Distribution-Valued Solution Concepts

Under its conventional positive interpretation, game theory makes predictions about the mixed strategy profile of the players in a noncooperative game by specifying a “solution set” of such profiles, e.g., the set of Nash equilibria of that game. Profiles outside of that set are implicitly assigned probability zero, and relative probabilities of profiles in that set are not provided. In contrast, Bayesian analysis does not make predictions about the state of a system by specifying a set of possible states of that system. Rather it provides a probability density over all those states, conditioned on all relevant information we have concerning the system. So when the “state of a system” is the strategy profile of the players of a game, and our information comprises the game specification, a Bayesian analysis would result in a posterior density over the set of all profiles, conditioned on the game specification. Evidently the very form of a prediction provided by a Bayesian analysis is different from the form provided by solution sets. In this paper, we show how to construct a Bayesian posterior density over profiles and discuss its practical advantages over solution sets. As an example, by combining this posterior density with a loss function of the scientist making the prediction, standard decision theory fixes the unique Bayes-optimal prediction of the profile conditioned on the game specification. So it provides a universal refinement. As another example, Bayesian regulators of the players involved in a game would use such a posterior density to make Bayes-optimal choices of a mechanism to control player behavior (and thereby fully adhere to Savage’s axioms). In particular, a regulator can do this in situations where conventional mechanism design cannot provide advice. We illustrate all of this numerically with Cournot duopoly games.