Total Reward Semi-Markov Mean-Field Games with Complementarity Properties

We study a class of dynamic games with a continuum of atomless players where each player controls a semi-Markov process of individual states, while the global state of the game is the aggregation of individual states of all the players. The model differs from standard models of dynamic games with continuum of players known as mean field or anonymous games in that the moments when the decisions are made are discrete, but different for each of the players. As a result, the individual states of each player follow a continuous time Markov chain, but the global state follows an ordinary differential equation. Games of this type were introduced by Gomes et al. (Appl Math Optim 68:99–143, 2013) and received some attention in the literature in last few years. In our paper we introduce a novel model of this type where players maximize their cumulative payoffs over their lifetime. We show that the payoffs of the players using any stationary strategy of a certain class in a game with continuum of players are close to those obtained in n-person counterparts of this game for n large enough. This implies that equilibrium strategies in the anonymous model can well approximate equilibria in related games with large finite number of players. In the rest of the paper we concentrate on a subclass of games where the payoff and transition probability functions exhibit some strategic complementarities between players. In that case we prove that the game possesses a stationary equilibrium. Moreover, largest and smallest equilibrium strategies are nondecreasing in the states. It also turns out that these equilibria can be well approximated using a distributed iterative procedure.