Distributionally consistent price taking equilibria in stochastic dynamic games

Abstract

We consider a non-cooperative multi-stage game with discrete-time state dynamics. Players have their own decoupled state dynamics and each player wishes to minimize its own expected total cost. The salient aspect of our model is that each player's stage cost includes a payment (e.g., to a public utility) proportional to the magnitude of the player's decision. The coefficient multiplying each player's decision, called the price, is the same for all players and is determined as a function of the average of all player's decisions at that stage. Hence, each player's cost depends on the decisions of the other players only through the price. Here, we provide a stochastic and dynamic generalization of an equilibrium concept adopted in the economics literature, called the price-taking equilibrium, at which each player has no incentive to unilaterally deviate from its equilibrium strategy provided that the player ignores the effect of its own decisions on the price. In our setup, we allow for stochasticity in the price process and players observe only the past price realizations in addition to their own state realizations and their own past decisions. At a price-taking equilibrium, if players are given the distribution of the price process as if the price process is exogenous, they would have no incentive to unilaterally deviate from their equilibrium strategies. The main contribution of this paper is to establish such a stochastic and dynamic game generalization of price taking equilibria. We first derive the conditions for the existence of a price-taking equilibrium in the special case where the state dynamics are linear, the stage cost are quadratic, and the price function is linear. In this special case, our existence results are constructive for both finite-horizon and infinite horizon-problems. In the case where the number of players is taken to infinity, a price taking equilibrium exists which in turn is a mean-field equilibrium and is thus actually a Bayesian Nash equilibrium unlike the setup with a finite number of players. Finally, non-constructive existence results for price-taking equilibria and asymptotic equivalence with Nash equilibria are obtained for the case where the state and action sets are finite.