Feedback Stackelberg-Nash equilibria in difference games with quasi-hierarchical interactions and inequality constraints

Abstract: In this paper, we study a class of two-player deterministic finite-horizon difference games with coupled inequality constraints, where each player has two types of decision variables: one involving sequential interactions and the other simultaneous interactions. We refer to this class of games as quasi-hierarchical dynamic games and define a solution concept called the feedback Stackelberg-Nash (FSN) equilibrium. Under separability assumption on cost functions, we provide a recursive formulation of the FSN solution using dynamic programming. We show that the FSN solution can be derived from the parametric feedback Stackelberg solution of an associated unconstrained game involving only sequential interactions, with a specific choice of the parameters that satisfy certain implicit complementarity conditions. For the linear-quadratic case, we show that an FSN solution is obtained by reformulating these complementarity conditions as a single large-scale linear complementarity problem. Finally, we illustrate our results using a dynamic duopoly game with production constraints.