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Near-Optimal Best-of-Both-Worlds Fairness for Few Agents

Moshe Babaioff, Gefen Frosh

TL;DR

This work studies BoBW fairness for indivisible goods among a small number of agents with additive valuations, introducing the IMMX criterion to balance MMS-based and envy-based guarantees. For three agents, it proves the existence of a BoBW distribution with ex-ante proportionality, where every supported allocation is Epistemic EFX and yields at least $9/10$ of each agent’s MMS, while the remaining agent (if not at MMS) is EFX-satisfied; this combination is near-optimal given known MMS upper bounds. It further provides a polynomial-time approximation scheme (FPTAS) that computes a BoBW distribution preserving these ex-post guarantees and achieving near-exact ex-ante proportionality (up to $1-\varepsilon$) and near-MMS guarantees, with exact ex-ante proportionality recoverable by dropping EEFX. A complementary two-agent result achieves ex-ante envy-freeness and ex-post EFX with $(1-\varepsilon)$-MMS in polynomial time, and this serves as a building block for the three-agent FPTAS. Overall, the results tighten the trade-off between ex-ante and ex-post fairness in BoBW settings, provide verifiable certificates for complex guarantees, and propose a practical framework for fair division with few agents that is near-optimal in theory and computationally feasible in practice.

Abstract

We consider the problem of fair allocation of indivisible goods among agents with additive valuations, aiming for Best-of-Both-Worlds (BoBW) fairness: a distribution over allocations that is ex-ante fair, and additionally, it is supported only on deterministic allocations that are ex-post fair. We focus on BoBW for few agents, and our main result is the design of the first BoBW algorithms achieving near-optimal fairness for three agents. For three agents, we prove the existence of an ex-ante proportional distribution whose every allocation is Epistemic EFX (EEFX) and guarantees each agent at least $\tfrac{9}{10}$ of her MMS. As MMS allocations do not exist for three additive agents, in every allocation at least one agent might not be getting her MMS. To compensate such an agent, we also guarantee that if an agent is not getting her MMS then she is EFX-satisfied - giving her the strongest achievable envy-based guarantee. Additionally, using an FPTAS for near-MMS partitions, we present an FPTAS to compute a BoBW distribution preserving all envy-based guarantees, and also preserving all value-based guarantees up to $(1-\varepsilon)$. We further show that exact ex-ante proportionality can be restored when dropping EEFX. To do so, we first design, for two agents and any $\varepsilon > 0$, a Fully Polynomial-Time Approximation Scheme (FPTAS) that outputs a distribution which is ex-ante envy-free (and thus proportional) and ex-post envy-free up to any good (EFX), while guaranteeing each agent at least a $(1-\varepsilon)$-fraction of her maximin share (MMS). We then leverage this two-agent FPTAS algorithm as a subroutine to obtain, for three agents, the FPTAS guaranteeing exact ex-ante proportionality. We note that our result for two agents essentially matches the strongest fairness and efficiency guarantees achievable in polynomial time, and thus might be of independent interest.

Near-Optimal Best-of-Both-Worlds Fairness for Few Agents

TL;DR

This work studies BoBW fairness for indivisible goods among a small number of agents with additive valuations, introducing the IMMX criterion to balance MMS-based and envy-based guarantees. For three agents, it proves the existence of a BoBW distribution with ex-ante proportionality, where every supported allocation is Epistemic EFX and yields at least of each agent’s MMS, while the remaining agent (if not at MMS) is EFX-satisfied; this combination is near-optimal given known MMS upper bounds. It further provides a polynomial-time approximation scheme (FPTAS) that computes a BoBW distribution preserving these ex-post guarantees and achieving near-exact ex-ante proportionality (up to ) and near-MMS guarantees, with exact ex-ante proportionality recoverable by dropping EEFX. A complementary two-agent result achieves ex-ante envy-freeness and ex-post EFX with -MMS in polynomial time, and this serves as a building block for the three-agent FPTAS. Overall, the results tighten the trade-off between ex-ante and ex-post fairness in BoBW settings, provide verifiable certificates for complex guarantees, and propose a practical framework for fair division with few agents that is near-optimal in theory and computationally feasible in practice.

Abstract

We consider the problem of fair allocation of indivisible goods among agents with additive valuations, aiming for Best-of-Both-Worlds (BoBW) fairness: a distribution over allocations that is ex-ante fair, and additionally, it is supported only on deterministic allocations that are ex-post fair. We focus on BoBW for few agents, and our main result is the design of the first BoBW algorithms achieving near-optimal fairness for three agents. For three agents, we prove the existence of an ex-ante proportional distribution whose every allocation is Epistemic EFX (EEFX) and guarantees each agent at least of her MMS. As MMS allocations do not exist for three additive agents, in every allocation at least one agent might not be getting her MMS. To compensate such an agent, we also guarantee that if an agent is not getting her MMS then she is EFX-satisfied - giving her the strongest achievable envy-based guarantee. Additionally, using an FPTAS for near-MMS partitions, we present an FPTAS to compute a BoBW distribution preserving all envy-based guarantees, and also preserving all value-based guarantees up to . We further show that exact ex-ante proportionality can be restored when dropping EEFX. To do so, we first design, for two agents and any , a Fully Polynomial-Time Approximation Scheme (FPTAS) that outputs a distribution which is ex-ante envy-free (and thus proportional) and ex-post envy-free up to any good (EFX), while guaranteeing each agent at least a -fraction of her maximin share (MMS). We then leverage this two-agent FPTAS algorithm as a subroutine to obtain, for three agents, the FPTAS guaranteeing exact ex-ante proportionality. We note that our result for two agents essentially matches the strongest fairness and efficiency guarantees achievable in polynomial time, and thus might be of independent interest.
Paper Structure (84 sections, 38 theorems, 86 equations, 6 algorithms)

This paper contains 84 sections, 38 theorems, 86 equations, 6 algorithms.

Table of Contents

  1. Introduction
  2. Fundamental fairness benchmarks.
  3. Relaxations of ex-post fairness.
  4. Ex-ante fairness.
  5. Best-of-Both-Worlds fairness.
  6. Limitations of existing BoBW results.
  7. Computational considerations.
  8. Guarantees for few agents.
  9. Our Results
  10. The "Individually MMS-satisfying or EFX-satisfying (IMMX)" Criterion: Motivation and Interpretation
  11. Near-Optimal BoBW for Three Additive Agents
  12. Computation.
  13. Polynomial-Time Approximation via FPTAS
  14. Optimal Poly-time BoBW for Two Additive Agents
  15. Our Techniques
  16. ...and 69 more sections

Key Result

Theorem 1.1

Consider settings with three agents with additive valuations over a set $M$ of indivisible goods. There exists a procedure that given three additive valuations $(v_1,v_2,v_3)$, constructs a probability distribution $\mu$ over at most six deterministic allocations such that:

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Maximin share (MMS)
  • Definition 2.2: Envy-freeness-up-to-any-item (EFX) PlautRoughgarden2020
  • Definition 2.3: Epistemic EFX (EEFX) and EEFX certificates
  • Definition 2.4: Minimum EFX share and MXS allocations
  • Theorem 2.1: $EEFX_+$ $\Rightarrow$ $MXS_+$ (caragiannis2022new Theorem 1)
  • Corollary 2.1: $EEFX \Rightarrow EEFX_+$
  • Theorem 3.1
  • Lemma 3.1
  • ...and 60 more