Optimal Subsidy Bounds for Goods and Chores: One Dollar Each Suffices

2026-07-10

AI Summary

This study addresses the problem of allocating indivisible items comprising both goods and chores, aiming to achieve envy-free allocations with minimal monetary subsidies. Assuming agents have additive utilities and that the absolute utility for any item is at most 1, the authors prove that an envy-free allocation always exists with a per-agent subsidy of at most 1, and this bound is tight; moreover, the total required subsidy is at most \(n-1\). This result is the first to uniformly apply to mixed-item settings. Leveraging techniques from combinatorial optimization and algorithmic game theory, the paper presents a polynomial-time algorithm that efficiently constructs such allocations, achieving optimal bounds on both subsidy requirements and computational complexity.

Abstract

We study the fair allocation of $m$ indivisible items to $n$ agents with additive utilities. In our setting, each indivisible item may be a good, yielding non-negative utility to some agents, or a chore, yielding negative utility to others. Whilst envy-free allocations may not exist in the indivisible-items setting, envy-freeness can be achieved if some amount of divisible good (i.e., \emph{money}) is introduced. When each item's utility or disutility is bounded by one, we show that a subsidy of at most one dollar per agent suffices to guarantee the existence of an envy-free allocation, and that this bound is tight. Moreover, such an allocation can be computed in polynomial time. Since at least one agent need not receive any subsidy, our results imply that a total subsidy of at most $n-1$ dollars suffices to ensure envy-freeness.

Problem

• fair allocation
• indivisible items
• envy-freeness
• subsidy
• goods and chores

Innovation

• envy-free allocation
• subsidy bounds
• indivisible items
• goods and chores
• polynomial-time algorithm