Equilibrium strategies in dynamic games with multi-levels of hierarchy

This paper considers noncooperative equilibria of three-player dynamic games with three levels of hierarchy in decision making. In this context, first a general definition of a hierarchical equilibrium solution is given, which also accounts for nonunique responses of the players who are not at the top of the hierarchy. Then, a general theorem is proven which provides a set of sufficient conditions for a triple of strategies to be in hierarchical equilibrium. When applied to linear-quadratic games, this theorem provides conditions under which there exists a linear one-step memory strategy for the player (say, J1) at the top of the hierarchy, which forces the other two players to act in such a way so as to jointly minimize the cost function of J1. Furthermore, there exists a linear one-step memory strategy for the second-level player (say, J2), which forces the remaining player to jointly minimize the cost function of J2 under the declared equilibrium strategy of J1. A numerical example included in the paper illustrates the results and the convergence property of the equilibrium strategies, as the number of stages in the game becomes arbitrarily large.