On the structure of Pareto optimal cake partitions

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

“…5 We note that the idea of considering ratios of players' pdfs to find efficient divisions of a cake is not new. It was used by Weller (1985) and by Barbanel (2000Barbanel ( , 2005, although the geometric structure used there, which makes no continuity assumptions, is rather different from the one we use here. (It was called the "RadonNikodym Set" in Barbanel (2000Barbanel ( , 2005.)…”

“…Thus, player 1's portion of the original cake (i.e., the line interval [0,1]) is [0,0.4), and player 2's portion is [0. 4,1]. Using the players' pdfs, we find that player 1's valuation of his portion is (0.4)(1.5) = 0.6, and player 2's valuation of her portion is (0.6)(1.25) = 0.75.…”

“…5. If the areas are not equal, the player who receives the larger sum (say, player 1) gives subintervals, or portions thereof (i.e., sub-subintervals), to player 2 according to the ratio of player 1's pdf to player 2's pdf, starting with the subinterval in which this ratio is the smallest (possibly 1 if the players pdfs are equal on some subinterval) and letting this ratio increase. 6.…”

“…We assume that f 1 Let G be the graph of f. G need not be connected; because f 1 and f 2 are each piecewise continuous, however, so is f, and hence G comprises a finite number of connected pieces. Since f provides a bijection between the cake C and the graph G, we can identify C with G. Any division of the graph G into portions corresponds to a division of the cake C into portions.…”

“…Recent algorithms, involving both discrete and continuous (moving-knife) procedures, are analyzed in Barbanel andBrams (2004, 2011) and Brams, Jones, and Klamler (2006Klamler ( , 2011aKlamler ( , 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.…”