Maximin Share Guarantees via Limited Cost-Sensitive Sharing

TL;DR

This paper explores fair allocation of indivisible goods with limited sharing, introducing new guarantees and algorithms that extend the classic maximin share concept to sharing scenarios, with theoretical proofs and limitations.

Contribution

It introduces the concept of sharing maximin share (SMMS), proves existence under certain conditions, and develops algorithms for approximate MMS allocations in limited sharing settings.

Findings

Exact MMS guarantees when goods are shared among at least half the agents with even total agents.

A Shared Bag-Filling Algorithm providing a $(1 - C)(k - 1)$-approximate MMS allocation.

Existence of SMMS allocations for identical utilities and two-agent cases.

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.