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Fair Artificial Currency Incentives in Repeated Weighted Congestion Games: Equity vs. Equality

Leonardo Pedroso, Andrea Agazzi, W. P. M. H. Heemels, Mauro Salazar

TL;DR

This work addresses fairness in repeated weighted congestion games by introducing an artificial currency mechanism (AC) and two optimal pricing schemes. It formalizes equity (equal outcomes across weights) and equality (equal opportunity per unit weight) and proves that both can be achieved with convergence to system-optimal allocations, though exact equality may be limited by weight distribution. The equity design uses weight-independent pricing to attain perfect equity with PoA -> 1 and InEqt -> 0, while the equality design uses a bracketed, theta-parameterized approach to approximate equality as closely as possible, with PoA still converging to near-optimal levels. The results offer practical guidance for fair, scalable resource sharing under repeated interactions, with numerical experiments illustrating trade-offs and showcasing convergence to the system optimum under both schemes.

Abstract

When users access shared resources in a selfish manner, the resulting societal cost and perceived users' cost is often higher than what would result from a centrally coordinated optimal allocation. While several contributions in mechanism design manage to steer the aggregate users choices to the desired optimum by using monetary tolls, such approaches bear the inherent drawback of discriminating against users with a lower income. More recently, incentive schemes based on artificial currencies have been studied with the goal of achieving a system-optimal resource allocation that is also fair. In this resource-sharing context, this paper focuses on repeated weighted congestion game with two resources, where users contribute to the congestion to different extents that are captured by individual weights. First, we address the broad concept of fairness by providing a rigorous mathematical characterization of the distinct societal metrics of equity and equality, i.e., the concepts of providing equal outcomes and equal opportunities, respectively. Second, we devise weight-dependent and time-invariant optimal pricing policies to maximize equity and equality, and prove convergence of the aggregate user choices to the system-optimum. In our framework it is always possible to achieve system-optimal allocations with perfect equity, while the maximum equality that can be reached may not be perfect, which is also shown via numerical simulations.

Fair Artificial Currency Incentives in Repeated Weighted Congestion Games: Equity vs. Equality

TL;DR

This work addresses fairness in repeated weighted congestion games by introducing an artificial currency mechanism (AC) and two optimal pricing schemes. It formalizes equity (equal outcomes across weights) and equality (equal opportunity per unit weight) and proves that both can be achieved with convergence to system-optimal allocations, though exact equality may be limited by weight distribution. The equity design uses weight-independent pricing to attain perfect equity with PoA -> 1 and InEqt -> 0, while the equality design uses a bracketed, theta-parameterized approach to approximate equality as closely as possible, with PoA still converging to near-optimal levels. The results offer practical guidance for fair, scalable resource sharing under repeated interactions, with numerical experiments illustrating trade-offs and showcasing convergence to the system optimum under both schemes.

Abstract

When users access shared resources in a selfish manner, the resulting societal cost and perceived users' cost is often higher than what would result from a centrally coordinated optimal allocation. While several contributions in mechanism design manage to steer the aggregate users choices to the desired optimum by using monetary tolls, such approaches bear the inherent drawback of discriminating against users with a lower income. More recently, incentive schemes based on artificial currencies have been studied with the goal of achieving a system-optimal resource allocation that is also fair. In this resource-sharing context, this paper focuses on repeated weighted congestion game with two resources, where users contribute to the congestion to different extents that are captured by individual weights. First, we address the broad concept of fairness by providing a rigorous mathematical characterization of the distinct societal metrics of equity and equality, i.e., the concepts of providing equal outcomes and equal opportunities, respectively. Second, we devise weight-dependent and time-invariant optimal pricing policies to maximize equity and equality, and prove convergence of the aggregate user choices to the system-optimum. In our framework it is always possible to achieve system-optimal allocations with perfect equity, while the maximum equality that can be reached may not be perfect, which is also shown via numerical simulations.
Paper Structure (17 sections, 6 theorems, 23 equations, 1 figure)

This paper contains 17 sections, 6 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Artificial Currency Mechanism
  4. Players' Decision Model
  5. Efficiency
  6. Fairness: Equality and Equity
  7. Artificial Currency Pricing Policy Design Problem
  8. Pricing Policy Design
  9. Well-posedness
  10. Design for Equity
  11. Design for Equality
  12. Numerical Results
  13. Conclusions
  14. Proof of Theorem \ref{['thm:NE']}
  15. Proof of Theorem \ref{['thm:Eqt']}
  16. ...and 2 more sections

Key Result

Theorem 1

At time $t$, given $W$, $K_t$, $U_t$, and a pricing policy $\mathbf{p}: \mathbb{R}_{>0} \to \mathbb{R}^2$, the set of NE is nonempty and essentially unique, i.e., $\mathcal{A}^{\mathrm{NE}}_t \neq \emptyset$ and $\forall A_t^1, A_t^2 \in \mathcal{A}^{\mathrm{NE}}_t \; \mathbf{w}^{A_t^1} = \mathbf{w}

Figures (1)

  • Figure 1: Numerical simulation results with comparison between the AC policy design for equity and equality.

Theorems & Definitions (17)

  • Definition 1: NE
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • ...and 7 more