An existence theorem for single leader multi-follower games with direct preference maps
Yutthakan Chummongkhon, Poom Kumam, Parin Chaipunya
TL;DR
The paper addresses the existence of equilibria in a single-leader multi-follower game (SLMFG) where followers solve a lower-level abstract economy described by direct preference maps, without requiring numerical objective functions. It develops a stability framework for the lower-level problem, proving upper semicontinuity of the follower response map in a metric space of constraint-preference profiles and introducing a gap function to characterize equilibria. By establishing regularity and completeness results, it shows the follower equilibrium correspondence $\mathsf{NE}(\lambda)$ is upper semicontinuous with respect to the profile $\lambda$, and that the gap function is lower semicontinuous, ensuring robustness to perturbations. Finally, it combines these results with a Weierstrass argument to prove the existence of a solution to the SLMFG with direct preference maps under suitable continuity and regularity assumptions, thereby extending prior existence results for SLMFGs that rely on objective functions in the lower level.
Abstract
This paper concerns with an existence of a solution for a single leader multi-follower game (SLMFG), where the followers jointly solve an abstract economy problem. Recall that an abstract economy problem is an extension of a generalized Nash equilibrium problem (GNEP) in the sense that the preference of each player can be described without numerical criteria. The results of this paper therefore extend the known literature concerning an existence of a solution of a SLMFG. Due to the lack of criterion functions in the lower-level, the technique we used is quite different from those that deal with GNEPs. In particular, we argue that the follower's abstract economy profiles, {\itshape i.e.,} constraint-preference couples, belong to a particular metric space where the response map is proved to be upper semicontinuous.