Mean field asymptotics of Markov decision evolutionary games and teams

We introduce Markov Decision Evolutionary Games with N players, in which each individual in a large population interacts with other randomly selected players. The states and actions of each player in an interaction together de-termine the instantaneous payoff for all involved players. They also determine the transition probabilities to move to the next state. Each individual wishes to maximize the to-tal expected discounted payoff over an infinite horizon. We provide a rigorous derivation of the asymptotic behavior of this system as the size of the population grows to infin-ity. We show that under any Markov strategy, the random process consisting of one specific player and the remaining population converges weakly to a jump process driven by the solution of a system of differential equations. We char-acterize the solutions to the team and to the game prob-lems at the limit of infinite population and use these to con-struct almost optimal strategies for the case of a finite, but large, number of players. We show that the large popu-lation asymptotic of the microscopic model is equivalent to a (macroscopic) Markov decision evolutionary game in which a local interaction is described by a single player against a population profile. We illustrate our model to de-rive the equations for a dynamic evolutionary Hawk and Dove game with energy level. 1