Two-Person Cake-Cutting: The Optimal Number of Cuts

Abstract

A cake is a metaphor for a heterogeneous, divisible good. When two players divide such a good, there is always a perfect division-one that is efficient (Pareto-optimal), envyfree, and equitable-which can be effected with a finite number of cuts under certain mild conditions; this is not always the case when there are more than two players (Brams, Jones, and Klamler, 2011b). We not only establish the existence of such a division but also provide an algorithm for determining where and how many cuts must be made, relating it to an algorithm, "Adjusted Winner" Taylor, 1996, 1999), that yields a perfect division of multiple homogenous goods.

Envy-Free Cake-Cutting among Families

“…In the impossibility results, it is implicitly assumed that at least one family contains at least 2 members (which implies n > k). 3 A property of cake partitions is called feasible if for every k families and n agents there exists an allocation satisfying this property. Otherwise, the property is called infeasible.…”

“…In this case, the components are sub-intervals and their number is one plus the number of cuts. Hence, the number of components is minimized by minimizing the number of cuts [28,37,32,2,3].…”

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

“…However, it is not o-efficient, because each player receives more ( 2 3 instead of 1 2 ) when the aforementioned two cuts are made. Barbanel and Brams [4] give a 2-player algorithm for finding an envy-free and equitable allocation that is o-efficient, which specifies (i) how many cuts are required and (ii) where they must be made to produce a perfect division of a cake.…”

“…It is worth pointing out that there is no o-efficient division, even for two players, if their value functions cross an infinite number of times (Barbanel and Brams [4]). The reason is that while adding another cut can always bring one closer to the maxsum allocation, it will never be achieved-and so give o-efficiency-with a finite number of cuts.…”

“…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.…”