Evidence on learning in coordination games

Evidence on Learning in Coordination Games

John B. Van Huyck, Raymond C. Battalio, Frederick W. Rankin

Revised October 2001

Abstract: This paper reports an experiment designed to detect the influence of strategic uncertainty on behavior in coordination games. Specifically, games in which a player's best response is an order statistic of the cohort's action combination. Unlike previous experiments using order statistic coordination games, the new experiment holds the payoff function constant and only changes cohort size and order statistic.

Key Words: coordination, equilibrium selection, learning, strategic uncertainty, adaptive behavior.

JEL classification: c72, c92.

Acknowledgments: Alison Aughinbaugh and John Wildenthal provided research assistance. Eric Battalio implemented the experimental design on the TAMU economics laboratory network. The National Science Foundation and Texas Advanced Research Program provided financial support. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Introduction

Most situations have multiple mutually consistent ways people can behave. Such situations involve a strategy coordination problem. What is best for a person depends on what others are doing. Strategic uncertainty arises even in situations where objectives, feasible strategies, institutions, and equilibrium conventions are completely specified and are common knowledge. Formally, strategic uncertainty exists whenever a player lacks perfect foresight about what other players are going to do and it influences behavior whenever a player does not have a strictly dominant strategy. Models that do not account for strategic uncertainty may not make accurate predictions about the consequences of environmental changes like group size.

This paper reports an experiment designed to detect the influence of strategic uncertainty on behavior in coordination games. The model of strategic uncertainty around which we have organized our thinking is Crawford (1995). The most innovative feature of his model is that "instead of fully endogenizing players’ beliefs and strategy choices (whose differences cannot be traced to differences in players’ information or other characteristics) the model characterizes them statistically, treating certain aspects of the process that describes how they evolve as exogenous parameters to be estimated on a case-by-case basis. This [feature] reflects the conviction ... that it is impossible to predict from rationality alone how people will respond to coordination problems." He used the data in Van Huyck, Battalio, and Beil (1990, 1991), henceforth VHBB, to estimate the parameters of his model.

Once the model has been estimated it can be used to determine the prior probability distribution over equilibria in the underlying coordination game.

The new experiment reported below can be seen as an out-of-sample test of his estimated model. Crawford is careful to point out that the estimated behavioral parameters change as payoff functions change. The VHBB (1991) median treatments changed the parameters of the payoff function rather than cohort size and order statistic. The comparison between the VHBB (1991) median treatments and the VHBB (1990) minimum treatments is also confounded by changes in the payoff function.

The new experiment holds the payoff function constant and only changes cohort size and order statistic. We focus on three aspects of behavior: subjects’ initial response to the experimental design, which provides evidence on the salience of deductive selection principles; subjects’ adaptive response to their experience in repeated play, which provides evidence on learning theories; and the pattern of equilibrium selection observed in the terminal continuation game, which provides evidence on inductive equilibrium selection theories.

The new experiment also uses the ‘blue box’ interface available on the TAMU Economics Laboratory network, which allows a much finer approximation of a continuous action space. Hence, it provides evidence on how important a coarse approximation is for obtaining VHBB’s (1990,1991) results. A finer approximation makes it less costly to explore locally and, hence, may destabilize inefficient mutual best response outcomes.

The initial conditions are similar to those observed by VHBB. Increasing the order statistic has a larger impact on initial behavior than increasing group size. Crawford’s model accurately predicts the ordering of the terminal empirical distribution functions. However, we find an anomalous bias towards efficiency that appears to be due to our using a finer approximation of a continuous action space.

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Conclusion

Three of four treatment comparisons provide support for Crawford's (1995) stochastic dominance propositions, see table 5. Changing the order statistic is more powerful than changing group size. The order statistic also influences initial behavior, which exaggerates the stochastic dominance relations, see table 2.

Subjects focus on the observed order statistic and then explore locally. This local exploration is skewed in the direction of efficiency. [1] Skewed local exploration interacting with the order statistic causes behavior to slowly creep towards efficiency in some treatments and, most surprisingly, this creeping is perfectly correlated with time in a few treatments. Since this creeping towards efficiency was not observed in VHBB (1991) and three of five initially inefficient OS[7,4] cohorts creep up to the efficient outcome within twenty periods, one must conclude that changing grid size has an important influence on behavior. Reducing the opportunity cost of local exploration increases subjects' propensity to experiment with actions slightly higher than last period's order statistic.

References

Bruno Broseta, "Strategic Uncertainty and Learning in Coordination Games," UCSD discussion paper, 93-34, August 1993.

Bruno Broseta, "Estimation of a Game-Theoretic Model of Learning: An ARCH approach," UCSD discussion paper 93-35, August 1993.

Gerard P. Cachon and Colin F. Camerer, "Loss-avoidance and Forward Induction in Experimental Coordination Games," The Quarterly Journal of Economics, 111(1) February 1996.

W.J. Conover, Practical Nonparametric Statistics, 2e. (New York, NY: J. Wiley & Sons, 1980).

R. Cooper, D.V. DeJong, R. Forsythe, and T.W. Ross, "Selection Criteria in Coordination Games: Some Experimental Results," American Economic Review, 80(1), March 1990.

Vincent P. Crawford, "Adaptive Dynamics in Coordination Games," Econometrica 63(1), January 1995, 103-144.

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Dale O. Stahl, “Evolution of Smartn Players,” Games and Economic Behavior, 5(4) October 1993, 604-617.

D. Teichroew, “Tables of expected values of order statistics and products of order statistics for samples of size twenty and less from the normal distribution.” Annals of Mathematical Statistics, 27, 1956, 410-26.

J.B. Van Huyck, R.C. Battalio, and R.O. Beil, "Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure," The American Economic Review 80(1), March 1990, 234-248.

J.B. Van Huyck, R.C. Battalio, and R.O. Beil, "Strategic Uncertainty, Equilibrium Selection, and Coordination Failure in Average Opinion Games," The Quarterly Journal of Economics, 106(3), August 1991, 885-911.

J.B. Van Huyck, J.P. Cook, and R.C. Battalio, "Adaptive Behavior and Coordination Failure," forthcoming in Journal of Economic Behavior and Organization.

J.B. Van Huyck, J.P. Cook, and R.C. Battalio, “Selection Dynamics, Asymptotic Stability, and Adaptive Behavior,” Journal of Political Economy, 102(5), October 1994, 975-1005.

Footnotes

[1] See also Van Huyck, Cook, and Battalio (1993) for similar phenomena.