Informationally Nonunique Equilibrium Solutions in Differential Games

Abstract

This paper is concerned with two-person deterministic nonzero-sum differential games (NZSDG) characterized by quadratic objective functionals and with state dynamics described by linear differential equations. It is first shown that such games admit uncountably many (informationally nonunique) noncooperative (Nash) equilibrium solutions when at least one of the players has access to dynamic information. We provide a characterization of all Nash equilibrium solutions to the problem for a particular dynamic information pattern, and propose an optimal unique selection of an element of the Nash equilibrium set, which exhibits a robust behavior by being insensitive to additive random perturbations in the state dynamics. We model these random perturbations as a local martingale process and obtain the abovementioned optimal Nash strategy pair as the unique noncooperative equilibrium solution of a related stochastic NZSDG. With regard to the latter, it is shown that the unique Nash equilibrium strategy of the player with dynamic closed-loop information can be realized by affine control laws.