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PREDICTING THE OUTCOME OF A GAME

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David Wolpert, NASA Ames Research Center
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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.