Evolutionary Stability of Polymorphic Population States in Continuous Games

TL;DR

This paper investigates the conditions under which finitely supported population states in continuous games are evolutionarily stable, proving that strong uninvadability implies asymptotic stability in the variational norm.

Contribution

It establishes that strongly uninvadable finitely supported states are asymptotically stable, complementing previous results about stability conditions in continuous strategy spaces.

Findings

Finitely supported states that are strongly uninvadable are asymptotically stable.

The paper provides a converse to existing results on stability of rest points in continuous games.

Stability is characterized with respect to the variational norm.

Abstract

In games with continuous strategy spaces, if a rest point of the replicator dynamics is asymptotically stable then the rest point must be finitely supported (van Veelen, M., Spreij, P., 2009. Evolution in games with a continuous action space. Econom. Theory 39 (3), 355-376). In this article, we address the converse question that is, we prove that a finitely supported population state is asymptotically stable with respect to the variational norm when it is strongly uninvadable.