Revisiting matrix games: the concept of neighborhood invader strategies

We extend the concept of neighborhood invader strategy (NIS) to finite-dimensional matrix games and compare this concept to the evolutionarily stable strategy (ESS) concept. We show that these two concepts are not equivalent in general. Just as ESS's may not be unique, NIS's may also not be unique. However, if there is an ESS and a NIS then these strategies must be the same. We show that an ESNIS (an ESS and NIS) for any matrix game is unique and that a mixed ESS with full support is a NIS. Thus a mixed ESS with full support is not invadable by any pure or mixed strategy and it can invade any pure or mixed strategy. An ESS which is an ESNIS, therefore, has better chance of being established evolutionarily through dynamic selection.