Generalized replicator dynamics based on mean-field pairwise comparison dynamic

TL;DR

This paper introduces a new class of replicator dynamics derived from mean field games, providing a novel interpretation and numerical methods for these models, with applications in potential games and energy management.

Contribution

It presents an inverse control approach linking mean field games to generalized replicator dynamics, offering new insights and computational techniques.

Findings

Derived generalized replicator dynamics from mean field game limits

Developed a finite difference method preserving probability properties

Conducted convergence studies on potential games and energy management

Abstract

The pairwise comparison dynamic is a forward ordinary differential equation in a Banach space whose solution is a time-dependent probability measure to maximize utility based on a nonlinear and nonlocal protocol. It contains a wide class of evolutionary game models, such as replicator dynamics and its generalization. We present an inverse control approach to obtain a replicator-type pairwise comparison dynamic from the large discount limit of a mean field game (MFG) as a coupled forward-backward system. This methodology provides a new interpretation of replicator-type dynamics as a myopic perception limit of the dynamic programming. The cost function in the MFG is explicitly obtained to derive the generalized replicator dynamics. We present a finite difference method to compute these models such that the conservation and nonnegativity of the probability density and bounds of the value function can be numerically satisfied. We conduct a computational convergence study of a large discount limit, focusing on potential games and an energy management problem under several conditions.