed behavior in the experiments wassystematic. The distribution of initial actions in a treatment variedsystematically with considerations of payoff-dominance and secu-rity. Given an initial median, the median in the remaining periodsof a treatment was perfectly predictable. The systematic and,hence, predictable behavior observed in the experiments suggeststhat it should be possible to construct an accurate equilibriumselection theory.

APPENDIX A: DUAL MARKET ORDER STATISTIC GAMES

In baseline experiments 4, 5, and 6, subjects participatedsimultaneously in two games: one set n equal to nine, and the otherset n equal to twenty-seven. In both games the column was chosenby the fifth-order statistic. The dual market technique allows us tostudy whether an individual subject behaves differently in gamesthat differ only according to the number of decision makers and,hence, only in the level of strategic uncertainty. This difference inbehavior is independent of subject heterogeneity and experience;see Battalio, Kogut, and Meyer [1990]. The hypothesis tested wasthat subjects would chose a more secure action in the game with 27subjects.

On average, subjects did chose a more secure action in thelarge game. In period 1, the mean diflFerence between the choice inthe small game less the choice in the large game was 0.22. Thisdifference is consistent with the expected effect, but it is notstatistically significant. In the large game the fifth-order statisticwas a four in every period of both the Gamma and Omegatreatments.

However, the mean difference between the predicted medianfor the small game less the predicted median for the large game was1.83, which was significantly different from zero at the 1 percentlevel. Subjects predicted that other subjects would chose a moresecure action in the large game. As reported in the text and unlikeTreatment Gsimma, most subjects did not chose an action equal tothe median of their predictions. Apparently, the dual markettechnique infiuenced subject behavior—specifically, the relation-ship between predictions and actions.


APPENDIX B: SUBJECT CHOICES BY EXPERIMENT

Subject

Exp. 112345

789

Exp, 2

234

OS

C

T

789

Exp, 3

23456789

Exp. 4123456789

1 2 3 4 5

Treatment Geimma

3

3

CT

C

T

t

5

1

54

6

5

44

5

5

54

4

2

44

3 4 3 4 4Treatment Gamma

65657

46

735

CT

O

S

46

555

CT

O

S

56

65665

65

655

CT

05

56

Treatment Gamma

57744747

66544725

56555755

56555555

57555555

6

446

444

555

CT

O

S

56

57555555

Treatment Gammadm647445353

544444344

544544443

444544444

444544444

444444444

7

444

44

55565

56

56555555

444444444

8

443

44

555

CT

CT

55

55555655

444444444

Period

9

4

4

4

555

CT

CT

55

C

66555556

444444444

10

4

4

655

CT

CT

CT

CT

5

C

55555555

444444444

11 12 13 14

Treatment Omega4 7 7 7

6

7

7

7

7

7

7

7

Treatment Omega

7777

CT

CT

5

7777

777

7777

777

7777

777

Treatment Omegac c c c

56575757

75555757

55555555

55555555

Treatment Omega477777747

777777777

77777777

tr-

7

tr-

7777777

15

7

7

7

7

7777

777

55555555

777777777


APPENDIX B : (CONTINUED)

Subject

Exp, 51234456789

Exp, 6123456789

Exp, 7123456789

Exp. 8123456789

1 2 3 4 5 6

Treatment Gammadm

4 4 4 4 4 4Treatment Gammadm354755744

445654554

655554554

644554554

755555555

Treatment Omega

4 6 7 7 77 7 7 7 76 7 6 7 7


574436355

554565355

555555555

555555555

555554555

7

555555555

7

4

655554555

7

555555555

8

4

555555555

7

555555555

Period

9

555555555

555555555

10

555555555

555555555

11 12 13 14

Treatment Omega7 7 7 77 7 7 77 7 7 77 7 7 7

7 7 7 75 7 7 74 7 7 7


555555555

555555555

555555555

Treatment Gamma7 7 7 77 7 7 7

7 7 7 77 7 7 77 7 7 77 7 7 77 7 7 7

Treatment Gamma776577756

777777757

777777757

777777777

15

777777777

555555555

77

777777

777777777


APPENDIX B: (CONTINUED)

Subject

Exp.123

45678

Exp.1

3456789

Exp.123456789

Exp.123456789

9

10

11

12

1 2 3 4 5

Treatment Omega1 1 1 1 1

5 1 1 1 1

1

57

b

47

7

77

7

77

Treatment Phi4 4 4 4

4564

44

t T

f T

O

TO

lO

44

445

4

45

4Treatment Phi445

55555

534454555

544555555

555555555

Treatment Phi733454676

544455555

555455555

555555555

7

7

4

4

4

en e

nen e

n

55555

555555555

6

7

7

7

4

4

4

55

en e

n

55555

555555555

7

7

7

7

4

5

4

en e

nen e

n

55555

555555555

8

7

7

7

4

4

4

en e

n

555555

555555555

Period

9

7

4

5

4

en e

n

5555555

555555555

10

7

4

4

4

55

en e

n

55555

555555555

11 12 13 14

Treatment Gamma

7 7 7 7

77777

77777

77777

77777

Treatment Gamma

6547447

6566

Ol Ol en

6666677

6666666

Treatment Gamma6

577777

7

7

Ir-

77

Ir-

7

7

777777

7

777777

Treatment Geimma776577677

777777777

777777777

777777777

15

7

77777

6666

CD

CO

C

O

7

777777

777777777

9 0 8 QUARTERLY JOURNAL OF ECONOMICS

APPENDIX C INSTRUCTIONS

This is an experiment in the economics of market decisionmaking. The instructions are simple. If you follow them closely Eindmake appropriate decisions, you may make an appreciahle amountof money. These earnings will be paid to you, in cash, at the end ofthe experiment.

