Equilibrium solutions in two-person quadratic decision problems with static information structures

Abstract

This paper is concerned with the class of two-person two-objective decision problems characterized by quadratic cost functions and static information structures. The primitive random variables are assumed to have a priori known but arbitrary probability, distributions with finite second-order moments. For this class of problems, sufficient conditions are derived for the existence of a unique pair of equilibrium solutions. These sufficient conditions are independent of the probabilistic structure of the problem. When the underlying probability, distribution is Gaussian and the observations of the decision makers are linear in the primitive random variables, then it is shown that these unique equilibrium solutions are linear and can be found as the unique solution of a Lyapunov-type matrix equation. For the two special extreme cases known as minimax and team problems, these linear policies constitute a saddle-point solution and a globally-optimal team solution, respectively-thus being in agreement with the existing results in the literature for these two extreme cases.