Oscillatory equilibrium in asymmetric evolutionary games: Generalizing evolutionarily stable strategy

TL;DR

This paper extends the concept of evolutionarily stable strategies to include oscillatory behaviors in asymmetric populations, linking stability to periodic orbits and information theory.

Contribution

It introduces the notions of two-species heterogeneity stable orbit and two-species information stable orbit for asymmetric evolutionary games.

Findings

Generalization of stability to oscillatory states in asymmetric games

Connection between stability and periodic orbits in replicator dynamics

Information-theoretic interpretation of evolutionary stability

Abstract

The concept of evolutionarily stability and its relation with the fixed points of the replicator equation are important aspects of evolutionary game dynamics. In the light of the fact that oscillating state of a population and individuals (or players) of different roles are quite natural occurrences, we ask the question how the concept of evolutionarily stability can be generalized so as to associate game-theoretic meaning to oscillatory behaviours of players asymmetrically interacting, i.e., if there are both intraspecific and interspecific interactions between two subpopulations in the population. We guide our scheme of generalization such that the evolutionary stability is related to the dynamic stability of the corresponding periodic orbits of a time-discrete replicator dynamics. We name the generalization of evolutionarily stable state as two-species heterogeneity stable orbit. Furthermore, we invoke the principle of decrease of relative entropy in order to associate the generalization of evolutionary stability with an information-theoretic meaning. This particular generalization is aptly termed as two-species information stable orbit.