Truthful Fair Mechanisms for Allocating Mixed Divisible and Indivisible Goods

TL;DR

This paper investigates the design of mechanisms that are both truthful and fair for allocating mixed divisible and indivisible goods, revealing fundamental impossibilities and proposing solutions under specific conditions.

Contribution

It establishes impossibility results for general settings and introduces mechanisms that achieve fairness and truthfulness in restricted scenarios.

Findings

No EFM and truthful mechanism exists in the general case.

Proposed EFM and truthful mechanisms for binary valuations with a single divisible good.

Nash welfare maximization cannot guarantee EFM and truthfulness simultaneously.

Abstract

We study the problem of designing truthful and fair mechanisms when allocating a mixture of divisible and indivisible goods. We first show that there does not exist an EFM (envy-free for mixed goods) and truthful mechanism in general. This impossibility result holds even if there is only one indivisible good and one divisible good and there are only two agents. Thus, we focus on some more restricted settings. Under the setting where agents have binary valuations on indivisible goods and identical valuations on a single divisible good (e.g., money), we design an EFM and truthful mechanism. When agents have binary valuations over both divisible and indivisible goods, we first show there exist EFM and truthful mechanisms when there are only two agents or when there is a single divisible good. On the other hand, we show that the mechanism maximizing Nash welfare cannot ensure EFM and truthfulness simultaneously.