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The Limits of Fairness of the Variational Generalized Nash Equilibrium

Sophie Hall, Florian Dörfler, Heinrich H. Nax, Saverio Bolognani

TL;DR

This paper analyzes fairness in generalized Nash equilibrium problems, focusing on the variational GNE ($v$-GNE) and its implicit unit comparability assumption. It introduces a flexible solution concept, $f$-GNE, defined via a bilevel optimization that minimizes a pre-specified fairness metric $f$ over the GNE set, and discusses special cases where $f$ aligns with $v$-GNE under decoupled costs. The authors establish invariance properties of GNE and $v$-GNE under cost transformations (CNC, CUC, CFC), and illustrate the fragility of $v$-GNE fairness with an electric vehicle charging game, demonstrating how different fairness metrics yield different allocations. They also show that $f$-GNE coincides with $v$-GNE in certain structured cases (e.g., fully decoupled costs with utilitarian $f$) and discuss computational tractability for problems with a single coupling constraint, while noting that higher-dimensional cases remain an open area for future work.

Abstract

Generalized Nash equilibrium (GNE) problems are commonly used to model strategic interactions between self-interested agents who are coupled in cost and constraints. Specifically, the variational GNE, a refinement of the GNE, is often selected as the solution concept due to its non-discriminatory treatment of agents by charging a uniform ``shadow price" for shared resources. We study the fairness concept of v-GNEs from a comparability perspective and show that it makes an implicit assumption of unit comparability of agent's cost functions, one of the strongest comparability notions. Further, we introduce a new solution concept, f-GNE in which a fairness metric is chosen a priori which is compatible with the comparability at hand. We introduce an electric vehicle charging game to demonstrate the fragility of v-GNE fairness and compare it to the f-GNE under various fairness metrics.

The Limits of Fairness of the Variational Generalized Nash Equilibrium

TL;DR

This paper analyzes fairness in generalized Nash equilibrium problems, focusing on the variational GNE (-GNE) and its implicit unit comparability assumption. It introduces a flexible solution concept, -GNE, defined via a bilevel optimization that minimizes a pre-specified fairness metric over the GNE set, and discusses special cases where aligns with -GNE under decoupled costs. The authors establish invariance properties of GNE and -GNE under cost transformations (CNC, CUC, CFC), and illustrate the fragility of -GNE fairness with an electric vehicle charging game, demonstrating how different fairness metrics yield different allocations. They also show that -GNE coincides with -GNE in certain structured cases (e.g., fully decoupled costs with utilitarian ) and discuss computational tractability for problems with a single coupling constraint, while noting that higher-dimensional cases remain an open area for future work.

Abstract

Generalized Nash equilibrium (GNE) problems are commonly used to model strategic interactions between self-interested agents who are coupled in cost and constraints. Specifically, the variational GNE, a refinement of the GNE, is often selected as the solution concept due to its non-discriminatory treatment of agents by charging a uniform ``shadow price" for shared resources. We study the fairness concept of v-GNEs from a comparability perspective and show that it makes an implicit assumption of unit comparability of agent's cost functions, one of the strongest comparability notions. Further, we introduce a new solution concept, f-GNE in which a fairness metric is chosen a priori which is compatible with the comparability at hand. We introduce an electric vehicle charging game to demonstrate the fragility of v-GNE fairness and compare it to the f-GNE under various fairness metrics.

Paper Structure

This paper contains 10 sections, 5 theorems, 21 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized Nash equilibrium problems
  4. Inter-Agent Cost Comparability
  5. Comparability in Generalized Nash Equilibrium Problems
  6. Fair Generalized Nash Equilibrium (f-GNE)
  7. Case study: Electric vehicle charging game
  8. Transformations
  9. Fairness metrics
  10. Discussion and Conclusion

Key Result

Proposition 1

Let Assumptions ass:convexJi and ass:constraintSets hold. The solution set $\mathcal{S}^\text{\tiny GNE}(x^*)$ of the GNE problem eq:GNEP1 is generally not invariant under transformations $\phi_{\text{\tiny CNC}} (\cdot)$ defined in eq:CNC.

Figures (4)

  • Figure 1: Pictorial representation of the effect of the high level of interpersonal comparability assumed by the v-GNE solution concept. Additional information (that can be unmeasurable, misreported, or deemed extraneous) affects the fairness of the resulting allocation (cf. Section \ref{['ssec:EVtransformations']}).
  • Figure 2: Solution concepts for competitive games without (NE) and with coupling constraints (GNE, v-GNE) - each with their respective comparability assumptions.
  • Figure 3: Comparison of the v-GNE solution and how it changes under transformations of cost scaling, initial condition and cost function structure.
  • Figure 4: Various fairness metrics, the v-GNE, and how their solution changes with scaling $a_1 = \{1,3\}$, left to right. Note that on the left all solutions overlap.

Theorems & Definitions (12)

  • Definition 1: Generalized Nash Equilibrium
  • Definition 2: Variational GNE
  • Proposition 1: Invariance of GNE under CNC
  • proof
  • Corollary 1: Invariance of NE under CNC
  • proof
  • Proposition 2: Invariance of v-GNE under CUC
  • proof
  • Proposition 3
  • proof
  • ...and 2 more