On Truthful Mechanisms for Maximin Share Allocations

TL;DR

This paper explores the design of truthful mechanisms for fair division of indivisible items, focusing on maximin share guarantees, and analyzes the limitations and possibilities across different information models, especially for two players.

Contribution

It introduces three models for eliciting preferences in maximin share allocations and provides new positive and negative results on the existence and approximation of truthful mechanisms.

Findings

Positive results for some models in specific cases

Impossibility results highlighting limitations of truthfulness

Special focus on the two-player case revealing complex challenges

Abstract

We study a fair division problem with indivisible items, namely the computation of maximin share allocations. Given a set of $n$ players, the maximin share of a single player is the best she can guarantee to herself, if she would partition the items in any way she prefers, into $n$ bundles, and then receive her least desirable bundle. The objective then is to find an allocation, so that each player is guaranteed her maximin share. Previous works have studied this problem mostly algorithmically, providing constant factor approximation algorithms. In this work we embark on a mechanism design approach and investigate the existence of truthful mechanisms. We propose three models regarding the information that the mechanism attempts to elicit from the players, based on the cardinal and ordinal representation of preferences. We establish positive and negative (impossibility) results for each model and highlight the limitations imposed by truthfulness on the approximability of the problem. Finally, we pay particular attention to the case of two players, which already leads to challenging questions.