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Almost Envy-free Allocation of Indivisible Goods: A Tale of Two Valuations

Alireza Kaviani, Masoud Seddighin, AmirMohammad Shahrezaei

TL;DR

This work tackles the open challenge of existence guarantees for $EFX$ and approximate variants in fair division under two structured valuation families: $(p,q)$-bounded valuations and restricted additive valuations. It develops a symmetric theory linking the two settings via a rank$-$virtual-value framework built around the weighted envy graph, rankpaths, and updating rules that monotonically improve a global potential and terminate. The main results include a complete $EF2X$ allocation for $(\infty,1)$-bounded valuations, an $EFX$ allocation with at most $\lfloor n/2\rfloor-1$ discarded goods, and a $\frac{\sqrt{2}}{2}$-$EFX$ guarantee for both $(\infty,1)$-bounded subadditive and restricted additive valuations, as well as an $EFX$ allocation for restricted additive valuations with $p=2$. Together these contributions illuminate a symmetric structure across valuation classes and introduce new concepts such as virtual value, rankpath, and root, offering a path toward broader $EFX$ guarantees and generalizations in fair division.

Abstract

The existence of $\textsf{EFX}$ allocations stands as one of the main challenges in discrete fair division.In this paper, we present symmetrical results on the existence of $\textsf{EFX}$ and its approximate variations for two distinct valuations: restricted additive valuations and $(p,q)$-bounded valuations introduced by Christodoulou \etal \cite{christodoulou2023fair}. In a $(p,q)$-bounded instance, each good has relevance for at most $p$ agents, and any pair of agents shares at most $q$ common relevant goods. We show that instances with $(\infty,1)$-bounded valuations admit $\textsf{EF2X}$ allocations and $\textsf{EFX}$ allocations with at most $\lfloor {n}/{2} \rfloor - 1$ discarded goods, mirroring results for the restricted additive setting \cite{akrami2022ef2x}. We also present ${({\sqrt{2}}/{2})\textsf{-EFX}}$ algorithms for both restricted additive and $(\infty,1)$-bounded subadditive settings. The symmetry of these results suggests these valuations share symmetric structures. Building on this, we propose an $\textsf{EFX}$ allocation for restricted additive valuations when $p=2$ and $q=\infty$. To achieve these results, we further develop the rank concept introduced by Farhadi \etal \cite{farhadi2021almost} and introduce several new concepts such as virtual value, rankpath, and root, which advance the overall understanding of $\textsf{EFX}$ allocations. In addition, we suggest an updating rule based on the virtual values which we believe will lead to broader and more generalized results on $\textsf{EFX}$.

Almost Envy-free Allocation of Indivisible Goods: A Tale of Two Valuations

TL;DR

This work tackles the open challenge of existence guarantees for and approximate variants in fair division under two structured valuation families: -bounded valuations and restricted additive valuations. It develops a symmetric theory linking the two settings via a rankvirtual-value framework built around the weighted envy graph, rankpaths, and updating rules that monotonically improve a global potential and terminate. The main results include a complete allocation for -bounded valuations, an allocation with at most discarded goods, and a - guarantee for both -bounded subadditive and restricted additive valuations, as well as an allocation for restricted additive valuations with . Together these contributions illuminate a symmetric structure across valuation classes and introduce new concepts such as virtual value, rankpath, and root, offering a path toward broader guarantees and generalizations in fair division.

Abstract

The existence of allocations stands as one of the main challenges in discrete fair division.In this paper, we present symmetrical results on the existence of and its approximate variations for two distinct valuations: restricted additive valuations and -bounded valuations introduced by Christodoulou \etal \cite{christodoulou2023fair}. In a -bounded instance, each good has relevance for at most agents, and any pair of agents shares at most common relevant goods. We show that instances with -bounded valuations admit allocations and allocations with at most discarded goods, mirroring results for the restricted additive setting \cite{akrami2022ef2x}. We also present algorithms for both restricted additive and -bounded subadditive settings. The symmetry of these results suggests these valuations share symmetric structures. Building on this, we propose an allocation for restricted additive valuations when and . To achieve these results, we further develop the rank concept introduced by Farhadi \etal \cite{farhadi2021almost} and introduce several new concepts such as virtual value, rankpath, and root, which advance the overall understanding of allocations. In addition, we suggest an updating rule based on the virtual values which we believe will lead to broader and more generalized results on .
Paper Structure (25 sections, 34 theorems, 34 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 25 sections, 34 theorems, 34 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Basic Concepts
  3. Weighted Envy Graph, Rank, and Virtual Value.
  4. A General Framework of Our Allocation Algorithms
  5. Results Overview
  6. Further Related Work
  7. Finding approximate $\mathsf{EFX}$ allocations.
  8. Relaxing $\mathsf{EFX}$.
  9. $\mathsf{EFX}$ with Charity.
  10. A Warmup: A Partial $\mathsf{EFX}$ and a Complete $\mathsf{EF2X}$ Allocation for $(\infty,1)$-bounded Valuations
  11. Update Rule 1
  12. Update Rule 2
  13. Update Rule 3
  14. Update Rule 4
  15. Final Step
  16. ...and 10 more sections

Key Result

Lemma 2.5

Assume $\mathcal{I} = (N,M,\mathcal{V})$ is an allocation instance, and let $X$ be a basic feasible allocation. If for some agent $i$, $v_{i}(X_i) = 0$, then there exists another instance $\mathcal{I}' = (N',M',\mathcal{V}')$ with $|N'|<|N|$, such that any $\alpha$-$\mathsf{EFX}$ or $\alpha$-$\maths

Figures (4)

  • Figure 1: An example of valuations that are both restricted additive and $(2, 2)$-bounded
  • Figure 2: An allocation, and envy graphs $G_X$, $G_{0, X}$,$G_{0.5, X}$, $G_{1, X}$
  • Figure 3: Example of $(\infty,1)$-bounded valuations
  • Figure 4: General Framework for Allocation Algorithms

Theorems & Definitions (78)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • ...and 68 more