Two-Person Pie-Cutting: The Fairest Cuts

Julius B. Barbanel, Steven J. Brams
2011 The College Mathematics Journal  
Pie-cutting is different from cake-cutting. For one thing, a cake is usually rectangular, a pie circular. Cutting a cake into n ≥ 2 pieces typically involves making n-1 parallel, vertical cuts across the cake, whereas cutting a pie usually means cutting wedgeshaped pieces from the center, which requires n cuts. We can think of a cake as being obtained from a pie by making some initial cut, which determines the two ends of the cake, so its remaining division requires only n-1 additional cuts. It
more » ... is the initial cut that essentially differentiates pie-cutting from cake-cutting. We focus on pie-cutting here, although we return to cake-cutting to compare it with pie-cutting. We restrict our analysis to two players, because the properties we impose on the allocation of pie pieces are demanding and cannot all be satisfied if there are three or more players. Assumption and Properties We assume that the two players, player 1 and player 2, wish to divide a pie into two pieces, using two cuts from the center, with one piece allocated to each player. Before stating the properties that we want the allocation to satisfy, it is helpful to provide some mathematical formalism. We assume that the cuts of the pie are radial, so the pie is mathematically equivalent to a circle. Player 1 and player 2 use additive measures m 1 and m 2 , respectively, to assess the value of a piece of pie. These measures are nonatomic (any piece of pie of positive measure is divisible into two pieces, each of positive measure, and therefore any single point of pie has measure 0) and absolutely continuous with respect to each other (any piece of pie that has measure 0 according to one player's measure has measure 0 according to the other player's measure). We lose no generality
doi:10.4169/college.math.j.42.1.025 fatcat:sllbgpn5crhfzjpycnzzhfuh7y