# 2010-13 Distribution-valued solution concepts

Under its conventional positive interpretation, game theory predicts that the mixed strategy profile of players in a non cooperative game will satisfy some set-valued solution concept. Relative probabilities of profiles in that set are unspecified,and all profiles not satisfying it are implicitly assigned probability zero. However the axioms underlying Bayesian rationality say that we should reason about player behavior using a probability density over all mixed strategy profiles, not using a subset of all such profiles. Such a density over profiles can be viewed as a solution concept that is distribution-valued rather than set-valued. A distribution-valued concept provides a best single prediction for any non cooperative game, i.e., a uni-versal refinement. In addition, regulators can use a distribution-valued solution concept to make Bayes optimal choices of a mechanism, as required by 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.