stochastic-evolutionary games - Markov Decision...

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Markov Decision Evolutionary Games Eitan Altman and Yezekael Hayel March 28, 2008 Abstract We present a class of evolutionary games involving large populations that have many pairwise interactions between randomly selected players. The fitness of a player depends not only on the actions chosen in the inter- action but also on the individual state of the players. Players have finite life time and take during which they participate in several local interac- tions. The actions taken by a player determine not only the immediate fitness but also the transition probabilities to its next individual state. We define and characterize the Evolutionary Stable Strategies (ESS) for these games and propose a method to compute them. We illustrate the model and results through a networking problem. 1 Introduction Evolutionary games have been developed by J. Meynard Smith to model the evolution of population sizes as a result of competition between them that oc- curs through many local pairwise interactions, i.e. interactions between ran- domly chosen pairs of individuals. Central in evolutionary games is the concept of Evolutionary Stable Strategy, which is a distribution of (deterministic or mixed) actions such that if used, the population is immune against penetration of mutations. This notion is stronger than that of Nash equilibrium as ESS is robust against a deviation of a whole fraction of the population where as the Nash equilibrium is defined with respect to possible deviations of a single player. A second foundation of evolutionary games is the replicator dynamics that describes the dynamics of the sizes of the populations as a result of the fitness they receive in interactions. Meynard Smith formally introduced both, without needing an explicit modeling of stochastic features. We shall call this the deterministic evolutionary game. Randomness is implicitly hinted in the requirement of robustness against mutations, and indeed the ESS is defined through robustness against any muta- tion. Random aspects can be explicitly included in the modeling of evolutionary games. We first note that since deterministic evolutionary games deal with large populations, they may provide an interpretation of the deterministic game as a limit smaller games that included randomness that has been averaged out by some strong law of large numbers. Such an interpretation can be found in [9]. 1
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Yet, other sources of randomness have been introduced into evolutionary games. Some authors have added small noise to the replicator dynamics in order to avoid the problem of having the dynamics stuck in some local minimum, see [13, 10, 11] and references therein. The ESS can then be replaced by other notions such as the the SSE [10]. In this paper we introduce another class of stochastic evolutionary games, which we call ”Markov Decision Evolutionary Games” (MDEG). There are again many local interactions among individuals belonging to large populations of players. Each individual has a finite life time during which (i) it may move
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