A dynamic that evolves toward a Nash equilibrium

TL;DR

This paper introduces a novel dynamic based on Hedge that converges to Nash equilibria in symmetric bimatrix games without requiring monotonicity or potential functions, advancing understanding of equilibrium computation.

Contribution

It generalizes Hedge-based dynamics to symmetric bimatrix games with diminishing learning rates, showing convergence to Nash equilibria without traditional assumptions.

Findings

Weighted empirical averages converge to equilibrium

First dynamic shown to evolve toward Nash without monotonicity

Applicable to complex symmetric bimatrix games

Abstract

In this paper, we study an exponentiated multiplicative weights dynamic based on 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 approach a minimax equilibrium in zero-sum games. We generalize that result to show that a weighted version of the empirical average converges to an equilibrium in the class of symmetric bimatrix games for a diminishing learning rate parameter. Our dynamic is the first dynamical system (whether continuous or discrete) shown to evolve toward a Nash equilibrium without assuming monotonicity of the payoff structure or that a potential function exists. Although our setting is somewhat restricted, it is also general as the class of symmetric bimatrix games captures the entire computational complexity of the PPAD class (even to approximate an equilibrium).