Equilibrium strategies in dynamic games with multi-levels of hierarchy

Al~traet-This paper considers noncooperative equilibria of loop information structure, the optimal responses will almost three-player dynamic games with three levels of hierarchy in always be nonunique. This difficulty is of course circumvented decision making. In this context, first a general definition of a if one confines the analysis to open-loop strategy spaces hierarchical equilibrium solution is given, which also accounts (Medanic and Radojevic, 1978), or to linear feedback strategy for nonunique responses of the players who are not at the top spaces (Medanic, 1977) in linear-quadratic games; but if the of the hierarchy. Then, a general theorem is proven which players have access to closed-loop information structure, a provides a set of sufficient conditions for a triple of strategies more general definition that also accounts for nonunique to be in hierarchical equilibrium. When applied to linear-responses is unavoidable. Such a general definition has in fact quadratic games, this theorem provides conditions under been given in Bas, ar and Selbuz (1979) within the context of which there exists a linear one-step memory strategy for the dynamic games with only two-levels of hierarchy in decision player (say, ~1) at the top of the hierarchy, which forces the making, and in this paper we first extend this equilibrium other two players to act in such a way so as to jointly solution concept to three levels of hierarchy. Then, for the minimize the cost function of ~1. Furthermore, there exists a general class of three-player games with three levels of linear one-step memory strategy for the second-level player hierarchy, we prove a general theorem (Theorem 1) which (say, ~2), which forces the remaining player to jointly provides a set of sufficient conditions for a triple of strategies minimize the cost function of ~2 under the declared to be in hierarchical equilibrium. equilibrium strategy of ~1. A numerical example included in Inherent in the suffficiency conditions of Theorem 1 is an the paper illustrates the results and the convergence property important feature of the hierarchical equilibrium solution that of the equilibrium strategies, as the number of stages in the is akin to the one observed in Ba~ar and Selbuz (1979) for the game becomes arbitrarily large, class of two-player games. Specifically, by announcing an appropriate strategy, the player at the top of the hierarchy can achieve an optimal cost level that is equal to the global 1. Introduction minimum of his cost function, and …