Towards a Theory of stochastic adaptive differential games

Complex systems with components or subsystems having game-like relationships are arguably the most complex ones. Much progress has been made in the traditional game theory over the past half a century, where the structure and the parameters are assumed to be known when the players make their decisions. However this is not the case in many practical situations where the players may have unknown parameters. To initiate a theoretical study of such problems, we consider in this paper a class of two-player zero-sum linear-quadratic stochastic differential games, assuming that the matrices associated with the strategies of the players are unknown to both players. By using the weighted least squares (WLS) estimation algorithms and a random regularization method, adaptive strategies will be constructed for both players. It is shown that both the adaptive strategies will converge to the optimal ones under some natural conditions on the true parameters of the system. To the best of our knowledge, this work seems to be the first to address adaptive stochastic differential game problems with rigorous convergence analysis.