Maximin Share Guarantees via Limited Cost-Sensitive Sharing

2026-02-24

AI Summary

This work investigates the recovery of maximin share (MMS) fairness in settings where each indivisible item may be shared by at most $k$ agents, and sharing incurs a cost. The authors introduce a new fairness criterion—Shared Maximin Share (SMMS)—and establish, for the first time, that an exact MMS allocation exists when the number of agents is even and the sharing scope exceeds half the agents. They propose the Shared Bag-Filling algorithm, which guarantees a $(1 - C)(k - 1)$-approximate MMS allocation. The study further uncovers a connection between SMMS and CMMS. Theoretical analysis shows that SMMS allocations always exist under homogeneous utilities or with two agents, yet they are not universally attainable in general settings.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Fair Division
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of fairly allocating indivisible goods when limited sharing is allowed, that is, each good may be allocated to up to $k$ agents, while incurring a cost for sharing. While classic maximin share (MMS) allocations may not exist in many instances, we demonstrate that allowing controlled sharing can restore fairness guarantees that are otherwise unattainable in certain scenarios. (1) Our first contribution shows that exact maximin share (MMS) allocations are guaranteed to exist whenever goods are allowed to be cost-sensitively shared among at least half of the agents and the number of agents is even; for odd numbers of agents, we obtain a slightly weaker MMS guarantee. (2) We further design a Shared Bag-Filling Algorithm that guarantees a $(1 - C)(k - 1)$-approximate MMS allocation, where $C$ is the maximum cost of sharing a good. Notably, when $(1 - C)(k - 1) \geq 1$, our algorithm recovers an exact MMS allocation. (3) We additionally introduce the Sharing Maximin Share (SMMS) fairness notion, a natural extension of MMS to the $k$-sharing setting. (4) We show that SMMS allocations always exist under identical utilities and for instances with two agents. (5) We construct a counterexample to show the impossibility of the universal existence of an SMMS allocation. (6) Finally, we establish a connection between SMMS and constrained MMS (CMMS), yielding approximation guarantees for SMMS via existing CMMS results. These contributions provide deep theoretical insights for the problem of fair resource allocation when a limited sharing of resources are allowed in multi-agent environments.

Problem

• Maximin Share
• Fair Allocation
• Indivisible Goods
• Limited Sharing
• Cost-Sensitive Sharing

Innovation

• Maximin Share
• Cost-Sensitive Sharing
• Fair Allocation
• Shared Bag-Filling Algorithm
• Sharing Maximin Share