Predicting the outcome of a game

The positive interpretation of conventional game theory predicts that the joint (mixed) strategy of players in a game satisfies an equilibrium concept. The relative probabilities of such satisfying strategies are not specified, and all other strategies are deemed impossible. As an alternative, in this paper we use statistical inference to predict the joint strategy. The associated positive problem is to determine the density function over joint strategies. This Predictive Game Theory (PGT) typically assigns non-zero probability density to multiple joint strategies. A loss function can be used to distill that density to a single joint strategy via decision theory. This mapping of a game to a joint strategy constitutes an “equilibrium concept”. It typically produces a single joint strategy and therefore needs no refinements. We explore a Bayesian version of PGT based on the entropic prior and a likelihood that quantifies the rationalities of the players. We show that the local peaks of the posterior density and the game’s Quantal Response Equilibria (QRE’s) approximate each other. Some joint strategies are not expressible as QRE’s. In contrast, we show that every joint strategy has non-zero density for one (and only one) set of player rationalities.