An Existence Result for Hierarchical Stackelberg v/s Stackelberg Games

In a hierarchical Stackelberg v/s Stackelberg game, a collection of players called leaders play a Nash game constrained by the equilibrium conditions of a distinct Nash game played amongst another set of players, called followers. Generically, follower equilibria are non-unique as a function of leader strategies and the decision problems of leaders are highly nonconvex and lacking in standard regularity conditions. Consequently, the provision of sufficient conditions for the existence of global or even local equilibria remains a largely open question. In this paper, we present what is possibly the first general existence result for equilibria for this class of games. Importantly, we impose no single-valuedness assumption on the equilibrium of the follower-level game. Specifically, under the assumption that the objectives of the leaders admit a quasi-potential function, a notion introduced in this paper, the global and local minimizers of a suitably defined optimization problem are shown to be the global and local equilibria of the game. In effect, existence of equilibria can be guaranteed by the solvability of an optimization problem which holds under mild conditions. We motivate quasi-potential games through an application in communication networks.