Multiple-Population Discrete-Time Mean Field Games with Discounted and Total Payoffs: Approximation of Games with Finite Populations

In the paper we present a model of discrete-time mean-field game with multiple populations of players. Its main result shows that the equilibria obtained for the mean-field limit are approximate Markov–Nash equilibria for n-person counterparts of these mean-field games when the number of players in each population is large enough. We consider two payoff criteria: \(\beta \) $\beta$ -discounted payoff and total payoff. The existence of mean-field equilibria for games with both payoffs has been proven in our previous article, hence, the theorems presented here show in fact the existence of approximate equilibria in certain classes of stochastic games with large finite numbers of players. The results are provided under some rather general assumptions on one-step reward functions and individual transition kernels of the players. In addition, the results for total payoff case, when applied to a single population, extend the theory of mean-field games also by relaxing some strong assumptions used in the existing literature.