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Best-of-Both-Worlds Fair Allocation of Indivisible and Mixed Goods

Xiaolin Bu, Zihao Li, Shengxin Liu, Xinhang Lu, Biaoshuai Tao

Abstract

We study the problem of fairly allocating either a set of indivisible goods or a set of mixed divisible and indivisible goods (i.e., mixed goods) to agents with additive utilities, taking the best-of-both-worlds perspective of guaranteeing fairness properties both ex ante and ex post. The ex-post fairness notions considered in this paper are relaxations of envy-freeness, specifically, EFX for indivisible-goods allocation, and EFM for mixed-goods allocation. For two agents, we show that there is a polynomial-time randomized algorithm that achieves ex-ante envy-freeness and ex-post EFX / EFM simultaneously. For $n$ agents with bi-valued utilities, we show there exist randomized allocations that are (i) ex-ante proportional and ex-post EFM, and (ii) ex-ante envy-free, ex-post EFX, and ex-post fractionally Pareto optimal.

Best-of-Both-Worlds Fair Allocation of Indivisible and Mixed Goods

Abstract

We study the problem of fairly allocating either a set of indivisible goods or a set of mixed divisible and indivisible goods (i.e., mixed goods) to agents with additive utilities, taking the best-of-both-worlds perspective of guaranteeing fairness properties both ex ante and ex post. The ex-post fairness notions considered in this paper are relaxations of envy-freeness, specifically, EFX for indivisible-goods allocation, and EFM for mixed-goods allocation. For two agents, we show that there is a polynomial-time randomized algorithm that achieves ex-ante envy-freeness and ex-post EFX / EFM simultaneously. For agents with bi-valued utilities, we show there exist randomized allocations that are (i) ex-ante proportional and ex-post EFM, and (ii) ex-ante envy-free, ex-post EFX, and ex-post fractionally Pareto optimal.

Paper Structure

This paper contains 39 sections, 16 theorems, 6 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. EFX Existence
  5. BoBW Fairness
  6. Preliminaries
  7. Allocations
  8. Utilities
  9. Solution Concepts
  10. Two Agents
  11. BoBW Fairness with Mixed Goods: Ex-Ante PROP + Ex-Post EFM
  12. Technical Overview
  13. Baseline comparison.
  14. Minimal unmatchable group.
  15. Preparations
  16. ...and 24 more sections

Key Result

Lemma 3.1

LocalSearch$(A, B, u)$ returns in polynomial time an integral EFX allocation $(A', B')$ under utility function $u$ with $|u(B') - u(A')| \leq |u(B) - u(A)|$.

Theorems & Definitions (34)

  • Definition 2.1: EF1 LiptonMaMo04Budish11 and EFX CaragiannisKuMo19PlautRo20
  • Definition 2.2: EFM BeiLiLi21
  • Definition 2.3: fPO
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Lemma 4.2: BeiLiLi21
  • Theorem 4.3
  • proof
  • ...and 24 more