PREDICTING THE OUTCOME OF A GAME

David Wolpert, NASA Ames Research Center

Many machine learning problems involve predicting the joint strategy choice of some goal-directed ``players'' engaged in a noncooperative game. Conventional game theory predicts that that joint strategy satisfies an ``equilibrium concept''. The relative probabilities of the joint strategies satisfying that concept are not given, and all other joint strategies are given probability zero. As an alternative, I view this prediction problem as one of statistical inference, where the ``data'' includes the game specification. This replaces the game theory issue of how to specify a set of equilibrium joint strategies with the issue of how to specify a density function over joint strategies.

I explore a Bayesian version of such a Predictive Game Theory (PGT) using the entropic prior and a likelihood that quantifies the rationalities of the players. A popular game theory equilibrium concept parameterized by player rationalities is the Quantal Response Equilibrium concept (QRE). I show that for some games the local peaks of the posterior density over joint strategies approximate the associated QRE's, and derive the associated correction terms. I also discuss how to estimate parameters of the likelihood from observational data. I end by showing how PGT specifies a unique equilibrium of any game, thereby solving a long-standing problem of conventional game theory.