Forward-looking evolutionary game dynamics subject to exploration cost

TL;DR

This paper introduces a novel evolutionary game dynamic model that incorporates forward-looking behavior and exploration costs, linking it to mean field games and solving it under certain conditions.

Contribution

It extends classical models by integrating forward-looking behavior and exploration costs through a coupled Hamilton-Jacobi-Bellman framework.

Findings

The model admits a unique solution under specific conditions.

The framework links exploration costs to a Lagrangian multiplier in the equations.

Computational experiments demonstrate the model in 1D and 2D cases.

Abstract

We extend classical evolutionary game dynamics based on the momentary action choices of agents by accounting for two elements: forward-looking behavior and exploration cost. We focus on pairwise comparison protocols that cover major evolutionary game dynamics, such as replicator and logit models. In the proposed mathematical framework, agents update their actions by paying a cost so that a utility or its relative difference is maximized. We show that forward-looking behavior can be modeled as a coupling between the evolutionary game dynamic and static Hamilton-Jacobi-Bellman equation: a mean field game. The exploration cost and its constraint are naturally related to these equations as a function of the optimal Lagrangian multiplier serving as a relaxation parameter, and it is incorporated into the game as a constraint. We show that under certain conditions, our evolutionary game dynamic admits a unique solution. Finally, we computationally investigate one- and two-dimensional problems.