On the consistency of leaders' conjectures in hierarchical games

In multi-leader multi-follower games, a set of leaders compete in a Nash game, while anticipating the equilibrium arising from a game between a set of followers. Conventional formulations are complicated by several concerns. First, since follower equilibria need not be unique, conjectures made by leaders regarding follower equilibria may not be consistent at equilibrium. When the follower equilibrium is a physical quantity to be exchanged, one is led to ask whether an equilibrium without consistent conjectures is even sensible. Second, these games are often irregular and nonconvex and no general sufficiency conditions for existence of equilibria are known. Third, no globally convergent algorithms for computing equilibria are known. We show that these concerns are addressed en masse by a modified model we introduce in this paper. In this model each leader makes conjectures while also requiring that his conjectures are consistent with those made by other leaders. If leader payoff functions admit a potential function, then under mild conditions, this model admits an equilibrium. At equilibrium, the conjectures of leaders are necessarily consistent, and when there is a unique follower equilibrium, the equilibria of the original model are equilibria of the new model. Preliminary empirical evidence suggests that such equilibria are also significantly easier to compute.