Global Equilibria of Multi-leader Multi-follower Games with Shared Constraints

Multi-leader multi-follower games are a class of hierarchical games in which a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. Much of the extant research on this topic is either model specific or relies on weaker notions of equilibria. We observe that when this original game is modified to ensure that the leader problems have shared constraints, much can be said about the existence of equilibria. In particular, when the leader objectives also admit a potential function, equilibria exist under mild conditions. Although the conventional formulation of the multi-leader multi-follower game does not have shared constraints, we present certain modified formulations that do have shared constraints. We also identify conventional formulations that can be addressed by our theory without the need for modifications. When the leader objectives admit a potential function, we show that the global minima of this potential function over the shared constraint, solutions of a mathematical program with equilibrium constraints (MPEC), are equilibria of the shared constraint multi-leader multi-follower game. We also show that local minima, B-stationary points, strong-stationary points and second-order strong stationary points of this MPEC are local Nash equilibria, Nash B-stationary points, Nash strong-stationary points and Nash second-order strong stationary points of the associated multi-leader multi-follower game. We note through several examples that such potential multi-leader multi-follower games capture a breadth of application problems of interest. We also derive a general existence result that extends beyond potential games and clarifies the theoretical properties of shared-constraint games. Our results are supported by analytical and computational studies of Cournot-based multi-leader multi-follower games and spot-forward power markets under uncertainty, both of which are potential multi-leader multi-follower games. From such studies it emerges that equilibria of multi-leader multifollower games with shared constraints are often easier to compute by the Gauss-Seidel heuristic than their conventional counterparts. Furthermore, for potential games, solving an MPEC provides amongst the first convergent avenues for computing global equilibria of such games and such approaches appear to require far less effort than the Gauss-Seidel heuristic.