Abstract
We study the fair division of a continuous resource, such as a land-estate or a time-interval, among pre-specified groups of agents, such as families. Each family is given a piece of the resource and this piece is used simultaneously by all family members, while different members may have different value functions. Three ways to assess the fairness of such a division are examined. (a) Average Fairness means that each family’s share is fair according to the “family value function”, defined as the arithmetic mean of the value functions of the family members. (b) Unanimous Fairness means that all members in all families feel that their family’s share is fair according to their personal value function. (c) Democratic Fairness means that in each family, at least a fixed fraction (e.g. a half) of the members feel that their family’s share is fair. We compare these criteria based on the number of connected components in the resulting division and on their compatibility with Pareto-efficiency.
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Notes
The condition of receiving at least 1 / n of the total endowment was introduced by Steinhaus (1948). Economists often call it fair-share guarantee (Bogomolnaia et al. 2017). Computer scientists often call it proportionality (Robertson and Webb 1998). This term is motivated by a generalization of the fair division problem in which different agents may have different entitlements (see page 4 below). In this setting, proportionality guarantees to each agent a value lower-bound that is in direct proportion to his/her entitlement.
In economic terms, the allotted piece becomes a “club good” (Buchanan 1965).
The assumption that agents’ preferences can be represented by measures is a strong one. It implies that the model is applicable only for the special case of linear preferences—the sum of values of two disjoint pieces equals the value of their union. Equivalently, the agents have constant marginal utilities (Chambers 2005)—the marginal utility of a land-plot for an agent does not depend on the other land-plots owned by that agent. Although most papers on the cake-cutting problem assume linearity, there are some notable exceptions; they are surveyed in Sect. 8.5.
In contrast, average-fairness and unanimous-fairness cannot be computed by any finite protocol. See Remark 1 in page 9.
A fourth fairness criterion that could be considered is individual fairness. In particular, an allocation is individually-proportional if the allocation \(X=(X_{1},\ldots ,X_{k})\) admits a refinement \(Y=(Y_{1},\ldots ,Y_{n})\), where for each family \(F_{j}\), \(\cup _{i\in F_{j}}Y_{i}=X_{j}\), such that for each agent i, \(V_{i}(Y_{i})\ge 1/n\). Individually-fair allocations always exist and can be found by using any classic fair division procedure on the individual agents, disregarding their families. Individual-fairness makes sense if, after the division of the land among the families, each family intends to further divide its share among its members. However, often this is not the case. When an inherited land-estate is divided between two families, the members of each family intend to live and use their entire share together, rather than dividing it among them. Therefore, the happiness of each family member depends on the entire value of his family’s share, rather than on the value of a potential private share he would get in a hypothetic sub-division.
See Cseh and Fleiner (2018) for a recent account of fair division among individual agents with different entitlements.
This impossibility appears not only in our one-dimensional theoretic model but also in practical, two-dimensional land division situations. A striking example was the India-Bangladesh border. According to Wikipedia page India– Bangladesh enclaves, up to 2015, “Within the main body of Bangladesh were 102 enclaves of Indian territory, which in turn contained 21 Bangladeshi counter–enclaves, one of which contained an Indian counter–counter–enclave... within the Indian mainland were 71 Bangladeshi enclaves, containing 3 Indian counter–enclaves”. Another example is Baarle-Hertog—a Belgian municipality made of 24 separate parcels of land, most of which are exclaves in the Netherlands. For more details and examples see the Wikipedia page List of enclaves and exclaves. We are grateful to Ian Turton for the references.
The goal of minimizing the number of components is pursued not only in cake-cutting papers but also in real-life politics. Going back to India and Bangladesh, after many years of negotiations they finally started to exchange most of their enclaves during the years 2015–2016. This reduced the number of components from 200 to a more reasonable number.
The definition uses capital N and K to distinguish the parameters of exact division from the parameters of unanimous-fair division.
They prove that, if C is a circle, the number of connected components is \(n-1\). Hence, the number of cuts is \(2n-2\). This is also true when C is an interval, although the number of connected components in this case is n.
See the example in the beginning of Sect. 4. In that example \(W^{\text {med}}\) is identical to \(W^{\text {min}}\).
Suppose an agent \(i \in F_j\) thinks the division is envy-free. Then \(V_i(X_j)\) is equal to \(\max _{j'=1}^{k} V_i(X_{j'})\). The maximum is at least as large as the average, so \(V_i(X_j)\) is at least as large as an average value of a piece, which is \(V_i(C)/k\).
Suppose an agent \(i \in F_j\) thinks the division is proportional. Then \(V_i(X_j) \ge 1/2\). By additivity, for the other family \(j'\ne j\), \(V_i(X_{j'}) \le 1/2\). Hence \(V_i(X_j) \ge V_i(X_{j'})\), so agent i thinks the division is envy-free.
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This research was funded in part by the following institutions: The Doctoral Fellowships of Excellence Program at Bar-Ilan University, the Mordechai and Monique Katz Graduate Fellowship Program, and the Israel Science Foundation Grant 1083/13.
We are grateful to Galya Segal-Halevi, Yonatan Aumann, Avinatan Hassidim, Noga Alon, Christian Klamler, Ulle Endriss, Neill Clift and Sophie Bade for helpful discussions. We are grateful to the anonymous reviewers of Social Choice and Welfare for their helpful comments, which substantially improved both the contents and presentation of this paper.
This paper started with a discussion in the MathOverflow website at http://mathoverflow.net/questions/203060/fair-cake-cutting-between-groups. We are grateful to the members who participated in the discussion: Pietro Majer, Tony Huynh and Manfred Weis. Other members of the StackExchange network contributed useful answers and ideas: Alex Ravsky, Andrew D. Hwang, BKay, Christian Elsholtz, Daniel Fischer, David K, D.W., Hurkyl, Ittay Weiss, Kittsil, Michael Albanese, Raphael Reitzig, Real, Babou, Domotor Palvolgyi (domotorp), Ian Turton (iant) and ivancho.
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Segal-Halevi, E., Nitzan, S. Fair cake-cutting among families. Soc Choice Welf 53, 709–740 (2019). https://doi.org/10.1007/s00355-019-01210-9
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DOI: https://doi.org/10.1007/s00355-019-01210-9