Credible assignments in coordination games

Credible Assignments in Coordination Games

John B. Van Huyck, Ann B. Gillette, and Raymond C. Battalio

Revised August 1989

Abstract: A strict equilibrium point in a game is defined as an assignment to each player of a strategy that is a unique best response for him when the others use the strategies assigned to them. Typically, equilibrium analysis does not explain an assignment's origin. This paper examines assignments made by an arbiter: an individual, different from the players, who has the ability to make non-binding common information assignments. An assignment that corresponds to the outcome of the actual game is a credible assignment. This paper uses the experimental method to examine the ability of an arbiter to determine the outcome of two-person coordination games. All of the arbiter's assignments in the experiments are strict equilibrium points. However, the arbiter's equilibrium assignment may not satisfy one or more of the following deductive selection principles: payoff-dominance, symmetry, or security. The experiments test the hypothesis that an assignment to a strict equilibrium is a credible assignment. In general, human subjects do not find the mutual consistency of a strict equilibrium assignment to be a sufficient reason for implementing the assignment when it conflicts with deductive principles like payoff-dominance and symmetry.

Acknowledgments: Mike Baye, Martin Hellwig, Stan Reynolds, referees, and seminar participants at the 1988 Economic Science Association meetings and 1989 Econometric Society Summer Meetings made constructive comments on an earlier version of this paper. David Borden, Benjamin Contreras, Steve Hackett, Patsy Johnson, Sophon Khanti-Akom, Gretchen Larson, Kirsten Madsen, Melannie Marks, Sondip Mathur, Artie Powell, Andy Rettenmeier, and Jason Scott provided research assistance. The National Science Foundation (SES-8420240;SES-8911032), the Texas Agriculture Extension Service, the Texas Engineering Extension Service, and the Texas A&M Center for Mineral and Energy Research provided financial support.

Introduction

Equilibrium methods are powerful tools for analyzing situations that exhibit strategic interdependence. A strict equilibrium in a game is defined as an assignment to each player of a strategy that is a unique best response for him when the others use the strategies assigned to them. An equilibrium assignment has the desirable property that the prescribed behavior is individually rational and mutually consistent. Typically, equilibrium analysis does not explain an assignment's origin. When there is a unique equilibrium, one possibility is that the players simply deduce the equilibrium point from the description of the game.

Unfortunately, equilibrium analysis often fails to determine a unique equilibrium in many games and, hence, fails to prescribe rational behavior in the game or to predict the outcome of the game. In games with strict multiple equilibria, the rational player using deductive equilibrium concepts is uncertain which equilibrium strategy other players will use and this uncertainty will influence the rational player's behavior. Strategic uncertainty arises even in situations where objectives, feasible strategies, institutions, and equilibrium conventions are completely specified and are common knowledge.

For example, consider the tacit unanimity game A depicted by payoff table A, [following paragraph]. Three assignments in pure strategies--(1,1), (2,2), (3,3)--satisfy the strict mutual best response property. Hence, this simple game has three strict equilibrium points. [There are also four mixed strategy equilibria. [1]] In games with multiple equilibria, players confront a non-trivial coordination problem and may fail to coordinate on an equilibrium point.

Payoff Table A

	1	2
1	5 ; 5	0 ; 0	0 ; 0
2	0 ; 0	5 ; 5	0 ; 0
3	0 ; 0	0 ; 0	5 ; 5

An equilibrium selection principle identifies a subset of equilibrium points according to some distinctive characteristic. If the principle selects a unique equilibrium, it may resolve the coordination problem. It is useful to distinguish between selection principles based on the history of a pregame--the games context--and selection principles based on the description of the game itself. Inductive selection principles select equilibria based on common information about the history preceding play of the game. [2] Deductive selection principles select equilibria based on the description of the game itself. In games with multiple equilibria, players may be able to use some selection principle to identify a specific equilibrium point. Hence, the outcome of a game with multiple strict equilibria may nevertheless be a mutual best response outcome.

Conclusion

Section 4 reports evidence that without pre-play communication there can be no presumption that games with multiple equilibria will yield an equilibrium outcome and that the introduction of an arbiter can coordinate behavior to produce equilibrium outcomes. The remaining sections report evidence from experiments in which the equilibrium assignment did not satisfy one or more of the following deductive selection principle: payoff-dominance, symmetry, or security.

These results suggest that the credibility of a strict equilibrium assignment depends on the strategic details of the game, that is, subjects behave as if they form their expectations of their opponents behavior based in part on the description of the game. Specifically, payoff-dominated equilibrium points were not credible assignments in these games. Also, an assignment to an unequal-payoff equilibrium in a symmetric game gave mixed results: slightly more than half the subjects found the asymmetric assignment credible and slightly less than half found it incredible. The experiments found little evidence that the credibility of an arbiter's assignments depends on out-of-equilibrium payoffs.

…

The experiments in this paper present evidence that human subjects when confronted with multiple strict equilibria use both inductive and deductive selection principles to coordinate on a specific equilibrium. Sometimes these principles conflict, which may lead to a high frequency of disequilibrium outcomes. However, the observed behavior was more systematic than theories without mutual consistency conditions predict. When subjects defect from the arbiters assignment it is almost always to the Harsanyi/Selten solution.

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Footnotes

[1] Let a vector [p1,p2,p3] denote the probability assigned to a player's pure strategies {1,2,3}, then the four mixed strategy equilibrium points are ([1/2,1/2,0],[1/2,1/2,0]); ([1/2,0,1/2],[1/2,0,1/2]); ([0,1/2,1/2],[0,1/2,1/2]); and ([1/3,1/3,1/3],[1/3,1/3,1/3]).

[2] We use the term inductive in the logical, rather than mathematical, sense of reasoning from experience to a conclusion.