ESS, NIS And GIS For Multi-Player Game In Single Population

Evolutionary stable strategy (ESS) is the basic concept of Evolutionary Game Theory. ESS can successfully resist the invasion of other strategy. On the other hand, it is important for a strategy to enter the population occupied by other strategy if it can be the evolutionary stable choice by the population in the long run. Invader strategy is a strategy that be able to invade all established communities, so it also can be adopted by the individuals of population in the long run. We analyze the evolutionary stability from two aspects: resisting the invasion of other strategy and successfully invading the population.In this paper, we discuss the concepts of evolutionary stable strategy (ESS), neighborhood invader strategy (NIS) and global invader strategy (GIS) in matrix game and a kind of non-matrix game. The main content is the properties of ESS, NIS,GIS and relationship among them, hence some correspond conclusions.In multi-player matrix game, we show that: a GIS is always an ESS; a GIS cannot coexist with an ESS unless it is itself an ESS; If a GIS exists, then it is unique; If there is more than one ESS, then there are no GIS. We also get the conclusion that in a pair wise game MS is equivalent to ESS, GIS is globally superior and it is globally asymptotically stable in the dynamics of duplicator. Also, there are some results in multi-player games different from those in pair wise games.In non-matrix game, NIS is equivalent to ESS on the assumption that the payoff is linear. And we get the same properties as multi-player matrix game: a GIS is always an ESS; a GIS cannot coexist with an ESS unless it is itself an ESS; if a GIS exists, then it is unique; if there is more than one ESS, then there are no GIS.Finally, we discuss the invasion and non-invasion of the strategy, and get the equivalent conditions of the invasion and non-invasion in continuous-time dynamic and discrete-time dynamic.