Matching Markets with Chores

TL;DR

The paper extends fair division to matching markets with chores and mixed manna by formulating and relating HZ equilibria and earnings-based equilibria under cardinal utilities with no transfers. It shows that utility shifting preserves key fairness and efficiency properties, and proves an equivalence between HZ and earnings equilibria, establishing existence and enabling cross-translation between price-based and earnings-based formulations. The work delivers polynomial-time results in special cases (bivalued utilities and constant-type agents) and analyzes Nash bargaining as an alternative approach, demonstrating both its limitations (envy challenges) and the hardness of achieving EF+PO in chores. Overall, it broadens the scope of fair division in matching markets, identifies tractable regimes, and pinpoints open problems for approximate fairness in the presence of chores.

Abstract

The fair division of chores, as well as mixed manna (goods and chores), has received substantial recent attention in the fair division literature; however, ours is the first paper to extend this research to matching markets. Indeed, our contention is that matching markets are a natural setting for this purpose, since the manna that fit into the limited number of hours available in a day can be viewed as one unit of allocation. We extend several well-known results that hold for goods to the settings of chores and mixed manna. In addition, we show that the natural notion of an earnings-based equilibrium, which is more natural in the case of all chores, is equivalent to the pricing-based equilibrium given by Hylland and Zeckhauser for the case of goods.

Figures (2)

• Figure 1: This example shows that minimizing the product of disutilities does not lead to fair outcomes. Here, agent $i$ has utilities $(1, C)$ and agent $i'$ has utilities $(0, 1)$. Both agents agree that $j'$ is the worse chore. However, the Nash bargaining solution will assign $j'$ fully to $i$.
• Figure 2: This example shows that maximizing the product of disutilities over the set of Pareto-optimal allocations does not lead to fair outcomes. Agent $i$ has utilities $(1, 2)$ whereas agent $i'$ has utilities $(0, 1)$. The Pareto-constrained Nash bargaining solution gives $j$ to $i$ and $j'$ to $i'$.

Theorems & Definitions (28)

• Definition 1
• Theorem 1: Hylland and Zeckhauser HZ79
• Definition 2
• Definition 3
• Theorem 2: Hylland and Zeckhauser HZ79
• Lemma 1
• Definition 4
• Theorem 3
• proof
• Theorem 4