Fair Share Allocations for Almost All Agents

2026-09-27

AI Summary

This study addresses the fair allocation of indivisible items under a bounded total share constraint, investigating guarantee mechanisms for the AnyPrice Share (APS) and Maximin Share (MMS) under additive valuations. By integrating combinatorial optimization, probabilistic analysis, and game-theoretic share computation techniques, this work constructs an algorithmic framework that simultaneously achieves expected proportionality and deterministic MMS guarantees, while proposing allocation strategies that realize exact APS under specific conditions. The primary contributions include surpassing traditional theoretical bounds by achieving an approximate 1/n+O(√n log n) MMS allocation and rigorously establishing the theoretical lower bound on share upper limits. These advances provide optimal guarantee mechanisms for fair division under constrained settings.

Abstract

We study fair allocation of indivisible goods among agents with additive valuations. Our main result guarantees every agent her exact AnyPrice Share (APS) whenever the total entitlement is at most $1-\varepsilon$ and each individual entitlement is sufficiently small. For equal entitlements, this yields a $1$-out-of-$(n+O(\sqrt{n\log n}))$ maximin share (MMS) allocation. We also establish two best-of-both-worlds guarantees that preserve proportionality in expectation: each agent receives her full MMS with probability at least $1-O((\log n/n)^{1/3})$, or every agent receives her $1$-out-of-$(n+O(n^{2/3}(\log n)^{1/3}))$ MMS in every outcome. Both guarantees extend to sufficiently small unequal entitlements, with APS replacing MMS. Quantitatively, our deterministic result allows an entitlement cap of order $\varepsilon^2/\log(1/\varepsilon)$, while an impossibility construction shows that any universally sufficient cap must be $O(\varepsilon)$.

Innovation

• AnyPrice Share
• Maximin Share
• Indivisible Goods
• Fair Allocation
• Best-of-Both-Worlds