text: 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 no-envy criterion: (a) Average envy-freeness means that for each family, the average value of its share (averaged over all family members) is weakly larger than the average value of any other share; (b) Unanimous envy-freeness means that in each family, each member values the family's share weakly more than any other share; (c) Democratic envy-freeness means that in each family, at least half the members value the family's share weakly more than any other share. We study each of these definitions from both an existential and a computational perspective.
  @cite_27: Abstract In theoretical land economics the existence of a competitive equilibrium with an additive price is considered problematic. This paper studies the exchange and allocation of a heterogeneous divisible commodity such as land, which is modeled as a measurable space. In a [ ]land' trading economy with unordered convex preferences, the existence of a competitive equilibrium with an additive equilibrium price is proved. This paper demonstrates also the existence of a weak core and a fair allocation.
  @cite_31: A cake is a metaphor for a heterogeneous, divisible good. When two players divide such a good, there is always a perfect division — one that is efficient (Pareto-optimal), envy-free, and equitable — which can be effected with a finite number of cuts under certain mild conditions; this is not always the case when there are more than two players (Brams, Jones, and Klamler, 2011b). We not only establish the existence of such a division but also provide an algorithm for determining where and how many cuts must be made, relating it to an algorithm, “Adjusted Winner” (Brams and Taylor, 1996, 1999), that yields a perfect division of multiple homogenous goods.
  @cite_7: We consider upper and lower bounds for maxmin allocations of a completely divisible good in both competitive and cooperative strategic contexts. We then derive a subgradient algorithm to compute the exact value up to any fixed degree of precision.
  @cite_17: Abstract Consider a heterogeneous but divisible commodity, bundles of which are represented by the (measurable) subsets of the good. One such commodity might be land. The mathematics literature has considered agents with utilities that are nonatomic measures over the commodity (and hence are additive). The existence of ‘α-fair’ allocations, in which each agent receives a utility proportional to his utility of the endowment of the entire economy, was demonstrated there. Here we extend these existence results to α-fair efficient allocations, envy-free allocations, envy-free efficient allocations, group envy-free and nicely shaped allocations of these types. We examine utilities that are not additive and relate the mathematics literature to the economics literature. We find sufficient conditions for the existence of egalitarian-equivalent efficient allocations. Finally, we consider the problem of allocating a time interval (uses of a facility). Existence of an envy-free allocation had been demonstrated in earlier literature. We show that any envy-free allocation is efficient as well as group envy-free. We extend this last result to a more general setting.
summary: @cite_17 @cite_27 study the concept of cake-cutting. Their model is the standard cake-cutting model in which the cake is divided among (and not among families as in our model). They define a group-envy-free division as a division in which no coalition of individuals can take the pieces allocated to another coalition with the same number of individuals and re-divide the pieces among its members such that all members are weakly better-off. Coalitions are also studied by @cite_31 @cite_7 . In our setting, the families are pre-determined and the agents do not form coalitions on-the-fly. In an alternative model, in which agents allowed to form coalitions based on their preferences, the family-cake-cutting problem becomes easier. For instance, it is easy to achieve a division with connected pieces between two coalitions: ask each agent to mark its median line, find the median of all medians, then divide the agents to two coalitions according to whether their median line is to the left or to the right of the median-of-medians.