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Cake Division with Minimal Cuts: Envy-Free Procedures for 3 Person, 4 Persons, and Beyond


  • Barbanel, J. B.
  • Brams, S. J.


The minimal number of parallel cuts required to divide a cake into n pieces is n-1. A new 3-person procedure, requiring 2 parallel cuts, is given that produces an envy- free division, whereby each person thinks he or she receives at least a tied- for- largest piece. An extension of this procedure leads to a 4-person division, us ing 3 parallel cuts, that makes at most one player envious. Finally, a 4-person envy-free procedure is given, but it requires up to 5 parallel cuts, and some pieces may be disconnected. All these procedures improve on extant procedures by using fewer moving knives, making fewer people envious, or using fewer cuts. While the 4-person, 5-cut procedure is complex, endowing people with more information about others' preferences, or allowing them to do things beyond stopping moving knives, may yield simpler procedures for making envy- free divisions with minimal cuts, which are known always to exist

Suggested Citation

  • Barbanel, J. B. & Brams, S. J., 2001. "Cake Division with Minimal Cuts: Envy-Free Procedures for 3 Person, 4 Persons, and Beyond," Working Papers 01-07, C.V. Starr Center for Applied Economics, New York University.
  • Handle: RePEc:cvs:starer:01-07

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    References listed on IDEAS

    1. Francis Su, "undated". "Rental Harmony: Sperner's Lemma in Fair Division," Claremont Colleges Working Papers 1999-10, Claremont Colleges.
    2. Steven J. Brams & D. Marc Kilgour, 2001. "Competitive Fair Division," Journal of Political Economy, University of Chicago Press, vol. 109(2), pages 418-443, April.
    3. Elisha Peterson & Francis Su, 2000. "Four-Person Envy-Free Chore Division," Claremont Colleges Working Papers 2000-48, Claremont Colleges.
    4. Brams,Steven J. & Taylor,Alan D., 1996. "Fair Division," Cambridge Books, Cambridge University Press, number 9780521556446, May.
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    Cited by:

    1. Nicolò, Antonio & Yu, Yan, 2008. "Strategic divide and choose," Games and Economic Behavior, Elsevier, vol. 64(1), pages 268-289, September.

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    JEL classification:

    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement


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