Excess Payoff Evolutionary Dynamics With Strategy-Dependent Revision Rates: Convergence to Nash Equilibria for Potential Games

Evolutionary dynamics in the context of population games models the dynamic non-cooperative strategic interactions among many nondescript agents. Each agent follows one strategy at a time from a finite set. A game assigns a payoff to each strategy as a function of the so-called population state vector, whose entries are the proportions of the population adopting the available strategies. Each agent repeatedly revises its strategy according to a revision protocol. We focus on a well-known class of protocols that prioritizes strategies with higher excess payoffs relative to a population-weighted average. In contrast to existing work for these protocols, we allow each agent’s revision rate to depend explicitly on its current strategy. Motivated by applications and relevance to distributed optimization, we focus on potential games and investigate the population state’s convergence to the game’s Nash equilibria. Our contributions are twofold: (1) For the considered protocol class, prior work established conditions that ensure convergence under strategy-independent revision rates. We show that these conditions may be violated when the revision rates are strategy-dependent. (2) We prove that a minor, well-motivated modification of the considered protocol class satisfies these conditions for any strategy-dependent revision rates. We also illustrate our results using a distributed task allocation example.