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.
Equilibrium Selection in Replicator Equations Using Adaptive-Gain Control / Zino, Lorenzo; Ye, Mengbin; Calafiore, Giuseppe C.; Rizzo, Alessandro. - In: IEEE TRANSACTIONS ON AUTOMATIC CONTROL. - ISSN 0018-9286. - STAMPA. - 70:10(2025), pp. 6799-6814. [10.1109/tac.2025.3567253]
Equilibrium Selection in Replicator Equations Using Adaptive-Gain Control
Zino, Lorenzo;Calafiore, Giuseppe C.;Rizzo, Alessandro
2025
Abstract
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.File | Dimensione | Formato | |
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https://hdl.handle.net/11583/2999909