Optimal Strategy Revision in Population Games: A Mean Field Game Theory Perspective

Abstract

This paper investigates the design of optimal strategy revision in Population Games (PG) by establishing its connection to finite-state Mean Field Games (MFG). Specifically, by linking Evolutionary Dynamics (ED) -- which models agent decision-making in PG -- to the MFG framework, we demonstrate that optimal strategy revision can be derived by solving the forward Fokker-Planck (FP) equation and the backward Hamilton-Jacobi (HJ) equation, both central components of the MFG framework. Furthermore, we show that the resulting optimal strategy revision, which maximizes each agent's payoffs over a finite time horizon, satisfies two key properties: positive correlation and Nash stationarity, which are essential for ensuring convergence to the Nash equilibrium. This convergence is then rigorously analyzed and established. Additionally, we discuss how different design objectives for the optimal strategy revision can recover existing ED models previously reported in the PG literature. Numerical examples are provided to illustrate the effectiveness and improved convergence properties of the optimal strategy revision design.