Cake division with minimal cuts: envy-free procedures for three persons, four persons, and beyond

Barbanel, Julius B.; Brams, Steven J.

doi:10.1016/j.mathsocsci.2004.03.006

Cited by 58 publications

(44 citation statements)

References 15 publications

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“…In the fair division of a divisible good like land, players will often seek connected pieces, for which algorithms that use a minimal number of cuts suffice (Barbanel and Brams [2]). In other situations, however, in which one might seek different parts of a divisible good (e.g., time slots in a day that fit into one's schedule, different sections of a beach that an investor wants to develop), disconnectedness is not a problem.…”

Section: Discussionmentioning

confidence: 99%

“…Recent algorithms, involving both discrete and continuous (moving-knife) procedures, are analyzed in Barbanel and Brams [2,4], Brams, Jones, and Klamler [8,10], Caragiannis, Lai, and Procaccia [12], and Cohler et al [13]. There is also a growing literature on pie-cutting (Brams, Jones, and Klamler [9]; Barbanel, Brams, and Stromquist [5]; Barbanel and Brams [3]), in which radial cuts are made from the center of a pie rather than parallel cuts being made along the edge of a cake.…”

Section: Introductionmentioning

confidence: 99%

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N-Person Cake-Cutting: There May Be No Perfect Division

2011

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A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envy-free, and equitable. We give an example of a cake in which it is impossible to divide it among three players such that these three properties are satisfied, however many cuts are made. It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which is not envy-free, is more desirable (a 2-cut division can at best satisfy either envy-freeness or equitability but not both). We prove that no perfect division exists for an extension of the example for three or more players.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

N-Person Cake-Cutting: There May Be No Perfect Division

2011

Self Cite

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“…There are simpler algorithms for the same task, e.g. Robertson and Webb [1998][pages 77-78] and Barbanel and Brams [2004], but they also use moving-knives. A discrete and finite algorithm for envy-free division for 3 agents was constructed by Selfridge and Conway [Brams and Taylor 1996].…”

Section: Related Workmentioning

confidence: 99%

“…This proved that the problem of envy-free division is strictly more difficult than the problem of proportional division. Barbanel and Brams [2004] Con. 3 General 2 Infinite 0 1/3 * Su [1999] Con.…”

Section: Related Workmentioning

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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“…The general problem of fair division and the specific problem of cutting a cake fairly have received much attention in recent years (for overviews, see [1], [2], [4], [5], [8]). Cutting a pie into wedge-shaped sectors, by contrast, has received far less attention, though it would seem that the connection between cake-cutting and piecutting is close ( [3], [11]).…”

Section: Introductionmentioning

confidence: 99%