Mechanism Design for Fair Division

TL;DR

This paper studies fair division of divisible items among agents, designing truthful mechanisms that approximate proportional fairness, with theoretical bounds and mechanisms tailored for highly demanded items.

Contribution

It introduces new truthful mechanisms for fair division that guarantee specific approximation ratios and explores the limits of what can be achieved without monetary transfers.

Findings

A truthful mechanism guarantees at least 36.8% of proportional fairness.

No truthful mechanism can guarantee more than 50% of proportional fairness.

Mechanisms perform well when items are highly demanded.

Abstract

We revisit the classic problem of fair division from a mechanism design perspective, using {\em Proportional Fairness} as a benchmark. In particular, we aim to allocate a collection of divisible items to a set of agents while incentivizing the agents to be truthful in reporting their valuations. For the very large class of homogeneous valuations, we design a truthful mechanism that provides {\em every agent} with at least a $1/e\approx 0.368$ fraction of her Proportionally Fair valuation. To complement this result, we show that no truthful mechanism can guarantee more than a $0.5$ fraction, even for the restricted class of additive linear valuations. We also propose another mechanism for additive linear valuations that works really well when every item is highly demanded. To guarantee truthfulness, our mechanisms discard a carefully chosen fraction of the allocated resources; we conclude by uncovering interesting connections between our mechanisms and known mechanisms that use money instead.