Stochastic Adaptive Linear Quadratic Differential Games

Abstract

Game theory is playing more and more important roles in understanding complex systems and in investigating intelligent machines with various uncertainties. As a starting point, we consider the classical two-player zero-sum linear-quadratic stochastic differential games, but in contrast to most of the existing studies, the coefficient matrices of the systems are assumed to be unknown to both players, and consequently it is necessary to study adaptive strategies of the players, which may be termed as adaptive games and which has rarely been explored in the literature. In this paper, by introducing a suitable information structure for adaptive games, we will show that a theory can be established on adaptive strategies that are designed based on both the certainty equivalence principle and the diminishing excitation technique. Under almost the same physical structure conditions as those in the traditional known parameters case, it is shown that the closed-loop adaptive game systems will be globally stable and asymptotically reach the Nash equilibrium.