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ssrn-id1184325 - Statistical prediction of the outcome of a...

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Electronic copy available at: http://ssrn.com/abstract=1184325 Statistical prediction of the outcome of a noncooperative game David H. Wolpert NASA Ames Research Center MailStop 269-1, Moffett Field, CA 94035-1000 (650) 604-3362 (V), (650) 604-3594 (F), david.h.wolpert.nasa.gov James Bono Department of Economics, American University, Washington DC August 29, 2009 1
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Electronic copy available at: http://ssrn.com/abstract=1184325 Abstract Conventionally, game theory predicts that the joint mixed strat- egy of players in a noncooperative game will satisfy some equilibrium concept. Relative probabilities of the joint strategies satisfying the concept are unspecified, and all strategies not satisfying it are as- signed probability zero. As an alternative, we recast the prediction problem of game theory as statistically estimating the joint strategy, from “data” that consists of the game specification. This replaces the focus of game theory, on specifying a set of “equilibrium” mixed strategies, with a new focus, on specifying a probability density over all mixed strategies. We explore a Bayesian version of such a Predic- tive Game Theory (PGT). We show that for some games the peaks of the posterior over joint strategies approximate quantal response equi- libria. We also show how PGT provides a best single prediction for any noncooperative game, i.e., a universal refinement. We also show how regulators can use PGT to make optimal decisions in situations where conventional game theory cannot provide advice. Keywords: Quantal Response Equilibrium, Bayesian Statistics, En- tropic prior, Maximum entropy, Regret JEL Classification codes: C02, C11, C70, C72 2
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1 Introduction Consider a physical system whose state we wish to predict, based on some information / data / knowledge concerning that system. We wish to make that prediction using conventional statistical tools, since there are so many compelling axiomatic arguments underpinning those tools [10, 21, 39]. However unlike in typical prediction scenarios where the tools of statistics are used, say that our “data” is that the physical system comprises a set of goal-seeking agents playing a specified noncooperative game. Also unlike typical prediction scenarios, say that the “state variable” of the physical system that we wish to predict is the joint mixed strategy chosen by those agents. To apply conventional statistical tools to this type of prediction problem, we need a way to incorporate our unusual data — the game specification — into a statistical model of the system, just like we would for other, more conventional types of data. In this paper I present one way to do this. 1.1 Background on relevant work from several fields Many physical systems whose state we wish to predict can be viewed as a set of interacting goal-seeking agents. In some of these systems all the agents are artificial. Examples include distributed adaptive control, distributed rein- forcement learning (e.g., such systems involving multiple autonomous adap- tive rovers on Mars or multiple adaptive telecommunications routers), and
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