Fair Division Among Couples and Small Groups

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.