Multi-leader multi-follower games are a class of hierarchical games in whicha collection of leaders compete in a Nash game constrained by the equilibriumconditions of another Nash game amongst the followers. The resultingequilibrium problem with equilibrium constraints is complicated by nonconvexagent problems and therefore providing tractable conditions for existence ofglobal or even local equilibria for it has proved challenging. Consequently,much of the extant research on this topic is either model specific or relies onweaker notions of equilibria. We consider a modified formulation in which everyleader is cognizant of the equilibrium constraints of all leaders. Equilibriaof this modified game contain the equilibria, if any, of the original game. Thenew formulation has a constraint structure called shared constraints, and ourmain result shows that if the leader objectives admit a potential function, theglobal minimizers of the potential function over the shared constraint areequilibria of the modified formulation. We provide another existence resultusing fixed point theory that does not require potentiality. Additionally,local minima, B-stationary, and strong-stationary points of this minimizationare shown to be local Nash equilibria, Nash B-stationary, and Nashstrong-stationary points of the corresponding multi-leader multi-follower game.We demonstrate the relationship between variational equilibria associated withthis modified shared-constraint game and equilibria of the original game fromthe standpoint of the multiplier sets and show how equilibria of the originalformulation may be recovered. We note through several examples that suchpotential multi-leader multi-follower games capture a breadth of applicationproblems of interest and demonstrate our findings on a multi-leadermulti-follower Cournot game.
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