In this experiment you will participate in a market of ninepeople. There will he a numher of market periods. In each periodevery participant will pick a value of X. The values of X you maychoose are 1,2, 3,4, 5, 6, 7. The value you pick for Xand the medianvalue of X chosen will determine the payoff you receive for thatperiod.

(The median choice is determined as follows. The choices madehy the nine participants will he ordered from smallest to largest innumerical order. The median choice is the fifth from the hottom orthe fifth from the top of the ordered choices. For example, to findthe median of the nine numhers 33, 30, 30, 27, 34, 32, 34, 29, 32,arrange the numhers in ascending order—27, 29,30, 30, 32, 32, 33,34, 34—find the fifth choice counting either from the first numherforward or the last numher hack of the ordered choices and that isthe median vjilue. In this example the median choice is 32.)

You are provided with a tahle that tells you the potentialpayoffs you may receive. Please look at the table now. The earningsin each period may he found hy looking across from the value youchoose on the left-hand side of the table and down from the medianvalue chosen from the top of the table. For example, if you choose a4 and the median value chosen is a 3, you earn 85 cents that period.

At the beginning of every period, each participant will write ona reporting sheet their participant number and the value ofX theyhave chosen and hand it in to the experimenter. The median valueof X chosen will be announced, and each participant will then figureout their earnings for that period.

If you will now look at your record sheet, you will see thefollowing entries: BALANCE, YOUR CHOICE OF X, MEDIANVALUE OF X CHOSEN, and YOUR EARNINGS. In the firstperiod your BALANCE is $2. In the second period your BALANCEis the value of your earnings in the first period plus the $2heginning balance. In the third period your BALANCE is the valueof your BALANCE in the second period plus the value of yourearnings in the second period. Please keep accurate recordsthroughout the experiment.


To be sure that everyone understands the instructions, pleasefill out the sheet labeled questions and turn it in to the experi-menter. DO NOT PUT YOUR NAME OR PARTICIPANT NUM-BER ON THE QUESTION SHEET. If there are any mistakes onthe question sheet, the experimenter will go over the instructionsagain.

IF YOU HAVE ANY QUESTIONS, PLEASE ASK THEM ATTHIS TIME.

TEXAS A&M UNIVERSITY

TEXAS A&M UNIVERSITY

AUBURN UNIVERSITY

REFERENCES

Basar, Tamer, and Geert J. Olsder, Dynamic Noncooperative Game Theory (NewYork, NY: Academic Press, 1982).

Battedio, Raymond, Carl Kogut, and Donald Meyer, "The Effects of VaryingNumber of Bidders in First Price Private Value Auctions: An Application of aDual Market Bidding Technique," Advances in Behavioral Economics Vol. 2,(Northwood, NJ: Abler Publishing Corp., 1990).

Bryant, John, "A Simple Rational Expectations Keynes-TjTJe Model," QuarterlyJournal of Economics, XCVIII (1983), 525-28.

Cooper, Russell, Douglas DeJong, Robert Forsythe, and Thomas Ross, "SelectionCriteria in Coordination Games: Some Experimental Results," AmericanEconomic Review, LXXX (1990), 218-34,

Cooper, Russell, and Andrew John, "Coordinating Coordination Failures in Keynes-ian Models," Quarterly Journal of Economics, CIII (1988), 441-64.

Crawford, Vincent, "An 'Evolutionary' Interpretation of Van Huyck, Battalio, andBeil's Experimental Results on Coordination," Games and Economic Behavior,III (1991), 25-60.

Frydman, Roman, and Edmund Phelps, eds,. Individual Forecasting and AggregateOutcomes (Cambridge: Cambridge University Press, 1983).

Harsanyi, John, and Reinhard Selten, A General Theory of Equilibrium Selection inGames (Cambridge, MA: MIT Press, 1988).

Keynes, John Maynard, The General Theory of Employment, Interest, and Money(New York, NY: Harcourt, Brace, & World, Inc., 1936).

Lewis, David, Convention: A Philosophical Study (Cambridge, MA: HarvEirdUniversity Press, 1969).

Lucas, Robert, Jr., "Adaptive Behavior and Economic Theory," in Rational Choice,R, M. Hogarth and M. W. Reder, eds. (Chicago, IL: University of Chicago Press,1987).

Luce, Duncan, and Howard Raiffa, Games and Decisions (New York, NY: Wiley,1957).

Milgrom, Paul, and John Roberts, "RelationalizabiUty, Learning, and Equihbriumin Games with Strategic Complementarities," Econometrica, LVIII (1990),1255-78.

Schelling, Thomas, The Strategy of Conflict (Cambridge, MA: Harvard UniversityPress, 1980).

Sugden, Robert, "Spontaneous Order," Journal of Economic Perspectives, III(1989), 85-98.

Van Damme, Eric, Stability and Perfection of Nash Equilibria (Berlin: SpringerVerlag, 1987),


Van Huyck, John, Raymond Battalio, and Richard Beil, "Tacit CoordinationGames, Strategic Uncertainty, and Coordination Failure," America/i EconomicReview, LXXX (1990), 234-49.

Van Huyck, John, Ann Gillette, and Raymond Battalio, "Credible Assignments inCoordination Games," Games and Economic Behavior, forthcoming.

Von Neumann, John, and Oskar Morgenstern, Theory of Games and EconomicBehavior (Princeton, NJ: Princeton University Press, 1972).

strategic uncertainty, equilibrium selection,consequently, a theory of equilibrium selection would...

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