Stochastic multi-criteria decision problems with multi levels of hierarchy

This paper develops a solution method to obtain hierarchical noncooperative equilibria of a class of stochastic decision problems with more than two levels of hierarchy (in decision making), and wherein the decision makers (DMs) have access to nested dynamic information. In particular, we consider the three-person three-criteria quadratic decision problems wherein DM1 has access to the static observations and the controls of the other two DMs (in addition to his own static observation of the state of nature), and DM2 has access to the static observation and the control of DM3 (in addition to his own static observation). First DM1 announces his strategy and enforces it on both DM2 and DM3, and then DM2 announces his strategy and dictates it on DM3. Assuming that all three DMs are rational decision makers striving to minimize their expected costs, we first introduce the notion of hierarchical equilibrium for this decision problem, and then present a method for obtaining equilibrium strategies which exhibit the following feature: The equilibrium strategy of the decision maker (DM1) at the top of the hierarchy forces the other two decision makers to act in such a way so as to minimize jointly his expected cost function, while that of DM2 forces DM3 to minimize jointly the expected cost function of DM2 under the declared equilibrium strategy of DM1. An explicit derivation is given, in this framework, which leads to strategies that are linear in the dynamic part of the information available to DM1 and DM2. These strategies are, however, nonlinear functions of the static part of the information, even if the underlying statistics are Gaussian.