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Existence of EFX for Two Additive Valuations

Ryoga Mahara

TL;DR

The paper proves that complete EFX allocations exist when agents belong to two valuation types with identical additive valuations within each type, generalizing beyond the previously known cases of two agents or identical valuations. It uses a constructive, iterative approach: starting from an EFX allocation, it builds a Pareto-dominating partial EFX allocation by exploiting non-degenerate perturbations, minimum preferred sets, champions, and envy-graphs. The method combines a two-stage process—first boosting the utilities of two base agents and then removing residual EFX envy—to ensure both Pareto dominance and EFX in the augmented allocation. This result advances the understanding of EFX existence under restricted valuation diversity and paves the way for exploring more general multi-type additive valuations.

Abstract

Fair division of indivisible items is a well-studied topic in Economics and Computer Science. The objective is to allocate items to agents in a fair manner, where each agent has a valuation for each subset of items. Envy-freeness is one of the most widely studied notions of fairness. Since complete envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling one is envy-freeness up to any item (EFX), where no agent envies another agent after the removal of any single item from the other agent's bundle. However, despite significant efforts by many researchers for several years, it is known that a complete EFX allocation always exists only in limited cases. In this paper, we show that a complete EFX allocation always exists when each agent is of one of two given types, where agents of the same type have identical additive valuations. This is the first such existence result for non-identical valuations when there are any number of agents and items and no limit on the number of distinct values an agent can have for individual items. We give a constructive proof, in which we iteratively obtain a Pareto dominating (partial) EFX allocation from an existing partial EFX allocation.

Existence of EFX for Two Additive Valuations

TL;DR

The paper proves that complete EFX allocations exist when agents belong to two valuation types with identical additive valuations within each type, generalizing beyond the previously known cases of two agents or identical valuations. It uses a constructive, iterative approach: starting from an EFX allocation, it builds a Pareto-dominating partial EFX allocation by exploiting non-degenerate perturbations, minimum preferred sets, champions, and envy-graphs. The method combines a two-stage process—first boosting the utilities of two base agents and then removing residual EFX envy—to ensure both Pareto dominance and EFX in the augmented allocation. This result advances the understanding of EFX existence under restricted valuation diversity and paves the way for exploring more general multi-type additive valuations.

Abstract

Fair division of indivisible items is a well-studied topic in Economics and Computer Science. The objective is to allocate items to agents in a fair manner, where each agent has a valuation for each subset of items. Envy-freeness is one of the most widely studied notions of fairness. Since complete envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling one is envy-freeness up to any item (EFX), where no agent envies another agent after the removal of any single item from the other agent's bundle. However, despite significant efforts by many researchers for several years, it is known that a complete EFX allocation always exists only in limited cases. In this paper, we show that a complete EFX allocation always exists when each agent is of one of two given types, where agents of the same type have identical additive valuations. This is the first such existence result for non-identical valuations when there are any number of agents and items and no limit on the number of distinct values an agent can have for individual items. We give a constructive proof, in which we iteratively obtain a Pareto dominating (partial) EFX allocation from an existing partial EFX allocation.

Paper Structure

This paper contains 14 sections, 10 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Envy-freeness up to one item (EF1):
  4. Envy-freeness up to any item (EFX):
  5. Our Contribution
  6. Related Work
  7. Preliminaries
  8. Non-degenerate instances:
  9. Minimum preferred set and Most envious agent:
  10. Champions:
  11. Existence of EFX for Two Additive Valuations
  12. Improvement of the utilities of $\alpha_0$ and $\beta_0$
  13. Elimination of EFX envy
  14. Conclusion

Key Result

Theorem 1

Complete EFX allocations always exist when each agent is of one of two given types, where agents of the same type have identical additive valuation functions.

Figures (1)

  • Figure 1:

Theorems & Definitions (17)

  • Theorem 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 7 more