Distribution-Valued Solution Concepts

Under its conventional positive interpretation, game theory predicts that the mixed strategy pro?le of players in a noncooperative game will satisfy some setvalued solution concept. Relative probabilities of pro?les in that set are unspeci?ed, and all pro?les 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 pro?les, not using a subset of all such pro?les. Such a density over pro?les 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 noncooperative game, i.e., a universal re?nement. 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.