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Fair Allocation of Indivisible Goods with Variable Groups

Paul Gölz, Ayumi Igarashi, Pasin Manurangsi, Warut Suksompong

Abstract

We study the fair allocation of indivisible goods with variable groups. In this model, the goal is to partition the agents into groups of given sizes and allocate the goods to the groups in a fair manner. We show that for any number of groups and corresponding sizes, there always exists an envy-free up to one good (EF1) outcome, thereby generalizing an important result from the individual setting. Our result holds for arbitrary monotonic utilities and comes with an efficient algorithm. We also prove that an EF1 outcome is guaranteed to exist even when the goods lie on a path and each group must receive a connected bundle. In addition, we consider a probabilistic model where the utilities are additive and drawn randomly from a distribution. We show that if there are $n$ agents, the number of goods $m$ is divisible by the number of groups $k$, and all groups have the same size, then an envy-free outcome exists with high probability if $m = ω(\log n)$, and this bound is tight. On the other hand, if $m$ is not divisible by $k$, then an envy-free outcome is unlikely to exist as long as $m = o(\sqrt{n})$.

Fair Allocation of Indivisible Goods with Variable Groups

Abstract

We study the fair allocation of indivisible goods with variable groups. In this model, the goal is to partition the agents into groups of given sizes and allocate the goods to the groups in a fair manner. We show that for any number of groups and corresponding sizes, there always exists an envy-free up to one good (EF1) outcome, thereby generalizing an important result from the individual setting. Our result holds for arbitrary monotonic utilities and comes with an efficient algorithm. We also prove that an EF1 outcome is guaranteed to exist even when the goods lie on a path and each group must receive a connected bundle. In addition, we consider a probabilistic model where the utilities are additive and drawn randomly from a distribution. We show that if there are agents, the number of goods is divisible by the number of groups , and all groups have the same size, then an envy-free outcome exists with high probability if , and this bound is tight. On the other hand, if is not divisible by , then an envy-free outcome is unlikely to exist as long as .

Paper Structure

This paper contains 26 sections, 33 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Our Results
  3. Further Related Work
  4. Preliminaries
  5. EF1: Existence and Computation
  6. EF1: Adding Connectivity Constraints
  7. Envy-Freeness: Asymptotic Existence
  8. Divisible Case
  9. Non-Existence via First Moment Method.
  10. Existence via Second Moment Method.
  11. Part (i): Bounding $p_{(i, i')}$.
  12. Part (ii): Bounding $\Pr\left[X^1 + \cdots + X^n = \frac{n}{k}\cdot\mathbf{1}\right]$.
  13. Non-Divisible Case
  14. Conclusion
  15. Surjectivity of $\overline{f}$ in the proof of \ref{['lem:main-coloring-to-assignment']}
  16. ...and 11 more sections

Key Result

Theorem 1

For any instance with arbitrary monotonic utilities, there exists an EF1 outcome. Moreover, such an outcome can be computed in time $O(mn^3)$.

Theorems & Definitions (53)

  • Theorem 1
  • proof
  • Theorem 2
  • Lemma 1: Igarashi23
  • Lemma 2
  • Lemma 3: IgarashiMe24
  • proof : Proof of \ref{['lem:main-coloring-to-assignment']}
  • proof : Proof of Theorem \ref{['thm:EF1:group']}
  • Theorem 3
  • Theorem 4
  • ...and 43 more