Structure versus regularity of set-valued maps in convex generalized Nash equilibrium problems in Banach spaces

TL;DR

This paper investigates the existence of solutions in convex generalized Nash equilibrium problems within Banach spaces, proposing alternative conditions like graph convexity to replace traditional lower semicontinuity assumptions, thus broadening applicability.

Contribution

It introduces graph convexity and the KKM property as new conditions for equilibrium existence, unifying and extending previous theorems in the context of Banach spaces.

Findings

Replaces lower semicontinuity with graph convexity or KKM property for equilibrium existence

Unifies several existing theorems in the literature

Extends Rosen's uniqueness condition to Banach spaces

Abstract

A generalized Nash equilibrium problem (GNEP) in Banach space consists of $N>1$ optimal control problems with couplings in both the objective functions and, most importantly, in the feasible sets. We address the existence of equilibria for convex GNEPs in Banach space. We show that the standard assumption of lower semicontinuity of the set-valued constraint maps - foundational in the current literature on GNEPs - can be replaced by graph convexity or the so-called Knaster-Kuratowski-Mazurkiewicz (KKM) property. Lower semicontinuity is often essential for obtaining upper semicontinuity of best response maps, crucial for the existence theory based on Kakutani-Fan fixed-point arguments. However, in function spaces or in settings with partial differential equation (PDE) constraints, verifying lower semicontinuity becomes much more challenging (even in convex cases), whereas graph convexity, for example, is often straightforward to check. Our results unify several existence theorems in the literature and clarify the structural role of constraint maps. We also extend Rosen's uniqueness condition to Banach spaces using a multiplier bias framework.