An Experiment to Evaluate Bayesian Learning of Nash Equilibrium Play By

Some recent theoretical approaches to the question of how players might converge over time to a Nash equilibrium have assumed that the players update their beliefs about other players according to Bayes’ Rule. Jordan has shown in a Bayesian model of this kind that play will (theoretically) always converge to a complete-information Nash equilibrium, even though individual players will not generally attain complete information. We report on an experiment designed to evaluate the empirical implications of Jordan’s model. A finite version of the model is constructed which generates unique predictions of subjects’ choices in nearly all periods. The experimental data reveals that the theory does reasonably well at predicting the equilibria that subjects eventually play, even when there are multiple equilibria. The results thus suggest that Jordan’s Bayesian model can provide an empirically effective solution to the equilibrium selection problem when the players have beliefs with finite support. However, the model’s predictions about the path of play over time are not consisitent with the experimental data. We are grateful to the University of Arizona’s Economic Science Laboratory (ESL) for research support; to the National Science Foundation for financial support; and to Timothy O’Neill of the ESL for his extraordinary programming services. An Experiment to Evaluate Bayesian Learning of Nash Equilibrium Play James C. Cox, Jason Shachat, and Mark Walker The attempt to rationalize equilibrium analysis in games has recently shifted from arguments based on players’ introspection and common knowledge to the idea that equilibrium play is learned by the players of a game through repeated play. Some of the contributions to this learning-based approach to equilibrium have the character that the players are fully rational, in the sense that they have well-defined beliefs about one another that are updated in a Bayesian fashion in response to their experience. (See, for example, Jordan (1991) and Kalai & Lehrer (1993).) In this paper we describe an experiment that is designed to shed light on the idea that equilibrium play may be learned by players who behave as Bayesian learners. The experimental results provide some empirical support for the Bayesian learning approach to rationalizing equilibrium analysis. The experiment is based on the Bayesian learning model analyzed by Jordan (1991). In Jordan’s model, players are engaged in a noncooperative normal-form game. Each player knows his own payoff function but he has only a probabilistic belief about the other players’ payoff functions. In this game of incomplete information, the players are assumed to play a Bayesian Nash equilibrium, but this will not generally yield a Nash equilibrium of the “true game” they are playing – the game defined by the players’ actual payoff tables. The question Jordan addresses is whether, by playing the game repeatedly, the players will learn to play a Nash equilibrium of the true game. Specifically, Jordan assumes that at each stage of play the players play a Bayesian Nash equilibrium based upon their current beliefs, and that after each play has occurred each player updates his belief according to Bayes’ Rule by incorporating the play he has just observed. Jordan proves that under certain conditions this process does indeed converge to a Nash equilibrium of the true game. A further interesting feature of Jordan’s analysis is that while the players do eventually play a complete-information Nash equilibrium, in most cases they do so without actually becoming completely informed: each player