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Decentralized Fair Division

Joel Miller, Rishi Advani, Ian Kash, Chris Kanich, Lenore Zuck

TL;DR

This work investigates decentralized fair division as a complement to centralized resource allocation. By endowing agents with fixed utilities $e_i$ and additive valuations $v_i$, it defines a decentralized process whose converged state allocates each good to the agent maximizing $u_i(g)=\log(v_i(g)+e_i)-\log(e_i)$, effectively aligning decisions with the ratio $v_i(g)/e_i$ and approximating Nash welfare. The paper proves welfare guarantees that bound the decentralized solution's performance relative to the optimum, and introduces Endowment-Relative Envy-Freeness (EREF) as a fairness criterion, with probabilistic guarantees under common valuation distributions. It also provides extensive experiments showing the decentralized approach can outperform centralized allocations in the presence of endowment disparities and demonstrates the value of hybrid schemes that combine both paradigms. Altogether, the results illuminate the trade-offs between fairness and welfare in decentralized vs centralized fair division and offer guidance for real-world systems that blend local knowledge with centralized planning.

Abstract

Fair division is typically framed from a centralized perspective. We study a decentralized variant of fair division inspired by the dynamics observed in community-based targeting, mutual aid networks, and community resource management paradigms. We develop an approach for decentralized fair division and compare it with a centralized approach with respect to fairness and social welfare guarantees. In the context of the existing literature, our decentralized model can be viewed as a relaxation of previous models of sequential exchange in light of impossibility results concerning the inability of those models to achieve desirable outcomes. We find that in settings representative of many real world situations, the two models of resource allocation offer contrasting fairness and social welfare guarantees. In particular, we show that under appropriate conditions, our model of decentralized allocation can ensure high-quality allocative decisions in an efficient fashion.

Decentralized Fair Division

TL;DR

This work investigates decentralized fair division as a complement to centralized resource allocation. By endowing agents with fixed utilities and additive valuations , it defines a decentralized process whose converged state allocates each good to the agent maximizing , effectively aligning decisions with the ratio and approximating Nash welfare. The paper proves welfare guarantees that bound the decentralized solution's performance relative to the optimum, and introduces Endowment-Relative Envy-Freeness (EREF) as a fairness criterion, with probabilistic guarantees under common valuation distributions. It also provides extensive experiments showing the decentralized approach can outperform centralized allocations in the presence of endowment disparities and demonstrates the value of hybrid schemes that combine both paradigms. Altogether, the results illuminate the trade-offs between fairness and welfare in decentralized vs centralized fair division and offer guidance for real-world systems that blend local knowledge with centralized planning.

Abstract

Fair division is typically framed from a centralized perspective. We study a decentralized variant of fair division inspired by the dynamics observed in community-based targeting, mutual aid networks, and community resource management paradigms. We develop an approach for decentralized fair division and compare it with a centralized approach with respect to fairness and social welfare guarantees. In the context of the existing literature, our decentralized model can be viewed as a relaxation of previous models of sequential exchange in light of impossibility results concerning the inability of those models to achieve desirable outcomes. We find that in settings representative of many real world situations, the two models of resource allocation offer contrasting fairness and social welfare guarantees. In particular, we show that under appropriate conditions, our model of decentralized allocation can ensure high-quality allocative decisions in an efficient fashion.
Paper Structure (26 sections, 13 theorems, 92 equations, 15 figures, 1 table)

This paper contains 26 sections, 13 theorems, 92 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Community based targeting.
  3. Rotating savings and credit associations.
  4. Collective resource management.
  5. Mutual aid networks, gift economies, and non-com-petitive spheres of exchange.
  6. Overview of our Model
  7. Our Contributions
  8. Welfare guarantees.
  9. Fairness guarantees.
  10. Experiments.
  11. Related Work on Fair Division
  12. Our Decentralized Model as an Extension of the Literature.
  13. Model and Problem Setup
  14. Centralized Allocation
  15. Decentralized Allocation
  16. ...and 11 more sections

Key Result

Proposition 1

For all agents $i$, $u_i(A_i) \geq u^*_i(A_i)$.

Figures (15)

  • Figure 1: Approximation ratios of the allocative schemes as a function of disparity in endowments. Tick marks show standard error.
  • Figure 2: Approximation ratios for $n=10$ and $m=10$.
  • Figure 3: Approximation ratios for $n=10$ and $m=20$.
  • Figure 4: Approximation ratios for $n=10$ and $m=30$.
  • Figure 5: Approximation ratios for $n=10$ and $m=40$.
  • ...and 10 more figures

Theorems & Definitions (32)

  • Definition 1: Nash welfare and Nash welfare maximization
  • Definition 2: envy-freeness up to 1 good (EF1)
  • Definition 3: Endowment-relative envy-freeness (EREF)
  • Example 1: The decentralized solution can approximate the maximum Nash welfare arbitrarily poorly
  • Example 2: The centralized solution can approximate the maximum Nash welfare arbitrarily poorly
  • Proposition 1
  • proof
  • Proposition 2
  • Corollary 1
  • Theorem 1
  • ...and 22 more