Fair-Share Allocations for Agents with Arbitrary Entitlements

TL;DR

This paper introduces the AnyPrice share (APS), a new fairness measure for allocating indivisible goods among agents with arbitrary entitlements, and provides a polynomial-time algorithm guaranteeing each agent a 3/5 fraction of her APS.

Contribution

The paper proposes the APS as a novel fairness concept for unequal entitlements and offers the first polynomial-time algorithm achieving a 3/5 approximation of APS for additive valuations.

Findings

APS is always at least as large as MMS, often strictly larger.

A polynomial-time algorithm guarantees each agent at least 3/5 of her APS.

The algorithm also corresponds to strategies in a bidding game ensuring the same fairness guarantee.

Abstract

We consider the problem of fair allocation of indivisible goods to $n$ agents, with no transfers. When agents have equal entitlements, the well established notion of the maximin share (MMS) serves as an attractive fairness criterion, where to qualify as fair, an allocation needs to give every agent at least a substantial fraction of her MMS. In this paper we consider the case of arbitrary (unequal) entitlements. We explain shortcomings in previous attempts that extend the MMS to unequal entitlements. Our conceptual contribution is the introduction of a new notion of a share, the AnyPrice share (APS), that is appropriate for settings with arbitrary entitlements. Even for the equal entitlements case, this notion is new, and satisfies $APS \ge MMS$, where the inequality is sometimes strict. We present two equivalent definitions for the APS (one as a minimization problem, the other as a maximization problem), and provide comparisons between the APS and previous notions of fairness. Our main result concerns additive valuations and arbitrary entitlements, for which we provide a polynomial-time algorithm that gives every agent at least a $\frac{3}{5}$-fraction of her APS. This algorithm can also be viewed as providing strategies in a certain natural bidding game, and these strategies secure each agent at least a $\frac{3}{5}$-fraction of her APS.