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Fair Interventions in Weighted Congestion Games

Miriam Fischer, Martin Gairing, Dario Paccagnan

TL;DR

The paper addresses how fairness constraints affect the ability to improve equilibrium quality in weighted congestion games, proving an unconditional lower bound on PoA that depends only on latency class. It then provides a fair, optimal, polynomial-time taxation mechanism that matches this bound, and shows that no polynomially computable intervention can do better, even without fairness, establishing tight hardness results. The approach hinges on a Lifted LP relaxation inspired by the diseconomy-of-scale framework, a Poisson-based taxation recursion, and a polynomial-latency specialization yielding explicit, computable taxes with PoA bounded by Bell numbers. This work thus positions fair interventions as a powerful yet fundamentally constrained tool, offering both optimal mechanisms and a clear boundary set by computational hardness and latency structure, with practical implications for designing fair, scalable congestion-management policies.

Abstract

In this work we study the power and limitations of fair interventions in weighted congestion games. Specifically, we focus on interventions that aim at improving the equilibrium quality (price of anarchy) and are fair in a suitably defined sense. Within this setting, we provide three key contributions. First, we show that no fair intervention can reduce the price of anarchy below a given factor depending solely on the class of latencies considered. Interestingly, this lower bound is unconditional, i.e., it applies regardless of how much computation interventions are allowed to use. Second, we design a taxation mechanism that is fair and achieves a price of anarchy matching this unconditional lower bound, all the while being polynomial-time computable. Third, we show that no intervention (fair or not) can achieve a better approximation if polynomial computability is required. We do so by proving that the minimum social cost is NP-hard to minimize below a factor identical to the one previously introduced. In doing so, our work shows that the algorithm proposed by Makarychev and Sviridenko (Journal of the ACM, 2018) to tackle optimization problems with a "diseconomy of scale" is optimal, and provide a novel way to derandomize its solution via equilibrium computation.

Fair Interventions in Weighted Congestion Games

TL;DR

The paper addresses how fairness constraints affect the ability to improve equilibrium quality in weighted congestion games, proving an unconditional lower bound on PoA that depends only on latency class. It then provides a fair, optimal, polynomial-time taxation mechanism that matches this bound, and shows that no polynomially computable intervention can do better, even without fairness, establishing tight hardness results. The approach hinges on a Lifted LP relaxation inspired by the diseconomy-of-scale framework, a Poisson-based taxation recursion, and a polynomial-latency specialization yielding explicit, computable taxes with PoA bounded by Bell numbers. This work thus positions fair interventions as a powerful yet fundamentally constrained tool, offering both optimal mechanisms and a clear boundary set by computational hardness and latency structure, with practical implications for designing fair, scalable congestion-management policies.

Abstract

In this work we study the power and limitations of fair interventions in weighted congestion games. Specifically, we focus on interventions that aim at improving the equilibrium quality (price of anarchy) and are fair in a suitably defined sense. Within this setting, we provide three key contributions. First, we show that no fair intervention can reduce the price of anarchy below a given factor depending solely on the class of latencies considered. Interestingly, this lower bound is unconditional, i.e., it applies regardless of how much computation interventions are allowed to use. Second, we design a taxation mechanism that is fair and achieves a price of anarchy matching this unconditional lower bound, all the while being polynomial-time computable. Third, we show that no intervention (fair or not) can achieve a better approximation if polynomial computability is required. We do so by proving that the minimum social cost is NP-hard to minimize below a factor identical to the one previously introduced. In doing so, our work shows that the algorithm proposed by Makarychev and Sviridenko (Journal of the ACM, 2018) to tackle optimization problems with a "diseconomy of scale" is optimal, and provide a novel way to derandomize its solution via equilibrium computation.
Paper Structure (20 sections, 12 theorems, 56 equations, 1 table)

This paper contains 20 sections, 12 theorems, 56 equations, 1 table.

Table of Contents

  1. Introduction
  2. Our Contributions and Their Implications
  3. Techniques
  4. Related Work
  5. Weighted Congestion Games and Taxation Mechanisms
  6. Roadmap
  7. Lower bound on Price of Anarchy of Fair Interventions
  8. Fair, optimal, and efficiently-computable Taxes
  9. LP Relaxation of MinSC
  10. Parametrized Taxation Mechanisms
  11. An Optimal Taxation Mechanism
  12. Tight Polynomial Algorithms
  13. Polynomial Latency Functions
  14. NP-hardness of Approximation and Limitations of Unfair Interventions
  15. Appendix
  16. ...and 5 more sections

Key Result

Theorem 1

In congestion games where all resources have the same latency function $\ell$, there exists no fair intervention that induces a price of anarchy lower than $\rho_\ell - \varepsilon$, for any $\varepsilon>0$, where and we define $\ell(0)=0$. If $\rho_\ell=\infty$, the price of anarchy of any fair intervention is unbounded.

Theorems & Definitions (23)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • Corollary 1
  • Theorem 3
  • proof
  • Lemma 2
  • proof
  • ...and 13 more