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 familyperceives the division as fair ; the weaker one --- family fairness --- requires that the familyas 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 familiesare 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 .
Is efficiency consistent with fairness? Our approach to this question concerns the case where multiple individuals with diverse preferences are bound to consume the same bundle . Families are our lead example: the father, mother and children get to consume the same garden, kitchen, and vacations. We adapt each of the three most popular principles of fairness: envy-freeness, egalitarian-equivalence and the fair-share guarantee, in three different ways to the world of families. For any given criterion of fairness, an allocation is *unanimous-fair* if it is fair according to every individual member of each family, it is *aggregate-fair* if it is fair according to a particular aggregation of all family members' preferences, and it is *collective-fair* if it is fair according to the family's -- typically incomplete -- preferences, that rank a bundle above another bundle if and only if each member of the family ranks the first bundle above the second. While efficiency is generally incompatible with unanimous egalitarian equivalence, and incompatible with unanimous envy-freeness in economies with three or more families , unanimously envy-free efficient allocations always exist in economies with just two families. The unanimous fair share guarantee is easy to achieve: Under generic conditions the set of efficient allocations with the fair share guarantee contains some collectively envy-free and some collectively egalitarian equivalent allocations. We use modified versions of the traditional market equilibrium approach and lexicographic optimization to establish our results .
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Fair division theory mostly involves individual consumption. But resources are often allocated to groups, such as families or countries, whose members
Is efficiency consistent with fairness? Our approach to this question concerns the case where multiple individuals with diverse preferences are bound to
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but have different preferences. Do fair and efficient allocations exist in such an "economy of families"? We adapt three common notions
. Families are our lead example: the father, mother and children get to consume the same garden, kitchen, and vacations. We adapt each of the three most popular principles
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fair-share, no-envy and
envy-freeness,
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, to an economy
and the fair-share guarantee, in three different ways to the world
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The stronger adaptation --- individual fairness--- requires that each individual in each familyperceives the division as fair ; the weaker one --- family fairness --- requires that the familyas 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
For any given criterion of fairness, an allocation is *unanimous-fair* if it is fair according to every individual member of each family, it is *aggregate-fair* if it is fair according to a particular aggregation of all family members' preferences, and it is *collective-fair* if it is fair according to the family's -- typically incomplete -- preferences, that rank a bundle above another bundle if and only if each member of the family ranks the first bundle above the second. While efficiency is generally incompatible with unanimous egalitarian equivalence, and incompatible with unanimous envy-freeness in economies
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, unanimously envy-free efficient allocations always exist in economies with just two families. The unanimous fair share guarantee is easy to achieve: Under generic conditions the set of efficient allocations with the fair share guarantee contains some collectively envy-free and some collectively egalitarian equivalent allocations. We use modified versions of the traditional market equilibrium approach
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individual-egalitarian-equivalence with two or more familiesare 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
lexicographic optimization to establish our results