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Probability Density Function
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Cakecutting algorithms  be fair if you can
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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
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Cutting a Pie Is Not a Piece of Cake
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Citations: 7
)
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Julius B. Barbanel
,
Steven J. Brams
,
Walter Stromquist
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 βα or 1(βα), 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. 496514, 2009
DOI:
10.4169/193009709X470407
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Citation Context
(3)
...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 uptodate 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 envyfree, undominated allocation for two players and a procedure that produces an envyfree 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. Brams
,
et al.
Proportional piecutting
...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 envyfree, 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
References
(12)
Cake Division with Minimal Cuts: EnvyFree Procedures for 3 Person, 4 Persons, and Beyond
(
Citations: 12
)
Julius B. Barbanel
,
S. J. Brams
Journal:
Mathematical Social Sciences
, 2001
Fair division  from cakecutting to dispute resolution
(
Citations: 259
)
Steven J. Brams
,
Alan D. Taylor
Published in 1996.
Old and new movingknife schemes
(
Citations: 15
)
Steven J. Brams
,
Alan D. Taylor
,
William S. Zwicker
Journal:
Mathematical Intelligencer  MATH INTELL
, vol. 17, no. 4, pp. 3035, 1995
A MOVINGKNIFE SOLUTION TO THE FOURPERSON ENVYFREE CAKEDIVISION PROBLEM
(
Citations: 21
)
STEVEN J. BRAMS
,
ALAN D. TAYLOR
,
WILLIAM S. ZWICKER
Published in 1997.
Better Ways to Cut a Cake
(
Citations: 22
)
Steven J. Brams
,
Michael A. Jones
,
Christian Klamler
Sort by:
Citations
(7)
Fair Allocation Rules
(
Citations: 19
)
Thomson William
Journal:
Handbook of Social Choice and Welfare
, vol. 2, pp. 393506, 2011
DivideandConquer: A Proportional, MinimalEnvy CakeCutting Algorithm
STEVEN J. BRAMSy
,
MICHAEL A. JONESz
,
Christian Klamler
Published in 2011.
Math and Congressional Redistricting
James Case
Published in 2009.
Proportional piecutting
(
Citations: 7
)
Steven J. Brams
,
Michael A. Jones
,
Christian Klamler
Journal:
International Journal of Game Theory  INT J GAME THEORY
, vol. 36, no. 3, pp. 353367, 2008
A Pie That Can't Be Cut Fairly
Walter Stromquist
Published in 2007.