Proportional Cake-Cutting among Families

This paper extends the classic cake-cutting problem to a situation in which the "cake" is divided among families. Each piece of cake is owned and used simultaneously by all members of the family. A typical example of such a cake is land. We examine three ways to assess the fairness of such a division, based on the classic proportionality criterion: (a) Average proportionality means that for each family, the value of its share averaged over all family members is at least 1/k (where k is the number of families); (b) Unanimous proportionality means that in each family, each member values the family's share as at least 1/k; (c) Democratic proportionality means that in each family, at least half the members value the family's share as at least 1/k. We compare these criteria based on the number of connected components in the resulting division.

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