Choice-theoretic Solutions for Strategic Form Games

We model a player’s decision as a choice set and analyze equilibria in which each player’s choice set is a best response to the other players’ sets. We formalize the notion of best response by an abstract “choice structure,” which captures iteratively undominated strategies (for several definitions of dominance), rationalizability, and a number of formulations of choice sets. We investigate properties of choice structures and provide a general existence result for choice-theoretic solutions. We give sufficient conditions for uniqueness of a maximal solution, we provide a robust iterative procedure for computing this solution, and we show that it encompasses the strategy profiles possible under common knowledge of the choice structure. We also give sufficient conditions for uniqueness of a minimal solution for a class of games that includes two-player games with Pareto optimal payoffs and n-player games with a unique mixed strategy equilibrium. Our uniqueness results for maximal solutions explain many known features of iterative elimination of strictly dominated strategies, as well as regularities observed in the literature on rationalizability, and they apply to a number of new choice structures. Our uniqueness result for minimal solutions generalizes Shapley’s (1964) uniqueness result for the saddle of a two-player, zero-sum game, and it provides conditions under which there is a unique minimal rationalizable set. ∗This paper synthesizes and supersedes “Dominance-based Solutions for Strategic Form Games,” by the authors, and “A Non-binary Approach to Duggan and Le Breton’s Dominance-based Solutions,” by the first author. We thank seminar participants at the 1996 Canadian Economic Theory Meetings and at the University of Iowa, University of Chicago, and Rutgers University.