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" (Brams and Taylor, 1996, 1999), that yields a perfect division of multiple homogenous goods. Recent algorithms, involving both discrete and continuous (moving-knife) procedures, are analyzed in Barbanel and Brams (2004, 2011) and Brams, Jones, and Klamler (2006, 2011a, 2011b). There is also a growing literature on pie-cutting (Brams, Jones, and Klamler, 2008; Barbanel, Brams, and Stromquist, 2009; Barbanel and Brams, 2011), in which radial cuts are made from the center of a pie rather than parallel cuts being made along the edge of a cake.