The ESS for evolutionary matrix games under time constraints and its relationship with the asymptotically stable rest point of the replicator dynamics.

Abstract

Recently we interpreted the notion of ESS for matrix games under time constraints and investigated the corresponding state in the polymorphic situation. Now we give two further static (monomorphic) characterizations which are the appropriate analogues of those known for classical evolutionary matrix games. Namely, it is verified that an ESS can be described as a neighbourhood invader strategy independently of the dimension of the strategy space in our non-linear situation too, that is, a strategy is an ESS if and only if it is able to invade and completely replace any monomorphic population which totally consists of individuals following a strategy close to the ESS. With the neighbourhood invader property at hand, we establish a dynamic characterization under the replicator dynamics in two dimensions which corresponds to the strong stability concept for classical evolutionary matrix games. Besides, in some special cases, we also prove the stability of the corresponding rest point in higher dimensions.