IDEAS home Printed from https://ideas.repec.org/p/pra/mprapa/34263.html
   My bibliography  Save this paper

Two-person cake-cutting: the optimal number of cuts

Author

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

Abstract

A cake is a metaphor for a heterogeneous, divisible good. When two players divide such a good, there is always a perfect division—one that is efficient (Pareto-optimal), envy-free, and equitable—which can be effected with a finite number of cuts under certain mild conditions; this is not always the case when there are more than two players (Brams, Jones, and Klamler, 2011b). We not only establish the existence of such a division but also provide an algorithm for determining where and how many cuts must be made, relating it to an algorithm, “Adjusted Winner” (Brams and Taylor, 1996, 1999), that yields a perfect division of multiple homogenous goods.

Suggested Citation

  • Barbanel, Julius B. & Brams, Steven J., 2011. "Two-person cake-cutting: the optimal number of cuts," MPRA Paper 34263, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:34263
    as

    Download full text from publisher

    File URL: https://mpra.ub.uni-muenchen.de/34263/1/MPRA_paper_34263.pdf
    File Function: original version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Barbanel,Julius B. Introduction by-Name:Taylor,Alan D., 2005. "The Geometry of Efficient Fair Division," Cambridge Books, Cambridge University Press, number 9780521842488, October.
    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. 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.
    4. Barbanel, Julius B. & Brams, Steven J., 2010. "Two-person pie-cutting: The fairest cuts," MPRA Paper 22703, University Library of Munich, Germany.
    5. Weller, Dietrich, 1985. "Fair division of a measurable space," Journal of Mathematical Economics, Elsevier, vol. 14(1), pages 5-17, February.
    6. 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

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.

    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 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. Barbanel, Julius B. & Brams, Steven J., 2010. "Two-person pie-cutting: The fairest cuts," MPRA Paper 22703, University Library of Munich, Germany.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. Taylor, Alan D., 2005. "A paradoxical Pareto frontier in the cake-cutting context," Mathematical Social Sciences, Elsevier, vol. 50(2), pages 227-233, September.
    12. 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.
    13. Farhad Hüsseinov & Nobusumi Sagara, 2013. "Existence of efficient envy-free allocations of a heterogeneous divisible commodity with nonadditive utilities," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(4), pages 923-940, October.
    14. Thomson, William, 2011. "Chapter Twenty-One - Fair Allocation Rules," Handbook of Social Choice and Welfare, in: K. J. Arrow & A. K. Sen & K. Suzumura (ed.), Handbook of Social Choice and Welfare, edition 1, volume 2, chapter 21, pages 393-506, Elsevier.
    15. Sagara, Nobusumi, 2008. "A characterization of [alpha]-maximin solutions of fair division problems," Mathematical Social Sciences, Elsevier, vol. 55(3), pages 273-280, May.
    16. Radzvilas, Mantas, 2016. "Hypothetical Bargaining and the Equilibrium Selection Problem in Non-Cooperative Games," MPRA Paper 70248, University Library of Munich, Germany.
    17. Steven Brams & D. Kilgour, 2013. "Kingmakers and leaders in coalition formation," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(1), pages 1-18, June.
    18. Steven Brams & D. Kilgour & Christian Klamler, 2012. "The undercut procedure: an algorithm for the envy-free division of indivisible items," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(2), pages 615-631, July.
    19. Steven J. Brams & D. Marc Kilgour, 2014. "Satisfaction Approval Voting," Studies in Choice and Welfare, in: Rudolf Fara & Dennis Leech & Maurice Salles (ed.), Voting Power and Procedures, edition 127, pages 323-346, Springer.
    20. Thomson, William, 2005. "Divide-and-permute," Games and Economic Behavior, Elsevier, vol. 52(1), pages 186-200, July.

    More about this item

    Keywords

    Cake-cutting; fair division; envy-freeness; adjusted winner; heterogeneous good;
    All these keywords.

    JEL classification:

    • 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
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

    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:34263. 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.