A Predictive Theory of Games

Abstract

Conventional noncooperative game theory hypothesizes that the joint strategy of a set of players in a game must satisfy an "equilibrium concept". All other joint strategies are considered impossible; the only issue is what equilibrium concept is "correct". This hypothesis violates the desiderata underlying probability theory. Indeed, probability theory renders moot the problem of what equilibrium concept is correct - every joint strategy can arise with non-zero probability. Rather than a first-principles derivation of an equilibrium concept, game theory requires a first-principles derivation of a distribution over joint (mixed) strategies. This paper shows how information theory can provide such a distribution over joint strategies. If a scientist external to the game wants to distill such a distribution to a point prediction, that prediction should be set by decision theory, using their (!) loss function. So the predicted joint strategy - the "equilibrium concept" - varies with the external scientist's loss function. It is shown here that in many games, having a probability distribution with support restricted to Nash equilibria - as stipulated by conventional game theory - is impossible. It is also show how to: i) Derive an information-theoretic quantification of a player's degree of rationality; ii) Derive bounded rationality as a cost of computation; iii) Elaborate the close formal relationship between game theory and statistical physics; iv) Use this relationship to extend game theory to allow stochastically varying numbers of players.