Risk-sensitive mean field stochastic games

Recently, there has been much interest in understanding the behavior of large-scale systems in dynamic environment. The complexity of the analysis of large-scale systems is dramatically reduced by exploiting the mean field approach leading to macroscopic dynamical systems. Under regularity assumptions and specific time-scaling techniques the evolution of the mean field limit can be expressed in deterministic or stochastic equation or inclusion (difference or differential). In this paper, we study a risk-sensitive mean field stochastic game with discounted and total payoff criterion. We provide a risk-sensitive mean field system for the long-term total payoff and derive backward-forward mean field equations. In contrast to risk-neutral discounted case, we show the non-existence of stationary mean field response in a simple scenario with two actions for each generic player.