Equilibrium Selection in Replicator Equations Using Adaptive-Gain Control

TL;DR

This paper introduces an adaptive-gain control method within replicator equations to reliably steer populations toward desired equilibria in strategic games, overcoming limitations of traditional open-loop strategies.

Contribution

It presents a novel closed-loop control approach with adaptive gain for equilibrium selection in replicator dynamics, requiring minimal prior knowledge of the game.

Findings

Controller guarantees convergence to desired equilibrium in most 2-action games.

Limited prior information needed for controller design.

Numerical simulations validate theoretical results.

Abstract

In this paper, we deal with the equilibrium selection problem, which amounts to steering a population of individuals engaged in strategic game-theoretic interactions to a desired collective behavior. In the literature, this problem has been typically tackled by means of open-loop strategies, whose applicability is however limited by the need of accurate a priori information on the game and scarce robustness to uncertainty and noise. Here, we overcome these limitations by adopting a closed-loop approach using an adaptive-gain control scheme within a replicator equation -a nonlinear ordinary differential equation that models the evolution of the collective behavior of the population. For most classes of 2-action matrix games we establish sufficient conditions to design a controller that guarantees convergence of the replicator equation to the desired equilibrium, requiring limited a-priori information on the game. Numerical simulations corroborate and expand our theoretical findings.