Index and Uniqueness of Symmetric Equilibria

Abstract

This talk presents a result which combines insights from basic game theory, polyhedral geometry, and linear algebra. All concepts will be defined and introduced with examples. In a symmetric two-player game, a symmetric equilibrium can only be dynamically stable if it has positive index. The sum of the indices of all equilibria is 1, so a unique equilibrium has index 1. The index is a topological notion related to geometric orientation, and defined in terms of the sign of the determinant of the payoffs in the equilibrium support. We prove a simple strategic characterization of the index conjectured by Josef Hofbauer: In a nondegenerate symmetric game, an equilibrium has index 1 if and only if it is the unique equilibrium in a larger symmetric game obtained by adding further strategies (it suffices to add a linear number of strategies). Our elementary proof introduces "unit-vector games" where one player's payoff matrix consists of unit vectors, and applies in a novel way simplicial polytopes. In addition, we employ a very different known result that any matrix with positive determinant is the product of three P-matrices, a class of matrices important in linear complementarity. Joint work with Anne Balthasar.