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Characterization of LQG differential games with different information patterns

A discrete-time linear-quadratic-Gaussian (LQG) differential game is considered where the players have access to unshared control- and measurement-histories. The particular problem that is solved is where one adversary has access to only noisy partial information of the state while the other makes a perfect measurement of the state vector. The system dynamics are assumed linear with additive process noise. The solutions show a significant departure from previously published results. First, process noise is included in the dynamical system and a quadratic weighting in the state is included in the cost criterion. Secondly, no prior assumption is made about the structure of the strategies. Specifically, the equilibrium strategies of both players are shown to be finite-dimensional, not infinite-dimensional as was originally thought. Thirdly, it is assumed that the perfect-measurement adversary's control matrix is in the range space of the other adversary's measurement matrix. Then, by a limit of the linear-exponential-Gaussian game solution to the LQG game solution, it is seen that the partial information player avoids reproducing an estimated version of his adversary's strategy. A key feature of this problem solution is the filter structure of the player with partial measurements. In particular, the error of the filter is a Gaussian random variable whose statistics are independent of the opponent's control history. This filter also allows the partial measurement player to estimate the entire state without having to guess/estimate his opponent's strategy. It is noted that for this class of stochastic games, for all possible pure control strategy pairs, there always exist correlations using which each player can improve his performance. Since the error variance of the filter derived here is independent of the opponent's strategy and his control action, the improvements in cost for each player appears minimal. This property is used to extend the notion of a saddle point for deterministic games to that of a saddle interval in pure strategies for games with uncertainty.