Maximin Fair Allocation of Indivisible Items under Cost Utilities

TL;DR

This paper studies maximin fair share (MMS) allocations for indivisible goods under cost utilities, where each item has a cost and a agent’s value for a bundle is the cost of the intersection with their approval set. It proves MMS allocations exist for three agents, and more generally for laminar approvals across any number of agents, and it provides a strategyproof, Pareto-optimal MMS mechanism when the number of items $m$ equals $n+2$. The results demonstrate that cost utilities can yield MMS guarantees where general additive valuations do not, while also highlighting limits via NP-hardness of MMS value computation and a negative result for strategyproof MMS with few items. Overall, the work strengthens the case for cost-structured valuations as a productive framework for fair division, with practical implications for settings where item prices and approvals are easily elicited. The approaches combine constructive MMS allocations, structural laminarity arguments, and tailored strategyproof mechanisms to achieve MMS plus Pareto efficiency in several important regimes.

Abstract

We study the problem of fairly allocating indivisible goods among a set of agents. Our focus is on the existence of allocations that give each agent their maximin fair share--the value they are guaranteed if they divide the goods into as many bundles as there are agents, and receive their lowest valued bundle. An MMS allocation is one where every agent receives at least their maximin fair share. We examine the existence of such allocations when agents have cost utilities. In this setting, each item has an associated cost, and an agent's valuation for an item is the cost of the item if it is useful to them, and zero otherwise. Our main results indicate that cost utilities are a promising restriction for achieving MMS. We show that for the case of three agents with cost utilities, an MMS allocation always exists. We also show that when preferences are restricted slightly further--to what we call laminar set approvals--we can guarantee MMS allocations for any number of agents. Finally, we explore if it is possible to guarantee each agent their maximin fair share while using a strategyproof mechanism.

Figures (2)

• Figure 1: An illustration of the sets involved in the proof of Lemma \ref{['lem:lam-ind']}. The largest set is $M$---the set of goods. Note how the approval sets $A_1, \dots, A_k$ are equivalent to the whole set of goods, and the same holds for $A_{i^*}$ and $A'_i$. The approval set $A_i$ is "one level below" the sets approving all items. The bundle $B'_{i^*}$---represented in blue---is the bundle in the allocation $\boldsymbol{B}|_{N' \cup \{i\}}$ that is highest valued according to $v_i$.
• Figure 2: An illustration of the sets involved in Case 2 of the proof of Lemma \ref{['lem:lam-ind']}. The possible approval set of agent $j$ in each case is represented in green.

Theorems & Definitions (19)

• lemma thmcounterlemma
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