In this paper, we provide a theoretical prediction of the way in which adaptive players behave in the long run in games with strict Nash equilibria. In the model, each player picks the action which has the highest assessment, which is a weighted average of past payoffs. Each player updates his assessment of the chosen action in an adaptive manner. Almost sure convergence to a Nash equilibrium is shown under one of the following conditions: (i) that, at any non-Nash equilbrium action profile, there exists a player who can find another action which gives always better payoffs than his current payoff, (ii) that all non-Nash equilibrium action profiles give the same payoff. We show almost sure convergence to a Nash equilibrium in the following games: pure coordination games; the battle of the sexes games; the stag hunt game; and the first order static game. In the game of chicken and market entry games, players may end up playing a maximum action profile.
