IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v60y2012i6p1461-1476.html
   My bibliography  Save this article

Algorithmic Solutions for Envy-Free Cake Cutting

Author

Listed:
  • Xiaotie Deng

    (Department of Computer Science, University of Liverpool, Liverpool L69 3BX, United Kingdom; and Department of Computer Science, City University of Hong Kong, Hong Kong)

  • Qi Qi

    (Department of Industrial Engineering and Logistics Management, Hong Kong University of Science and Technology, Hong Kong)

  • Amin Saberi

    (Department of Management Science and Engineering, Stanford University, Stanford, California 94305)

Abstract

We study the problem of finding an envy-free allocation of a cake to d + 1 players using d cuts. Two models are considered, namely, the oracle-function model and the polynomial-time function model. In the oracle-function model, we are interested in the number of times an algorithm has to query the players about their preferences to find an allocation with the envy less than (epsilon). We derive a matching lower and upper bound of (theta)(1/(epsilon)) d - 1 for players with Lipschitz utilities and any d > 1. In the polynomial-time function model, where the utility functions are given explicitly by polynomial-time algorithms, we show that the envy-free cake-cutting problem has the same complexity as finding a Brouwer's fixed point, or, more formally, it is PPAD-complete. On the flip side, for monotone utility functions, we propose a fully polynomial-time algorithm (FPTAS) to find an approximate envy-free allocation of a cake among three people using two cuts.

Suggested Citation

  • Xiaotie Deng & Qi Qi & Amin Saberi, 2012. "Algorithmic Solutions for Envy-Free Cake Cutting," Operations Research, INFORMS, vol. 60(6), pages 1461-1476, December.
    • Handle: RePEc:inm:oropre:v:60:y:2012:i:6:p:1461-1476
    DOI: 10.1287/opre.1120.1116
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.1120.1116
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.1120.1116?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Katerina Sherstyuk, 1998. "How to gerrymander: A formal analysis," Public Choice, Springer, vol. 95(1), pages 27-49, April.
    2. Abdelghani A. Elimam & Maurice Girgis & Samir Kotob, 1996. "The Use of Linear Programming in Disentangling the Bankruptcies of Al-Manakh Stock Market Crash," Operations Research, INFORMS, vol. 44(5), pages 665-676, October.
    3. Cloutier, John & Nyman, Kathryn L. & Su, Francis Edward, 2010. "Two-player envy-free multi-cake division," Mathematical Social Sciences, Elsevier, vol. 59(1), pages 26-37, January.
    4. Xiaotie Deng & Qi Qi & Amin Saberi & Jie Zhang, 2011. "Discrete Fixed Points: Models, Complexities, and Applications," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 636-652, November.
    5. Frédéric Meunier, 2008. "Discrete Splittings of the Necklace," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 678-688, August.
    6. Shahar Dobzinski & Noam Nisan & Michael Schapira, 2010. "Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders," Mathematics of Operations Research, INFORMS, vol. 35(1), pages 1-13, February.
    7. Simmons, Forest W. & Su, Francis Edward, 2003. "Consensus-halving via theorems of Borsuk-Ulam and Tucker," Mathematical Social Sciences, Elsevier, vol. 45(1), pages 15-25, February.
    8. Scarf, Herbert E., 1993. "The computation of equilibrium prices: An exposition," Handbook of Mathematical Economics, in: K. J. Arrow & M.D. Intriligator (ed.), Handbook of Mathematical Economics, edition 4, volume 2, chapter 21, pages 1007-1061, Elsevier.
    9. Hylland, Aanund & Zeckhauser, Richard, 1979. "The Efficient Allocation of Individuals to Positions," Journal of Political Economy, University of Chicago Press, vol. 87(2), pages 293-314, April.
    10. 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.
    11. John Winsor Pratt & Richard Jay Zeckhauser, 1990. "The Fair and Efficient Division of the Winsor Family Silver," Management Science, INFORMS, vol. 36(11), pages 1293-1301, November.
    12. Hervé Moulin, 2007. "Minimizing the Worst Slowdown: Offline, Online," Operations Research, INFORMS, vol. 55(5), pages 876-889, October.
    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. Sagrario Lantarón & Mariló López & Susana Merchán & Javier Rodrigo & José Samuel Rodríguez, 2021. "Envy-Free Allocation by Sperner’s Lemma Adapted to Rotation Shifts in a Company," Mathematics, MDPI, vol. 9(9), pages 1-12, April.
    2. 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.
    3. 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.
    4. 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.
    5. 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.

