Index Theory for Strategic-Form Games With an Application to Extensive-Form Games

Abstract

Whenever equivalent mixed strategies of a player are identified (topologically) in a normal-form game, the resulting space may not be a simplex anymore but is a general polytope. We show that an index/degree theory of equilibria can be developed in full generality for games in which the strategy sets of the players are general polytopes and their payoff functions are multiaffine. Index and degree theories work as a tool that helps identify equilibria that are robust to payoff perturbations of the game. Because the strategy set of each player is the result of the identification of equivalent mixed strategies, the resulting polytope is of lower dimension than the original mixed strategy simplices. This, together with an index theory, has algorithmic applications for checking for robustness of equilibria as well as finding equilibria in extensive-form games.