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N-Person cake-cutting: there may be no perfect division

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  • Brams, Steven J.
  • Jones, Michael A.
  • Klamler, Christian

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 in which it is impossible to divide it among three players such that these three properties are satisfied, however many 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 an extension of the example for three or more players.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 34264.

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Date of creation: 22 Oct 2011
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Handle: RePEc:pra:mprapa:34264

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Related research

Keywords: Cake-cutting; fair division; efficiency; envy-freeness; equitability; heterogeneous good;

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  1. Barbanel, Julius B. & Brams, Steven J., 2004. "Cake division with minimal cuts: envy-free procedures for three persons, four persons, and beyond," Mathematical Social Sciences, Elsevier, vol. 48(3), pages 251-269, November.
  2. Barbanel, Julius B. & Brams, Steven J., 2011. "Two-person cake-cutting: the optimal number of cuts," MPRA Paper 34263, University Library of Munich, Germany.
  3. Barbanel, Julius B. & Brams, Steven J., 2010. "Two-person pie-cutting: The fairest cuts," MPRA Paper 22703, University Library of Munich, Germany.
  4. Nurmi, Hannu, 1996. "Fair division: From cake-cutting to dispute resolution : Steven J. Brams and Alan D. Taylor, (Cambridge University Press, Cambridge, 1995) pp. xiv + 272, US$ 54.95 (hardcover), US$ 18.95 (paper)," European Journal of Political Economy, Elsevier, vol. 12(1), pages 169-172, April.
  5. I. D. Hill, 2008. "Mathematics and Democracy: Designing Better Voting and Fair-division Procedures," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 171(4), pages 1032-1033.
  6. Brams,Steven J. & Taylor,Alan D., 1996. "Fair Division," Cambridge Books, Cambridge University Press, number 9780521556446, November.
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