On the evolutionary selection of sets of Nash equilibria

Abstract

It is well known for the common multi-population evolutionary dynamics applied to normal form games that a pure strategy combination is asymptotically stable if and only if it is a strict equilibrium point. We extend this result to sets and show the following. For certain regular selection dynamics every connected and closed asymptotically stable set of rest points containing a pure strategy combination is a strict equilibrium set and hence a Nash equilibrium component. A converse statement holds for two person games, for convex strict equilibrium sets and for the standard replicator dynamic.