Fair and Efficient Division among Families

Fair division theory mostly involves individual consumption. But resources are often allocated to groups, such as families or countries, whose members consume the same bundle but have different preferences. Do fair and efficient allocations exist in such an "economy of families"? We adapt three common notions of fairness: fair-share, no-envy and egalitarian-equivalence, to an economy of families. The stronger adaptation --- individual fairness --- requires that each individual in each family perceives the division as fair; the weaker one --- family fairness --- requires that the family as a whole, treated as a single agent with (typically) incomplete preferences, perceives the division as fair. Individual-fair-share, family-no-envy and family-egalitarian-equivalence are compatible with efficiency under broad conditions. The same holds for individual-no-envy when there are only two families. In contrast, individual-no-envy with three or more families and individual-egalitarian-equivalence with two or more families are typically incompatible with efficiency, unlike the situation in an economy of individuals. The common market equilibrium approach to fairness is of limited use in economies with families. In contrast, the leximin approach is broadly applicable: it yields an efficient, individual-fair-share, and family-egalitarian-equivalent allocation.