Adaptive-Gain Control for Equilibrium Selection in the Logit Dynamics

We study the problem of controlling evolutionary game-theoretic dynamics when agents follow sophisticated learning rules, in particular the logit protocol. Much previous work focused on settings where agents are less sophisticated learners following imitative protocols that leads to the well-known replicator dynamic. Here, we consider adaptive control schemes for the logit dynamics with the objective of steering the population to a desired equilibrium by modifying the agents’ payoff functions in a 2-action coordination game. Through the analysis of the controlled dynamics, we establish sufficient conditions for global convergence to the desired equilibrium. We find that the conditions to control the logit system have fewer requirements than those to control the replicator equation: Adaptive-gain controllers that are successful in performing their task in the logit system may fail in the replicator system. We then provide numerical simulations to illustrate and compare the amount of control effort needed to achieve the objective in the logit system versus the replicator system.