Abstract
A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envy-free, and equitable. We give an example of a cake that is impossible to divide among three players, so that these three properties are satisfied, however many (finite) cuts are made. It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which is not envy-free, is more desirable (a 2-cut division can at best satisfy either envy-freeness or equitability, but not both). We prove that no perfect division exists for more than 4 cuts and for an extension of this example to more than three players.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 35-47 |
| Number of pages | 13 |
| Journal | American Mathematical Monthly |
| Volume | 120 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jan 2013 |
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ASJC Scopus subject areas
- Mathematics(all)
Cite this
N-person cake-cutting : There may be no perfect division. / Brams, Steven; Jones, Michael A.; Klamler, Christian.
In: American Mathematical Monthly, Vol. 120, No. 1, 01.2013, p. 35-47.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - N-person cake-cutting
T2 - There may be no perfect division
AU - Brams, Steven
AU - Jones, Michael A.
AU - Klamler, Christian
PY - 2013/1
Y1 - 2013/1
N2 - A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envy-free, and equitable. We give an example of a cake that is impossible to divide among three players, so that these three properties are satisfied, however many (finite) cuts are made. It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which is not envy-free, is more desirable (a 2-cut division can at best satisfy either envy-freeness or equitability, but not both). We prove that no perfect division exists for more than 4 cuts and for an extension of this example to more than three players.
AB - A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envy-free, and equitable. We give an example of a cake that is impossible to divide among three players, so that these three properties are satisfied, however many (finite) cuts are made. It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which is not envy-free, is more desirable (a 2-cut division can at best satisfy either envy-freeness or equitability, but not both). We prove that no perfect division exists for more than 4 cuts and for an extension of this example to more than three players.
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U2 - 10.4169/amer.math.monthly.120.01.035
DO - 10.4169/amer.math.monthly.120.01.035
M3 - Article
VL - 120
SP - 35
EP - 47
JO - American Mathematical Monthly
JF - American Mathematical Monthly
SN - 0002-9890
IS - 1
ER -