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Cake Cutting – Fair and Square

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  • Erel Segal-Halevi
  • Shmuel Nitzan

    ()
    (Bar-Ilan University)

Abstract

The classic fair cake-cutting problem [Steinhaus, 1948] is extended by introducing geometric constraints on the allocated pieces. Specifically, agents may demand to get their share as a square or a rectangle with a bounded length/width ratio. This is a plausible constraint in realistic cake-cutting applications, notably in urban and agricultural economics where the “cake” is land. Geometric constraints greatly affect the classic results of the fair division theory. The existence of a proportional division, giving each agent 1/n of his total cake value, is no longer guaranteed. We prove that it is impossible to guarantee each agent more than 1/(2n-1) of his total value. Moreover, we provide procedures implementing partially proportional division, giving each agent 1/(An-B) of his total value, where A and B are constants depending on the shape of the cake and its pieces. Fairness and social welfare implications of these procedures are analyzed in various scenarios.

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Bibliographic Info

Paper provided by Bar-Ilan University, Department of Economics in its series Working Papers with number 2014-01.

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Length: 44 pages
Date of creation: Jan 2014
Date of revision:
Handle: RePEc:biu:wpaper:2014-01

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Keywords: fair division; cake cutting; land division; geometry; non-additive utilities; social welfare;

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References

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  1. Dall'Aglio, Marco & Maccheroni, Fabio, 2009. "Disputed lands," Games and Economic Behavior, Elsevier, vol. 66(1), pages 57-77, May.
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  12. Z. Landau & O. Reid & I. Yershov, 2009. "A fair division solution to the problem of redistricting," Social Choice and Welfare, Springer, vol. 32(3), pages 479-492, March.
  13. Steven Brams & Michael Jones & Christian Klamler, 2008. "Proportional pie-cutting," International Journal of Game Theory, Springer, vol. 36(3), pages 353-367, March.
  14. 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.
  15. Francis Su, . "Rental Harmony: Sperner's Lemma in Fair Division," Claremont Colleges Working Papers 1999-10, Claremont Colleges.
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