Distribution-Valued Solution Concepts

Under its conventional positive interpretation, game theory predicts that the mixed strategy profile of players in a particular noncooperative game will fall within some set determined by the game, 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 given. In contrast, Bayesian rationality does not tell us to predict which state a system has by specifying a subset of its possible states. Rather it tells us to specify a posterior probability density over all those states, conditioned on all relevant information we have. So in particular, when the “state of a system” is the mixed strategy profile of the players of a game, and our information comprises the game specification, Bayesian rationality tells us to predict that profile with a posterior density over the set of all profiles, conditioned on the game specification. Here we introduce such a posterior density over profiles. Via standard Bayesian decision theory, this density fixes the unique optimal prediction of the profile of any noncooperative game, i.e., it provides a universal refinement. In addition, regulators can use such a density to make Bayes optimal choices of a mechanism, thereby fully adhering to Savage’s axioms. In particular, they can do this in strategic situations where conventional mechanism design cannot provide advice. We illustrate all of this on a Cournot duopoly game