On solving the closed-loop setpoint regulation problem for the replicator equation via nonlinear MPC

We address the setpoint regulation problem in evolutionary game-theoretic dynamics within population games, aiming to design an algorithm that guides the population toward a desired collective behavior - particularly, an equilibrium point. In detail, we focus on a discrete-time replicator equation, which models the collective behavior of a population engaged in two-player, two-action matrix games with every other member of the population. To tackle the problem, we develop an optimal control strategy that manipulates the payoff matrix in a closed-loop manner by adding a nonnegative gain to one of its entries. This strategy is intended to direct the population behavior toward the desired equilibrium. The control problem is resolved using nonlinear model predictive control, ensuring both closed-loop stability and recursive feasibility through appropriate selection of the terminal ingredients.