Average testing and Pareto efficiency

We propose a simple adaptive procedure for playing strategic games: average testing. In this procedure each player sticks to her current strategy if it yields a payoff that exceeds her average payoff by at least some fixed ε>0; otherwise she chooses a strategy at random. We consider generic two-person games where both players play according to the average testing procedure on blocks of k-periods. We demonstrate that for all k large enough, the pair of time-average payoffs converges (almost surely) to the 3ε-Pareto efficient boundary.