The residual maximin share

TL;DR

This paper introduces the residual maximin share (RMMS), a new fairness measure for allocating indivisible goods, unifying and improving upon existing fairness concepts for various valuation types.

Contribution

It defines the RMMS, proves its feasibility and self-maximizing properties, and shows how it unifies and enhances previous fairness results in allocation problems.

Findings

RMMS is feasible and self-maximizing.

RMMS value is at least as large as MXS and 2/3-MMS.

Partial and complete allocations satisfying RMMS with EFX and EFL exist.

Abstract

We consider fair allocations of indivisible goods to agents with general monotone valuations. We observe that it is useful to introduce a new share-based fairness notion, the {\em residual maximin share} (RMMS). This share is {\em feasible} and {\em self maximizing}. Its value is at least as large as the MXS for monotone valuations, and at least as large as $\frac{2}{3}$-MMS for additive valuations. Known techniques easily imply the existence of partial allocations that are both RMMS and EFX, and complete allocations that are both RMMS and EFL. This unifies and somewhat improves upon several different results from previous papers.