Fair and square: Cake-cutting in two dimensions

Segal-Halevi, Erel; Nitzan, Shmuel; Hassidim, Avinatan; Aumann, Yonatan

doi:10.1016/j.jmateco.2017.01.007

Cited by 29 publications

(26 citation statements)

References 67 publications

(120 reference statements)

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“…Also, it would be interesting to obtain analogues of procedures such as sequential allocation and round-robin that respect the connectivity constraints and still produce desirable allocations. Finally, we may consider placing constraints on the 'shapes' of players' pieces, e.g., by requiring that the size of each piece is large relative to its diameter; similar ideas have been recently explored by Segal-Halevi et al [2015] in the context of the land division problem (i.e., cutting a 2-dimensional cake).…”

Section: Discussionmentioning

confidence: 99%

Fair Division of a Graph

Bouveret¹,

Cechlárová

Elkind

et al. 2017

Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence

129

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We consider fair allocation of indivisible items under an additional constraint: there is an undirected graph describing the relationship between the items, and each agent's share must form a connected subgraph of this graph. This framework captures, e.g., fair allocation of land plots, where the graph describes the accessibility relation among the plots. We focus on agents that have additive utilities for the items, and consider several common fair division solution concepts, such as proportionality, envy-freeness and maximin share guarantee. While finding good allocations according to these solution concepts is computationally hard in general, we design efficient algorithms for special cases where the underlying graph has simple structure, and/or the number of agents-or, less restrictively, the number of agent types-is small. In particular, despite non-existence results in the general case, we prove that for acyclic graphs a maximin share allocation always exists and can be found efficiently.

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Section: Discussionmentioning

confidence: 99%

Fair Division of a Graph

Bouveret¹,

Cechlárová

Elkind

et al. 2017

Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence

129

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“…4. Geometric constraints: For example, when the cake is square and the pieces must be square, it is impossible to guarantee an r-proportional allocation for any r ≥ 1/2, but there is an algorithm that guarantees a 1/4-proportional allocation [Segal-Halevi et al, 2017;Segal-Halevi et al, 2015] .…”

Section: Related Workmentioning

confidence: 99%

“…4. The cake is 2-dimensional and the pieces must be squares or fat polygons [Segal-Halevi et al, 2017].…”

Section: Geometric Cake Modelsmentioning

confidence: 99%

“…Fair division of land and other resources among agents with different preferences has been an important issue since Biblical times. Today it is an active area of research in the interface of computer science [Robertson and Webb, 1998;Procaccia, 2015;Brânzei, 2015;Segal-Halevi, 2017] and economics [Moulin, 2004;Young, 1995]. Its applications range from politics [Brams and Taylor, 1996;Brams, 2007] to multi-agent systems [Chevaleyre et al, 2006].…”

Section: Introductionmentioning

confidence: 99%

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Redividing the Cake

Segal-Halevi

2018

Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence

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A heterogeneous resource, such as a land-estate, is already divided among several agents in an unfair way. It should be re-divided among the agents in a way that balances fairness with ownership rights. We present re-division protocols that attain various trade-off points between fairness and ownership rights, in various settings differing in the geometric constraints on the allotments: (a) no geometric constraints; (b) connectivity -the cake is a one-dimensional interval and each piece must be a contiguous interval; (c) rectangularity -the cake is a two-dimensional rectangle or rectilinear polygon and the pieces should be rectangles; (d) convexity -the cake is a two-dimensional convex polygon and the pieces should be convex. Our re-division protocols have implications on another problem: the price-of-fairness -the loss of social welfare caused by fairness requirements. Each protocol implies an upper bound on the price-of-fairness with the respective geometric constraints.

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“…A third scenario in which free disposal is required is when the pieces must have a pre-specified geometric shape, such as a square [Segal-Halevi et al 2015a].…”

Section: Legendmentioning

confidence: 99%

Waste Makes Haste

Segal-Halevi

Hassidim

Aumann

2016

ACM Trans. Algorithms

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We consider the classic problem of envy-free division of a heterogeneous good ("cake") among several agents. It is known that, when the allotted pieces must be connected, the problem cannot be solved by a finite algorithm for 3 or more agents. The impossibility result, however, assumes that the entire cake must be allocated. In this paper we replace the entire-allocation requirement with a weaker partial-proportionality requirement: the piece given to each agent must be worth for it at least a certain positive fraction of the entire cake value. We prove that this version of the problem is solvable in bounded time even when the pieces must be connected. We present simple, bounded-time envy-free cake-cutting algorithms for: (1) giving each of n agents a connected piece with a positive value; (2) giving each of 3 agents a connected piece worth at least 1/3; (3) giving each of 4 agents a connected piece worth at least 1/7; (4) giving each of 4 agents a disconnected piece worth at least 1/4; (5) giving each of n agents a disconnected piece worth at least (1−ǫ)/n for any positive ǫ.

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Fair and square: Cake-cutting in two dimensions

Cited by 29 publications

References 67 publications

Fair Division of a Graph

Fair Division of a Graph

Redividing the Cake

Waste Makes Haste

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