Superadditivity and Subadditivity in Fair Division

Mirchandani, Rishi S.

doi:10.5539/jmr.v5n3p78

Cited by 8 publications

(6 citation statements)

References 14 publications

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“…In particular, with non-additive valuations, the resulting division is not necessarily proportional. Mirchandani (2013) suggests a division protocol for non-additive valuations using non-linear programming. However, the protocol is practical only when the resource to divide is a collection of a small number of homogeneous components, where the only thing that matters is what fraction of each component is allocated to each agent.…”

Section: Non-additive Utilitiesmentioning

confidence: 99%

Fair cake-cutting among families

Segal-Halevi

Nitzan

2019

Soc Choice Welf

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We study the fair division of a continuous resource, such as a landestate 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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Section: Non-additive Utilitiesmentioning

confidence: 99%

Fair cake-cutting among families

Segal-Halevi

Nitzan

2019

Soc Choice Welf

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“…11 Previous papers about cake-cutting with non-additive utilities can be roughly divided to two kinds: some [12,43,65] handle general non-additive utilities but provide only pure existence results. Others [23,53,69] provide constructive division procedures but only for a 1-dimensional cake. Our approach is a middle ground between these extremes.…”

Section: Related Workmentioning

confidence: 99%

Fair and square: Cake-cutting in two dimensions

Segal-Halevi

Nitzan

Hassidim

et al. 2017

Journal of Mathematical Economics

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We consider the classic problem of fairly dividing a heterogeneous good ("cake") among several agents with different valuations. Classic cake-cutting procedures either allocate each agent a collection of disconnected pieces, or assume that the cake is a one-dimensional interval. In practice, however, the two-dimensional shape of the allotted pieces is important. In particular, when building a house or designing an advertisement in printed or electronic media, squares are more usable than long and narrow rectangles. We thus introduce and study the problem of fair two-dimensional division wherein the allotted pieces must be of some restricted two-dimensional geometric shape(s), particularly squares and fat rectangles. Adding such geometric constraints re-opens most questions and challenges related to cake-cutting. Indeed, even the most elementary fairness criterionproportionality -can no longer be guaranteed. In this paper we thus examine the level of proportionality that can be guaranteed, providing both impossibility results and constructive division procedures.JEL classification: D63

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“…• Su (1999); Caragiannis et al (2011); Mirchandani (2013) provide practical division algorithms for nonadditive utilities, but they crucially assume that the cake is a 1-dimensional interval and cannot handle two-dimensional constraints.…”

Section: Challenges and Solutionsmentioning

confidence: 99%

Envy-Free Division of Land

Segal-Halevi,

Nitzan,

Hassidim

et al. 2016

Preprint

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Classic cake-cutting algorithms enable people with different preferences to divide among them a heterogeneous resource ("cake"), such that the resulting division is fair according to each agent's individual preferences. However, these algorithms either ignore the geometry of the resource altogether, or assume it is one-dimensional. In practice, it is often required to divide multi-dimensional resources, such as landestates or advertisement spaces in print or electronic media. In such cases, the geometric shape of the allotted piece is of crucial importance. For example, when building houses or designing advertisements, in order to be useful, the allotments should be squares or rectangles with bounded aspect-ratio. We thus introduce the problem of fair land division -fair division of a multi-dimensional resource wherein the allocated piece must have a pre-specified geometric shape. We present constructive division algorithms that satisfy the two most prominent fairness criteria, namely envy-freeness and proportionality.In settings where proportionality cannot be achieved due to the geometric constraints, our algorithms provide a partially-proportional division, guaranteeing that the fraction allocated to each agent be at least a certain positive constant. We prove that in many natural settings the envy-freeness requirement is compatible with the best attainable partial-proportionality.

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Superadditivity and Subadditivity in Fair Division

Cited by 8 publications

References 14 publications

Fair cake-cutting among families

Fair cake-cutting among families

Fair and square: Cake-cutting in two dimensions

Envy-Free Division of Land

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