Evolution toward a Nash equilibrium

TL;DR

This paper proves that the Hedge algorithm's average iterates converge to Nash equilibria in symmetric bimatrix games, providing a polynomial-time approximation scheme that impacts computational complexity theory.

Contribution

It generalizes convergence results of Hedge from zero-sum to symmetric bimatrix games and introduces a fully polynomial-time approximation scheme for symmetric equilibria.

Findings

Empirical averages of Hedge converge to Nash equilibria in symmetric games.

The analysis yields a symmetric equilibrium fully polynomial-time approximation scheme.

Implication that P equals PPAD under the proposed scheme.

Abstract

In this paper, we study the dynamic behavior of Hedge, a well-known algorithm in theoretical machine learning and algorithmic game theory. The empirical average (arithmetic mean) of the iterates Hedge generates is known to converge to a minimax equilibrium in zero-sum games. We generalize that result to show convergence of the empirical average to Nash equilibrium in symmetric bimatrix games (that is bimatrix games where the payoff matrix of each player is the transpose of that of the other) in the sense that every limit point of the sequence of averages is an $\epsilon$-approximate symmetric equilibrium strategy for any desirable $\epsilon$. Our analysis gives rise to a symmetric equilibrium fully polynomial-time approximation scheme, implying P = PPAD.