On the Equivalence of Evolutionary Stable Strategies

Evolutionary game theory (EGT) has been widely adopted in various research fields and recently drawn great attentions in communications and networking. It is considered as an effective tool to analyze how a group of players converges to a stable equilibrium after a period of strategic interactions. Such an equilibrium strategy is defined as the evolutionarily stable strategy (ESS). There are two formal definitions for the ESS, where one of them is a literally mathematical interpretation and the other is a refined Nash equilibrium illustration, and both of them are widely used in the literature. However, the equivalence of these two definitions has not been well investigated. In this letter, we theoretically prove the equivalence of them under some sufficient condition. Moreover, we also show that the sufficient condition is general in communications and networking areas by introducing two examples.