    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. Shell, Karl & Wright, Randall, 1993. "Indivisibilities, Lotteries, and Sunspot Equilibria," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 3(1), pages 1-17, January.
    2. Doğan, Battal, 2016. "Nash-implementation of the no-envy solution on symmetric domains of economies," Games and Economic Behavior, Elsevier, vol. 98(C), pages 165-171.
    3. Xiaotie Deng & Qi Qi & Amin Saberi & Jie Zhang, 2011. "Discrete Fixed Points: Models, Complexities, and Applications," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 636-652, November.
    4. Anna Bogomolnaia & Hervé Moulin & Fedor Sandomirskiy & Elena Yanovskaya, 2017. "Competitive Division of a Mixed Manna," Econometrica, Econometric Society, vol. 85(6), pages 1847-1871, November.
    5. Ketelaars, Martijn & Borm, Peter & Herings, P.J.J., 2023. "Duality in Financial Networks," Other publications TiSEM 26750293-9599-4e05-9ae1-8, Tilburg University, School of Economics and Management.
    6. Scott Duke Kominers & Alexander Teytelboym & Vincent P Crawford, 2017. "An invitation to market design," Oxford Review of Economic Policy, Oxford University Press and Oxford Review of Economic Policy Limited, vol. 33(4), pages 541-571.
    7. Battal Doğan & M. Bumin Yenmez, 2023. "When does an additional stage improve welfare in centralized assignment?," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 76(4), pages 1145-1173, November.
    8. Julien Combe & Vladyslav Nora & Olivier Tercieux, 2021. "Dynamic assignment without money: Optimality of spot mechanisms," Working Papers 2021-11, Center for Research in Economics and Statistics.
    9. Robert Scherf & Matthew Weinzierl, 2020. "Understanding Different Approaches to Benefit‐Based Taxation," Fiscal Studies, John Wiley & Sons, vol. 41(2), pages 385-410, June.
    10. Ivan Balbuzanov & Maciej H. Kotowski, 2019. "Endowments, Exclusion, and Exchange," Econometrica, Econometric Society, vol. 87(5), pages 1663-1692, September.
    11. Miralles, Antonio & Pycia, Marek, 2021. "Foundations of pseudomarkets: Walrasian equilibria for discrete resources," Journal of Economic Theory, Elsevier, vol. 196(C).
    12. Bogomolnaia, Anna & Moulin, Herve, 2015. "Size versus fairness in the assignment problem," Games and Economic Behavior, Elsevier, vol. 90(C), pages 119-127.
    13. Nolan Miller & Alexander Wagner & Richard Zeckhauser, 2013. "Solomonic separation: Risk decisions as productivity indicators," Journal of Risk and Uncertainty, Springer, vol. 46(3), pages 265-297, June.
    14. Korpela, Ville & Lombardi, Michele & Saulle, Riccardo D., 2024. "Designing rotation programs: Limits and possibilities," Games and Economic Behavior, Elsevier, vol. 143(C), pages 77-102.
    15. Fu, Hu & Kleinberg, Robert & Lavi, Ron & Smorodinsky, Rann, 2017. "Job security, stability and production efficiency," Theoretical Economics, Econometric Society, vol. 12(1), January.
    16. Bettina Klaus & David F. Manlove & Francesca Rossi, 2014. "Matching under Preferences," Cahiers de Recherches Economiques du Département d'économie 14.07, Université de Lausanne, Faculté des HEC, Département d’économie.
    17. Han, Xiang, 2024. "On the efficiency and fairness of deferred acceptance with single tie-breaking," Journal of Economic Theory, Elsevier, vol. 218(C).
    18. Eun Jeong Heo & Vikram Manjunath, 2017. "Implementation in stochastic dominance Nash equilibria," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 48(1), pages 5-30, January.
    19. Katharina Huesmann & Achim Wambach, 2015. "Constraints on Matching Markets Based on Moral Concerns," CESifo Working Paper Series 5356, CESifo.
    20. Kesten, Onur, 2009. "Why do popular mechanisms lack efficiency in random environments?," Journal of Economic Theory, Elsevier, vol. 144(5), pages 2209-2226, September.

    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:inm:oropre:v:60:y:2012:i:6:p:1461-1476. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.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.