On the existence of stable equilibria in monotone games

Although the local stability of some equilibrium can be seen as a weak requirement for a game to possess reasonable equilibrium predictions, conditions under which such equilibria generally exist remain an open question, as we give examples of monotone games in which no stable equilibrium exists. Our main result shows that in monotone games, very weak assumptions on the local strictness of best responses imply the existence of an equilibrium satisfying a rather strong form of local stability, in that all possible adaptive learning process beginning within a neighborhood of that equilibrium remain in that neighborhood and eventually converge back. In the special case of two-player games, no conditions on local strictness are required at all. Finally, we show how these results lead naturally to results on equilibrium selection, in the sense that in a game with two equilibria, exactly one is locally stable. Relying only on the monotonicity of best responses, we provide examples in which our methods apply, but standard differential methods do not.