  In the paper we present a model of discrete-time mean-field game with several
populations of players. Mean-field games with multiple populations of the
players have only been studied in the literature in the continuous-time
setting. The main results of this article are the first stationary and Markov
mean-field equilibrium existence theorems for discrete-time mean-field games of
this type. We consider two payoff criteria: discounted payoff and total payoff.
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.
