Neighborhood strong superiority and evolutionary stability of polymorphic profiles in asymmetric games

Abstract

<p style='text-indent:20px;'>In symmetric evolutionary games with continuous strategy spaces, Cressman [<xref ref-type="bibr" rid="b6">6</xref>] has proved an interesting stability result for the associated replicator dynamics relating the concepts of neighborhood superiority and neighborhood attracting for polymorphic states with respect to the weak topology. Similar stability results are also established for monomorphic profiles in 2-player asymmetric games [<xref ref-type="bibr" rid="b8">8</xref>]. In the present paper, we use the model of asymmetric evolutionary games introduced by Mendoza-Palacios and Hernández-Lerma [<xref ref-type="bibr" rid="b17">17</xref>] and obtain a stability result for polymorphic profiles in <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-player asymmetric evolutionary games with continuous action spaces using the concept of neighborhood strong superiority (Definition 2.3). In particular, we prove that neighborhood strong superior polymorphic profiles are neighborhood attracting. It is also shown that a polymorphic neighborhood strong superior profile is in fact a vector of Dirac measures. Moreover, we establish that the notion of neighborhood strong superiority does not imply strong uninvadability and vice-versa.</p>