Equilibrium Selection in Replicator Equations Using Adaptive-Gain Control

In this article, 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 limited by the need of accurate a priori information on the game and a lack of 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.