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Unformatted text preview: A Stochastic Evolutionary Game Approach to Energy Management in a Distributed Aloha Network Eitan Altman * , Yezekael Hayel * INRIA, 2004 Route des Lucioles, 06902 Sophia Antipolis, France. E-mail: firstname.lastname@example.org LIA, Universite dAvignon 339, chemin des Meinajaries, Agroparc BP 1228, 84911 Avignon Cedex 9, France. E-mail: email@example.com Abstract A major contribution of biology to competitive decision making is the area of evolutionary games. It describes the evolution of sizes of large populations as a result of many local interactions, each involving a small number of randomly selected individuals. An individual plays only once; it plays in a one shot game against another randomly selected player with the goal of maximizing its utility (fitness) in that game. We introduce here a new more general type of games: a Stochastic Evolutionary Game where each player may be in different states; the player may be involved in several local interactions during his life time and his actions determine not only the utilities but also the transition probabilities and his life duration. This is used to study a large population of mobiles forming a sparse ad-hoc network, where mobiles compete with their neighbors on the access to a radio channel. We study the impact of the level of energy in the battery on the aggressiveness of the access policy of mobiles at equilibrium. We obtain properties of the ESS (Evolutionary Stable Strategy) equilibrium which, Unlike the Nash equilibrium concept, is robust against deviations of a whole positive fraction of the population. We further study dynamical properties of the system when it is not in equilibrium. I. INTRODUCTION The evolutionary games formalism is a central mathematical tools developed by biologists for predicting population dynam- ics in the context of interactions between populations . This formalism identifies and studies two concepts: The ESS (for Evolutionary Stable Strategy ), and the Replicator Dynamics . The ESS is characterized by a property of robustness against invaders (mutations). More specifically, (i) if an ESS is reached, then the proportions of each population do not change in time. (ii) at ESS, the populations are immune from being in- vaded by other small populations. This notion is stronger than Nash equilibrium in which it is only requested that a single user would not benefit by a change (mutation) of its behavior. ESS has first been defined in 1972 by M. Smith , who further developed it in his seminal text Evolution and the Theory of Games , followed shortly by Axelrods famous work . Although ESS has been defined in the context of biological systems, it is highly relevant to engineering as well (see ). In the biological context, the replicator dynamics is a model for the change of the size of the population(s) as biologist observe, where as in engineering, we can go beyond characterizing and modelling existing evolution. The evolution of protocols canmodelling existing evolution....
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- Spring '09