Team Variance Optimization of n-Player Stochastic Games with Separately Controlled Chains

In this paper, we study a subclass of n-player stochastic games, in which each player has their own internal state controlled only by their own action and their objective is a common goal called team variance which measures the total variation of the random rewards of all players. It is assumed that players cannot observe each others' state/action. Thus, players' internal chains are controlled separately by their own action and they are coupled through the objective of team variance. Since the variance metric is not additive or Markovian, the dynamic programming principle fails in this problem. We study this problem from the viewpoint of sensitivity-based optimization. A difference formula and a derivative formula for team variance with respect to policy perturbations are derived, which provide sensitivity information to guide decentralized optimization. The existence of a stationary pure Nash equilibrium policy is derived. We further develop a bilevel optimization algorithm that iteratively updates the team mean at the outer level and minimizes the team variance at the inner level, where the team mean serves as a signal to coordinate the optimization of n players in a decentralized manner. We prove that the algorithm can converge to a strictly local minimum or a first-order stationary point in the space of mixed policies. Finally, we demonstrate the effectiveness of our approach using a numerical experiment of energy management in smart grid, where the assumption of separately controlled chains holds naturally.