Fair and Truthful Mechanism with Limited Subsidy

TL;DR

This paper investigates the design of truthful resource allocation mechanisms for indivisible goods, demonstrating that with matroidal valuations, fairness and efficiency can be achieved with minimal subsidies, unlike the general case.

Contribution

It introduces a mechanism achieving envy-freeness and optimality with limited subsidies for matroidal valuations, contrasting with impossibility results for general valuations.

Findings

Mechanism achieves envy-freeness and optimality with at most $1 subsidy per agent.

Impossibility of achieving fairness, truthfulness, and optimality simultaneously for general valuations.

Matroidal valuations enable mechanisms leveraging M-convexity for fair and efficient allocations.

Abstract

The notion of \emph{envy-freeness} is a natural and intuitive fairness requirement in resource allocation. With indivisible goods, such fair allocations are unfortunately not guaranteed to exist. Classical works have avoided this issue by introducing an additional divisible resource, i.e., money, to subsidize envious agents. In this paper, we aim to design a truthful allocation mechanism of indivisible goods to achieve both fairness and efficiency criteria with a limited amount of subsidy. Following the work of Halpern and Shah, our central question is as follows: to what extent do we need to rely on the power of money to accomplish these objectives? For general valuations, the impossibility theorem of combinatorial auction translates to our setting: even if an arbitrarily large amount of money is available for use, no mechanism can achieve truthfulness, envy-freeness, and utilitarian optimality simultaneously when agents have general monotone submodular valuations. By contrast, we show that, when agents have matroidal valuations, there is a truthful allocation mechanism that achieves envy-freeness and utilitarian optimality by subsidizing each agent with at most $1$, the maximum marginal contribution of each item for each agent. The design of the mechanism rests crucially on the underlying matroidal M-convexity of the Lorenz dominating allocations.