Index and Robustness of Mixed Equilibria: An Algebraic Approach

TL;DR

This paper introduces an algebraic method to compute the index of completely mixed equilibria in finite games, exploring its relation to fixed points and payoff-robustness, with implications for game stability analysis.

Contribution

It develops a new algebraic approach for calculating the index of mixed equilibria and links it to robustness and fixed point properties in finite games.

Findings

Any integer can be the index of an isolated completely mixed equilibrium.

In monogenic equilibria, the index is limited to 0, +1, or -1.

Non-zero index correlates with payoff-robustness.

Abstract

We present a new method for computation of the index of completely mixed equilibria in finite games, based on the work of Eisenbud et al.(1977). We apply this method to solving two questions about the relation of the index of equilibria and the index of fixed points, and the index of equilibria and payoff-robustness: any integer can be the index of an isolated completely mixed equilibrium of a finite game. In a particular class of isolated completely mixed equilibria, called monogenic, the index can be $0$, $+1$ or $-1$ only. In this class non-zero index is equivalent to payoff-robustness. We also discuss extensions of the method of computation to extensive-form games, and cases where the equilibria might be located on the boundary of the strategy set.