Simple learning in weakly acyclic games and convergence to Nash equilibria

We study the evolution of action profiles played by players when they revise their actions using a simple better-reply update rule. We employ a continuous-time model, in which each player is allowed to update its strategy in accordance with a Poisson process. We demonstrate that when payoff information experiences no delay and players make decisions based on up-to-date information, action profiles generated by the players converge almost surely to a Nash equilibrium if and only if the game is weakly acyclic. Interestingly, we prove that, in some cases, if the payoff information is delayed, the action profiles are guaranteed to converge almost surely to a Nash equilibrium in a strictly larger class of games which includes weakly acyclic games as special cases.