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

Author

Listed:
  • 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.

Suggested Citation

  • Brams, Steven J. & Jones, Michael A. & Klamler, Christian, 2011. "N-Person cake-cutting: there may be no perfect division," MPRA Paper 34264, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:34264
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    File URL: https://mpra.ub.uni-muenchen.de/34264/1/MPRA_paper_34264.pdf
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    References listed on IDEAS

    as
    1. Brams, Steven J. & Jones, Michael A. & Klamler, Christian, 2010. "Divide-and-conquer: A proportional, minimal-envy cake-cutting algorithm," MPRA Paper 22704, University Library of Munich, Germany.
    2. 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, October.
    3. Barbanel, Julius B. & Brams, Steven J., 2011. "Two-person cake-cutting: the optimal number of cuts," MPRA Paper 34263, University Library of Munich, Germany.
    4. Barbanel, Julius B. & Brams, Steven J., 2010. "Two-person pie-cutting: The fairest cuts," MPRA Paper 22703, University Library of Munich, Germany.
    5. Barbanel,Julius B. Introduction by-Name:Taylor,Alan D., 2005. "The Geometry of Efficient Fair Division," Cambridge Books, Cambridge University Press, number 9780521842488.
    6. Brams,Steven J. & Taylor,Alan D., 1996. "Fair Division," Cambridge Books, Cambridge University Press, number 9780521556446.
    7. 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.
    8. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Soldatos, Gerasimos T., 2014. "On the Religion-Public Policy Correlation," MPRA Paper 60859, University Library of Munich, Germany.
    2. Orit Arzi & Yonatan Aumann & Yair Dombb, 2016. "Toss one’s cake, and eat it too: partial divisions can improve social welfare in cake cutting," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 46(4), pages 933-954, April.
    3. Maria Kyropoulou & Josu'e Ortega & Erel Segal-Halevi, 2018. "Fair Cake-Cutting in Practice," Papers 1810.08243, arXiv.org, revised Feb 2022.
    4. Ji-Won Park & Jaeup U. Kim & Cheol-Min Ghim & Chae Un Kim, 2021. "The Boltzmann fair division for distributive justice," Papers 2109.11917, arXiv.org, revised Nov 2021.
    5. Chèze, Guillaume, 2017. "Existence of a simple and equitable fair division: A short proof," Mathematical Social Sciences, Elsevier, vol. 87(C), pages 92-93.
    6. Marco Dall'Aglio & Camilla Di Luca & Lucia Milone, 2017. "Finding the Pareto optimal equitable allocation of homogeneous divisible goods among three players," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 27(3), pages 35-50.

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    More about this item

    Keywords

    Cake-cutting; fair division; efficiency; envy-freeness; equitability; heterogeneous good;
    All these keywords.

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
    • D30 - Microeconomics - - Distribution - - - General
    • D74 - Microeconomics - - Analysis of Collective Decision-Making - - - Conflict; Conflict Resolution; Alliances; Revolutions
    • D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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