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Regularized Proportional Fairness Mechanism for Resource Allocation Without Money

Sihan Zeng, Sujay Bhatt, Alec Koppel, Sumitra Ganesh

TL;DR

This work tackles fair and incentive compatible resource allocation without monetary transfers by balancing Nash social welfare and exploitability. It introduces Regularized Proportional Fairness Network (RPF-Net), a neural network based approach that regularizes PF with a learned term to deter misreports, and provides a differentiable framework to train end-to-end through the PF optimizer. The authors establish generalization and distribution shift guarantees and empirically demonstrate that RPF-Net substantially lowers PF exploitability while preserving NSW and efficiency, outperforming prior learning based baselines like ExS-Net. The results offer a practical path to high NSW in payment-free settings, albeit with notable computational demands due to the optimization based activation. Overall, RPF-Net advances the Pareto frontier between NSW and IC without money, with robust performance under distribution shifts.

Abstract

Mechanism design in resource allocation studies dividing limited resources among self-interested agents whose satisfaction with the allocation depends on privately held utilities. We consider the problem in a payment-free setting, with the aim of maximizing social welfare while enforcing incentive compatibility (IC), i.e., agents cannot inflate allocations by misreporting their utilities. The well-known proportional fairness (PF) mechanism achieves the maximum possible social welfare but incurs an undesirably high exploitability (the maximum unilateral inflation in utility from misreport and a measure of deviation from IC). In fact, it is known that no mechanism can achieve the maximum social welfare and exact incentive compatibility (IC) simultaneously without the use of monetary incentives (Cole et al., 2013). Motivated by this fact, we propose learning an approximate mechanism that desirably trades off the competing objectives. Our main contribution is to design an innovative neural network architecture tailored to the resource allocation problem, which we name Regularized Proportional Fairness Network (RPF-Net). RPF-Net regularizes the output of the PF mechanism by a learned function approximator of the most exploitable allocation, with the aim of reducing the incentive for any agent to misreport. We derive generalization bounds that guarantee the mechanism performance when trained under finite and out-of-distribution samples and experimentally demonstrate the merits of the proposed mechanism compared to the state-of-the-art.

Regularized Proportional Fairness Mechanism for Resource Allocation Without Money

TL;DR

This work tackles fair and incentive compatible resource allocation without monetary transfers by balancing Nash social welfare and exploitability. It introduces Regularized Proportional Fairness Network (RPF-Net), a neural network based approach that regularizes PF with a learned term to deter misreports, and provides a differentiable framework to train end-to-end through the PF optimizer. The authors establish generalization and distribution shift guarantees and empirically demonstrate that RPF-Net substantially lowers PF exploitability while preserving NSW and efficiency, outperforming prior learning based baselines like ExS-Net. The results offer a practical path to high NSW in payment-free settings, albeit with notable computational demands due to the optimization based activation. Overall, RPF-Net advances the Pareto frontier between NSW and IC without money, with robust performance under distribution shifts.

Abstract

Mechanism design in resource allocation studies dividing limited resources among self-interested agents whose satisfaction with the allocation depends on privately held utilities. We consider the problem in a payment-free setting, with the aim of maximizing social welfare while enforcing incentive compatibility (IC), i.e., agents cannot inflate allocations by misreporting their utilities. The well-known proportional fairness (PF) mechanism achieves the maximum possible social welfare but incurs an undesirably high exploitability (the maximum unilateral inflation in utility from misreport and a measure of deviation from IC). In fact, it is known that no mechanism can achieve the maximum social welfare and exact incentive compatibility (IC) simultaneously without the use of monetary incentives (Cole et al., 2013). Motivated by this fact, we propose learning an approximate mechanism that desirably trades off the competing objectives. Our main contribution is to design an innovative neural network architecture tailored to the resource allocation problem, which we name Regularized Proportional Fairness Network (RPF-Net). RPF-Net regularizes the output of the PF mechanism by a learned function approximator of the most exploitable allocation, with the aim of reducing the incentive for any agent to misreport. We derive generalization bounds that guarantee the mechanism performance when trained under finite and out-of-distribution samples and experimentally demonstrate the merits of the proposed mechanism compared to the state-of-the-art.
Paper Structure (30 sections, 11 theorems, 74 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 30 sections, 11 theorems, 74 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries & Problem Formulation
  4. PF Mechanism Has a Non-Zero Exploitability
  5. Learning Approximate Payment-Free Mechanism
  6. Regularized Proportional Fairness Network
  7. Neural Network Design
  8. Training RPF-Net
  9. PF Mechanism Exploitability Evaluation & RPF-Net Differentiation
  10. Characterizing Sub-Differentiability of $f^{PF}$
  11. Composing (Sub)Gradients
  12. RPF-Net Differentiation
  13. Theoretical Guarantees
  14. Generalization Bounds
  15. Robustness to Distribution Mismatch
  16. ...and 15 more sections

Key Result

Theorem 1

Suppose that strict complementary slackness holds at solution $(a^{\star},\mu^{\star},\nu^{\star},\lambda^{\star})$ and that the demands are non-zero. When at least $NM-N$ inequality constraints in def:PF hold as equalities at $a^{\star}$, the matrix $\boldsymbol{M}$ is invertible, and the mapping f

Figures (5)

  • Figure 1: Exploitability of PF and learned RPF-Net.
  • Figure 2: RPF-Net Pipeline.
  • Figure 3: Mechanism performance in 2x2 and 10x3 systems (normalized with respect to PF mechanism)
  • Figure 4: Performance of RPF-Net trained under data containing untruthful report. Numbers normalized by NSW/exploitability/efficiency of PF mechanism.
  • Figure 5: Allocations from PF (top) and RPF-Net (bottom).

Theorems & Definitions (20)

  • Definition 1: Mechanism
  • Definition 2: Nash Social Welfare
  • Definition 3: Exploitability
  • Remark 1
  • Example 4.1
  • Theorem 1
  • Proposition 1
  • Theorem 2: Generalization Bound
  • Theorem 3
  • Definition 4
  • ...and 10 more