Cake-cutting with different entitlements: How many cuts are needed?

Segal-Halevi, Erel

doi:10.1016/j.jmaa.2019.123382

Cited by 19 publications

(8 citation statements)

References 15 publications

(14 reference statements)

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“…The negative result (b) follows immediately from an identical negative result for individual agents (Segal-Halevi 2019), by considering k one-member families.…”

Section: Average Fairnessmentioning

confidence: 86%

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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“…The negative result (b) follows immediately from an identical negative result for individual agents (Segal-Halevi 2019), by considering k one-member families.…”

Section: Average Fairnessmentioning

confidence: 86%

Fair cake-cutting among families

Segal-Halevi

Nitzan

2019

Soc Choice Welf

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“…While we trivially have

f (n) ⩾ n - 1

for

n ⩾ 2

, it is not hard to do better. The following construction from [7] shows that

f (n) ⩾ 2 n - 2

for all

n ⩾ 2

. Let

μ_{1}

be the uniform measure on [0,1] and for

1 ⩽ i ⩽ n - 1

, let

μ_{i + 1}

be the uniform measure supported on the (tiny) interval

[i / n - ε, i / n + ε]

, where

ε = 1 / {(100 n)}^{10}

.…”

Section: Resultsmentioning

confidence: 99%

“…What if we ask for an efficient disproportionate division with a bounded number of cuts? A beautiful topological result of Stromquist and Woodall [11] furnishes an answer; a folklore argument utilising this result shows that a disproportionate division for

n

agents with arbitrary demands may always be found with

O (n^{2})

cuts, and a more efficient rendition of this argument, recently discovered by Segal‐Halevi [7], shows that in fact

O (n \log n)

cuts always suffice. Our main result improves on these decades‐old topological arguments as follows.…”

Section: Introductionmentioning

confidence: 99%

Disproportionate division

Crew

Narayanan

Spirkl

2020

Bull. London Math. Soc.

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We study the disproportionate version of the classical cake-cutting problem: how efficiently can we divide a cake, here [0,1], among n 2 agents with different demands α1, α2,. .. , αn summing to 1? When all the agents have equal demands of α1 = α2 = • • • = αn = 1/n, it is well known that there exists a fair division with n − 1 cuts, and this is optimal. For arbitrary demands on the other hand, folklore arguments from algebraic topology show that O(n log n) cuts suffice, and this has been the state of the art for decades. Here, we improve the state of affairs in two ways: we prove that disproportionate division may always be achieved with 3n − 4 cuts, and also give an effective algorithm to construct such a division. We additionally offer a topological conjecture that implies that 2n − 2 cuts suffice in general, which would be optimal.

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“…Most of the literature on cake-cutting is about the search for a fair division into pieces that get allocated to the agents (such that, for example, no agent values someone else's pieces more than his own), as opposed to consensus division, as considered here. In the context of fair division, there is an "arbitrary proportions" analogue to the problem studied in this paper: Segal-Halevi [Seg19] and Crew et al [CNS20] have studied an analogous generalisation of fair division in which each agent has a (non-negative fractional) claim on the cake, all claims summing to 1. In common with consensus division, it is found that in the more general case of unequal proportions, more cuts may be required than in the special case of equal proportions.…”

Section: Background Related Workmentioning

confidence: 99%

Consensus Division in an Arbitrary Ratio

Goldberg¹,

Li²

2022

Preprint

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We consider the problem of partitioning a line segment into two subsets, so that n finite measures all has the same ratio of values for the subsets. Letting α ∈ [0, 1] denote the desired ratio, this generalises the PPA-complete consensushalving problem, in which α = 1 2 . It is known that for any α, there exists a solution using 2n cuts of the segment. Here we show that if α is irrational, that upper bound is almost optimal. We also obtain bounds that are nearly exact for a large subset of rational values α. On the computational side, we explore its dependence on the number of cuts available. More specifically, 1. when using the minimal number of cuts for each instance is required, the problem is NP-hard for any α;2. for a large subset of rational α = k , when k−1 k • 2n cuts are available, the problem is in the Turing closure of PPA-k; 3. when 2n cuts are allowed, the problem belongs to PPA for any α; furthermore, the problem belong to PPA-p for any prime p if 2(p − 1) • p/2 p/2 • n cuts are available.

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Cake-cutting with different entitlements: How many cuts are needed?

Cited by 19 publications

References 15 publications

Fair cake-cutting among families

Fair cake-cutting among families

Disproportionate division

Consensus Division in an Arbitrary Ratio

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