Excess Payoff Dynamics, Potential Dynamics, and Stable Games

We consider a model of evolution in games in which agents occasionally receive opportunities to switch strategies, choosing between them using a probabilistic rule. Both the rate at which revision opportunities arrive and the probabilities with which each strategy is chosen are functions of current normalized payoffs. W e call the aggregate dynamics induced by this model excess payoff dynamics. We prove that these dynamics satisfy existence, uniqueness, and continuity of solutions, respect a basic payoff monotonicity property, and have rest points at and only at the Nash equilibria of the underlying game. We show that the dynamics globally converge to Nash equilibrium in potential games. Finally, we introduce a new class of games called stable games, which include games with an interior ESS, zero sum games, and concave potential games as special cases. We show that while excess payoff dynamics can exhibit periodic behavior in stable games, a subset of these dynamics called potential dynamics always converge to equilibrium. We use this convergence result to prove that every stable game admits a unique component of Nash equilibria.