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EF2X Exists For Four Agents

Arash Ashuri, Vasilis Gkatzelis, Alkmini Sgouritsa

Abstract

We study the fair allocation of indivisible goods among a group of agents, aiming to limit the envy between any two agents. The central open problem in this literature, which has proven to be extremely challenging, is regarding the existence of an EFX allocation, i.e., an allocation such that any envy from some agent i toward another agent j would vanish if we were to remove any single good from the bundle allocated to j. When the agents' valuations are additive, which has been the main focus of prior works, Chaudhury et al. [2024] showed that an EFX allocation is guaranteed to exist for all instances involving up to three agents. Subsequently, Berger et al. [2022] extended this guarantee to nice-cancelable valuations and Akrami et al. [2023] to MMS-feasible valuations. However, the existence of EFX allocations for instances involving four agents remains open, even for additive valuations. We contribute to this literature by focusing on EF2X, a relaxation of EFX which requires that any envy toward some agent vanishes if any two of the goods allocated to that agent were to be removed. Our main result shows that EF2X allocations are guaranteed to exist for any instance with four agents, even for the class of cancelable valuations, which is more general than additive. Our proof is constructive, proposing an algorithm that computes such an allocation in pseudopolynomial time. Furthermore, for instances involving three agents we provide an algorithm that computes an EF2X allocation in polynomial time, in contrast to EFX, for which the fastest known algorithm for three agents is only pseudopolynomial.

EF2X Exists For Four Agents

Abstract

We study the fair allocation of indivisible goods among a group of agents, aiming to limit the envy between any two agents. The central open problem in this literature, which has proven to be extremely challenging, is regarding the existence of an EFX allocation, i.e., an allocation such that any envy from some agent i toward another agent j would vanish if we were to remove any single good from the bundle allocated to j. When the agents' valuations are additive, which has been the main focus of prior works, Chaudhury et al. [2024] showed that an EFX allocation is guaranteed to exist for all instances involving up to three agents. Subsequently, Berger et al. [2022] extended this guarantee to nice-cancelable valuations and Akrami et al. [2023] to MMS-feasible valuations. However, the existence of EFX allocations for instances involving four agents remains open, even for additive valuations. We contribute to this literature by focusing on EF2X, a relaxation of EFX which requires that any envy toward some agent vanishes if any two of the goods allocated to that agent were to be removed. Our main result shows that EF2X allocations are guaranteed to exist for any instance with four agents, even for the class of cancelable valuations, which is more general than additive. Our proof is constructive, proposing an algorithm that computes such an allocation in pseudopolynomial time. Furthermore, for instances involving three agents we provide an algorithm that computes an EF2X allocation in polynomial time, in contrast to EFX, for which the fastest known algorithm for three agents is only pseudopolynomial.

Paper Structure

This paper contains 27 sections, 37 theorems, 101 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results
  3. Additional Related Work
  4. Preliminaries
  5. Types of valuation functions.
  6. Sets of bundles and notation.
  7. {EF, EFX, EF2X}-envy.
  8. {EFX, EF2X}-feasibility.
  9. {EFX, EF2X}-best sets.
  10. The Plaut-Roughgarden (PR) algorithm.
  11. Computing EF$2$X Allocations for Four Agents
  12. Potential function and running time.
  13. Our Techniques
  14. EFkX-Best-Bundle.
  15. Swap-Optimization.
  16. ...and 12 more sections

Key Result

Theorem 3.1

For every instance involving four agents with cancelable valuation functions and any number of goods, there exists an EF2X allocation, and we can compute one in pseudo-polynomial time.

Figures (1)

  • Figure 1: Dashed edges correspond to transitions where the partition remains the same. Non-dashed edges correspond to transitions where the partition may change. The non-dashed edges are bold whenever the potential strictly increases during the transition and non-bold whenever the potential weakly increases.

Theorems & Definitions (95)

  • Theorem 3.1
  • Definition 3.2: potential function
  • Definition 3.3: stage A
  • Definition 3.4: stage B
  • Definition 3.5: in-stage partition
  • Definition 3.6: stage B1
  • Definition 3.7: swap-optimal sets
  • Definition 3.8: stage B2
  • Definition 3.9: stage B2i
  • Definition 3.10: stage B2ii
  • ...and 85 more