IDEAS home Printed from https://ideas.repec.org/p/pra/mprapa/12772.html
   My bibliography  Save this paper

Cutting a pie is not a piece of cake

Author

Listed:
  • Barbanel, Julius B.
  • Brams, Steven J.
  • Stromquist, Walter

Abstract

Is there a division among n players of a cake using n-1 parallel vertical cuts, or of a pie using n radial cuts, that is envy-free (each player thinks he or she receives a largest piece and so does not envy another player) and undominated (there is no other allocation as good for all players and better for at least one)? David Gale first asked this question for pies. We provide complete answers for both cakes and pies. The answers depend on the number of players (two versus three or more players) and whether the players' preferences satisfy certain continuity assumptions. We also give some simple algorithms for cutting a pie when there are two or more players, but these algorithms do not guarantee all the properties one might desire in a division, which makes pie-cutting harder than cake-cutting. We suggest possible applications and conclude with two open questions.

Suggested Citation

  • Barbanel, Julius B. & Brams, Steven J. & Stromquist, Walter, 2008. "Cutting a pie is not a piece of cake," MPRA Paper 12772, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:12772
    as

    Download full text from publisher

    File URL: https://mpra.ub.uni-muenchen.de/12772/1/MPRA_paper_12772.pdf
    File Function: original version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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. Introduction by-Name:Taylor,Alan D., 2005. "The Geometry of Efficient Fair Division," Cambridge Books, Cambridge University Press, number 9780521842488, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Brams, Steven & Landweber, Peter, 2018. "3 Persons, 2 Cuts: A Maximin Envy-Free and a Maximally Equitable Cake-Cutting Algorithm," MPRA Paper 84683, University Library of Munich, Germany.
    2. 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.
    3. 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.
    4. William Thomson, 2007. "Children Crying at Birthday Parties. Why?," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 31(3), pages 501-521, June.
    5. Barbanel, Julius B. & Brams, Steven J., 2011. "Two-person cake-cutting: the optimal number of cuts," MPRA Paper 34263, University Library of Munich, Germany.
    6. Park, Ji-Won & Kim, Chae Un & Isard, Walter, 2012. "Permit allocation in emissions trading using the Boltzmann distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(20), pages 4883-4890.
    7. Ji-Won Park & Chae Un Kim & Walter Isard, 2011. "Permit Allocation in Emissions Trading using the Boltzmann Distribution," Papers 1108.2305, arXiv.org, revised Mar 2012.
    8. Segal-Halevi, Erel & Nitzan, Shmuel & Hassidim, Avinatan & Aumann, Yonatan, 2017. "Fair and square: Cake-cutting in two dimensions," Journal of Mathematical Economics, Elsevier, vol. 70(C), pages 1-28.
    9. Barbanel, Julius B. & Brams, Steven J., 2010. "Two-person pie-cutting: The fairest cuts," MPRA Paper 22703, University Library of Munich, Germany.
    10. Marco LiCalzi & Antonio Nicolò, 2009. "Efficient egalitarian equivalent allocations over a single good," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 40(1), pages 27-45, July.
    11. Erel Segal-Halevi & Shmuel Nitzan, 2019. "Fair cake-cutting among families," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 53(4), pages 709-740, December.
    12. Bilò, Vittorio & Caragiannis, Ioannis & Flammini, Michele & Igarashi, Ayumi & Monaco, Gianpiero & Peters, Dominik & Vinci, Cosimo & Zwicker, William S., 2022. "Almost envy-free allocations with connected bundles," Games and Economic Behavior, Elsevier, vol. 131(C), pages 197-221.
    13. Chèze, Guillaume, 2017. "Existence of a simple and equitable fair division: A short proof," Mathematical Social Sciences, Elsevier, vol. 87(C), pages 92-93.
    14. Vittorio Bil`o & Ioannis Caragiannis & Michele Flammini & Ayumi Igarashi & Gianpiero Monaco & Dominik Peters & Cosimo Vinci & William S. Zwicker, 2018. "Almost Envy-Free Allocations with Connected Bundles," Papers 1808.09406, arXiv.org, revised May 2022.
    15. Nicolò, Antonio & Yu, Yan, 2008. "Strategic divide and choose," Games and Economic Behavior, Elsevier, vol. 64(1), pages 268-289, September.
    16. Fedor Sandomirskiy & Erel Segal-Halevi, 2019. "Efficient Fair Division with Minimal Sharing," Papers 1908.01669, arXiv.org, revised Apr 2022.
    17. Erel Segal-Halevi & Shmuel Nitzan & Avinatan Hassidim & Yonatan Aumann, 2020. "Envy-Free Division of Land," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 896-922, August.
    18. Edith Cohen & Michal Feldman & Amos Fiat & Haim Kaplan & Svetlana Olonetsky, 2010. "Envy-Free Makespan Approximation," Discussion Paper Series dp539, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    19. Taylor, Alan D., 2005. "A paradoxical Pareto frontier in the cake-cutting context," Mathematical Social Sciences, Elsevier, vol. 50(2), pages 227-233, September.
    20. Vittorio Bilò & Ioannis Caragiannis & Michele Flammini & Ayumi Igarashi & Gianpiero Monaco & Dominik Peters & Cosimo Vinci & William Zwicker, 2021. "Almost Envy-Free Allocations with Connected Bundles," Post-Print hal-03834506, HAL.

    More about this item

    Keywords

    Fair division; cake-cutting; pie-cutting; divisible good; envy-freeness; allocative efficiency;
    All these keywords.

    JEL classification:

    • D7 - Microeconomics - - Analysis of Collective Decision-Making
    • D6 - Microeconomics - - Welfare Economics
    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:pra:mprapa:12772. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Joachim Winter (email available below). General contact details of provider: https://edirc.repec.org/data/vfmunde.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.