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

Marcelo Bongarti, Michael Hintermüller

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.

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

Abstract

A generalized Nash equilibrium problem (GNEP) in Banach space consists of 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.

Paper Structure

This paper contains 7 sections, 5 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Generalized finite games in Banach spaces
  3. Potential lack of lower semicontinuity of constraint maps
  4. The Nikaido-Isoda Function
  5. Proof of \ref{['no_lowersemicontinuity_theo_0']}
  6. Multiplier bias and uniqueness of an equilibrium for jointly convex games
  7. Conclusions and Outlook

Key Result

Theorem 3.2

Assume that ${\bf X}$ is graph-convex and that ${\rm Fix}({\bf X})$, the set of feasible fixed points of ${\bf X}$, i.e., is nonempty. Then the game $G = ({\bf X},\mathcal{J})$ has an equilibrium.

Theorems & Definitions (18)

  • Definition 2.1: GNEP
  • Definition 2.2: GNE
  • Definition 2.3: Convex GNEP
  • Definition 3.1
  • Theorem 3.2
  • Example 3.3
  • Example 3.4: PDE-constrained spot market
  • Definition 3.5
  • Theorem 3.6
  • proof
  • ...and 8 more