Distributed Computation of Nash Equilibria in Linear Stochastic Differential Games †

Abstract

Abstract In this paper, we study a class of n-person stochastic linear-quadratic differential games under multiple probabilistic modeling, and with each player acquiring a noisy measurement of the initial state. We obtain conditions for the existence and uniqueness of Nash equilibrium, and provide a method for iterative distributed computation of the solution. The distributed algorithm involves learning in the policy space, and it does not require that the players know each other s perception of the probabilistic model underlying the decision process. For the finite horizon problem, such an iteration converges whenever the length of the time horizon is sufficiently small, and the limit in this case is an affine policy for all players if the underlying distributions are jointly Gaussian. When the horizon is infinite, and a discount factor is used in the cost functionals, the iteration converges under conditions depending on the magnitude of the discount factor, the limiting policies being affine in the case of Gaussian distributions.