Fair Division Among Couples and Small Groups

2025-08-18

AI Summary

This paper studies fair allocation of indivisible goods among small groups (e.g., couples), where members within each group share all allocated items. Focusing on two fairness notions—envy-freeness up to one good (EF1) and proportionality up to $k$ goods (PROP$k$)—the work yields three key contributions: (1) It proves that an EF1 allocation always exists for any two couples and can be computed in polynomial time; (2) it establishes a tight threshold by showing that EF1 may fail to exist for three or more couples; (3) it demonstrates that a PROP$k$ allocation is guaranteed to exist—and efficiently computable—when the maximum group size is $k$, and further provides a sufficient condition for achieving PROP1 for arbitrarily many couples. Methodologically, the paper integrates combinatorial optimization with fair division theory, and—novelty—it simultaneously guarantees hierarchical proportionality and fractional Pareto optimality in the small-group setting.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We study the fair allocation of indivisible goods across groups of agents, where each agent fully enjoys all goods allocated to their group. We focus on groups of two (couples) and other groups of small size. For two couples, an EF1 allocation -- one in which all agents find their group's bundle no worse than the other group's, up to one good -- always exists and can be found efficiently. For three or more couples, EF1 allocations need not exist. Turning to proportionality, we show that, whenever groups have size at most $k$, a PROP$k$ allocation exists and can be found efficiently. In fact, our algorithm additionally guarantees (fractional) Pareto optimality, and PROP1 to the first agent in each group, PROP2 to the second, etc., for an arbitrary agent ordering. In special cases, we show that there are PROP1 allocations for any number of couples.

Problem

• Fair allocation of indivisible goods among small groups
• Existence and computation of EF1 allocations for couples
• Proportionality guarantees with PROPk for small group sizes

Innovation

• EF1 allocation for couples
• PROPk allocation for groups
• Pareto optimality algorithm