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Cutting a Pie Is Not a Piece of Cake

Cutting a Pie Is Not a Piece of Cake,10.4169/193009709X470407,American Mathematical Monthly,Julius B. Barbanel,Steven J. Brams,Walter Stromquist

Cutting a Pie Is Not a Piece of Cake   (Citations: 7)
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Let's deem a pie mathematically equivalent to the circle S 1 = R / Z, with representative points in the unit interval (0,1) . A pair of real numbers α,β∈(0,1) with α<β can be used to specify either the sector of pie given by the subinterval (α,β) or its complement (0,1)\(α,β), which we shall denote by (β,α). Cutting along a diameter, for example, divides the pie into two sectors (α,α+0.5) and (α+0.5,α) for some α∈(0,1). By the "length" of a sector (α,β) or (β,α) for α<β, we mean the value β-α or 1-(β-α), respectively. Diameter cuts, for example, produce sectors of length 0.5. (One comment: Our notation does not allow for the possibility that one player receives a sector of length zero. We do allow for this possibility, but we will not need notation for it.) In order to assess values of pieces, let's assume that each player uses an integrable probability density function p:(0,1)→R satisfying p(0) = p(1) (and, of course, p(t)dt (0,1) ∫ =1 ). This player then assigns the value ("measure") p(t)dt (α,β) ∫ to the sector (α,β). We say that a player prefers one sector over another if her measure of the first sector is strictly greater than her measure of the second. Players' measures of sectors are continuous in the following sense: If (α n ) and (β n ) are sequences in S 1 converging to α∈S 1 and β∈S 1 , respectively, with respect to the absolute value function induced from R, then p(t)dt (α n ,β n ) ∫
Journal: American Mathematical Monthly - AMER MATH MON , vol. 116, no. 6, pp. 496-514, 2009
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    • ...For Gale, a “pie” is the perimeter of the disc { z: |z| ≤ 1/2 π}, and a “cake” is the unit interval I = [0,1]...
    • ...An up-to-date exposition of these results can be found in [1]...

    James Case. Math and Congressional Redistricting

    • ...(2007). Barbanel et al. (2007) also provide a procedure that produces an envy-free, undominated allocation for two players and a procedure that produces an envy-free but not necessarily efficient allocation for three players...
    • ...Whereas we restrict our attention to additive preferences, as Barbanel et al. (2007) do, Thomson (2007) considers more general preferences...
    • ...For two players, Barbanel et al. (2007) also rely on geometry, applying the approach developed in Barbanel (2005)...
    • ...For equal entitlements, Barbanel et al. (2007) showed this impossibility with a much complex example...
    • ...The first example is simple and intuitive, whereas the second example is more general but still simpler than the example with equal entitlements in Barbanel et al. (2007)...

    Steven J. Bramset al. Proportional pie-cutting

    • ...Brams and Barbanel [3] showed that such an allocation always exists when n = 2. They also found an example with n = 4 in which no such allocation exists, but the example relies on measures that are not absolutely continuous with respect to each other...
    • ...Points in the interval are defined mod 18, so that, for example, [15,3] and [15,21] both represent the same connected interval...
    • ...5. In any envy-free, undominated allocation, B’s piece cannot include any part of the sector [2,3]...
    • ...Proof: B would need to avoid most of [3,4]...
    • ...But this allocation is dominated by [3,9], [9,15], [15,3] which delivers (602,600,600)...
    • ...But this allocation is dominated by [3,9], [9,15], [15,3] which delivers (602,600,600)...
    • ...• Plan 3: [3,9], [9,15], [15,3] to A, B, C respectively...
    • ...• Plan 3: [3,9], [9,15], [15,3] to A, B, C respectively...
    • ...Plan 2. To do this, we freeze it in place with another set of “decorations,” and then shift 1 extra point of A’s measure into each sectors [3,4] and [8,9] to make them more attractive...

    Walter Stromquist. A Pie That Can't Be Cut Fairly

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