On the Existence of Stable Equilibria in Monotone Games

This paper shows that under very general conditions, there exists a locally stable Nash equilibrium in games of strategic complements (GSC), as well as in the more general case of games with non-decreasing best response correspondences. While it is well known that in such cases a unique equilibrium is globally stable, no equilibrium can be globally stable when multiple equilibria exist. However, the existence of a locally stable equilibrium remains an open question, as we give examples of GSC in which no stable equilibrium exists. One main advantage of our approach is that our results can be derived simply by exploiting the monotonicity properties of the game, and do not require any differentiability assumptions. Results on equilibrium refinement follow as a corollary under slightly stronger assumptions, in the sense that games with two equilibria possess exactly one locally stable equilibrium.