Best-of-Both-Worlds Fair-Share Allocations

TL;DR

This paper presents a polynomial-time algorithm for fair allocation of indivisible items among agents, ensuring both ex-ante proportionality and ex-post guarantees of at least half the maximin share, improving fairness guarantees beyond previous methods.

Contribution

It introduces a new deterministic algorithm that guarantees each agent at least half their maximin share ex-post and proportional share ex-ante, with novel guarantees based on the truncated proportional share.

Findings

Algorithm guarantees each agent at least half their MMS ex-post.

Allocations are ex-ante proportional and fair.

Guarantees are nearly optimal given theoretical bounds.

Abstract

We consider the problem of fair allocation of indivisible items among $n$ agents with additive valuations, when agents have equal entitlements to the goods, and there are no transfers. Best-of-Both-Worlds (BoBW) fairness mechanisms aim to give all agents both an ex-ante guarantee (such as getting the proportional share in expectation) and an ex-post guarantee. Prior BoBW results have focused on ex-post guarantees that are based on the "up to one item" paradigm, such as envy-free up to one item (EF1). In this work we attempt to give every agent a high value ex-post, and specifically, a constant fraction of his maximin share (MMS). The up to one item paradigm fails to give such a guarantee, and it is not difficult to present examples in which previous BoBW mechanisms give agents only a $\frac{1}{n}$ fraction of their MMS. Our main result is a deterministic polynomial time algorithm that computes a distribution over allocations that is ex-ante proportional, and ex-post, every allocation gives every agent at least his proportional share up to one item, and more importantly, at least half of his MMS. Moreover, this last ex-post guarantee holds even with respect to a more demanding notion of a share, introduced in this paper, that we refer to as the truncated proportional share (TPS). Our guarantees are nearly best possible, in the sense that one cannot guarantee agents more than their proportional share ex-ante, and one cannot guarantee agents more than a $\frac{n}{2n-1}$ fraction of their TPS ex-post.