Cake-Cutting Is Not a Piece of Cake

  • Malik Magdon-Ismail
  • Costas Busch
  • Mukkai S. Krishnamoorthy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2607)


Fair cake-cutting is the division of a cake or resource among N users so that each user is content. Users may value a given piece of cake differently, and information about how a user values different parts of the cake can only be obtained by requesting users to “cut” pieces of the cake into specified ratios. One of the most interesting open questions is to determine the minimum number of cuts required to divide the cake fairly. It is known that O(N logN) cuts suffices, however, it is not known whether one can do better.

We show that sorting can be reduced to cake-cutting: any algorithm that performs fair cake-division can sort. For a general class of cake-cutting algorithms, which we call linearly-labeled, we obtain an Ω(N logN) lower bound on their computational complexity. All the known cake-cutting algorithms fit into this general class, which leads us to conjecture that every cake-cutting algorithm is linearly-labeled. If in addition, the number of comparisons per cut is bounded (comparison-bounded algorithms), then we obtain an Ω(N logN) lower bound on the number of cuts. All known algorithms are comparison-bounded.

We also study variations of envy-free cake-division, where each user feels that they have more cake than every other user. We construct utility functions for which any algorithm (including continuous algorithms) requires Ω(N2) cuts to produce such divisions. These are the the first known general lower bounds for envy-free algorithms. Finally, we study another general class of algorithms called phased algorithms, for which we show that even if one is to simply guarantee each user a piece of cake with positive value, then Ω(N logN) cuts are needed in the worst case. Many of the existing cake-cutting algorithms are phased.


Utility Function Label Tree Label Algorithm Fair Division Division Scheme 
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  1. 1.
    J. Barbanel. Game-theoretic algorithms for fair and strongly fair cake division with entitlements. Colloquium Math., 69:59–53, 1995.zbMATHMathSciNetGoogle Scholar
  2. 2.
    J. Barbanel. Super envy-free cake division and independence of measures. J. Math. Anal. Appl., 197:54–60, 1996.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Steven J. Brams and Allan D Taylor. An envy-free cake division protocol. Am. Math. Monthly, 102:9–18, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Steven J. Brams and Allan D. Taylor. Fair Division: From Cake-Cutting to Dispute Resolution. Cambridge University Press, New York, NY, 1996.zbMATHGoogle Scholar
  5. 5.
    Stephen Demko and Theodore P. Hill. Equitable distribution of indivisible objects. Mathematical Social Sciences, 16(2):145–58, October 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    L. E. Dubins and E. H. Spanier. How to cut a cake fairly. Am. Math. Monthly, 68:1–17, 1961.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Jacob Glazer and Ching-to Albert Ma. Efficient allocation of a ‘prize’-King Solomon’s dilemma. Games and Economic Behavior, 1(3):223–233, 1989.CrossRefMathSciNetGoogle Scholar
  8. 8.
    C-J Haake, M. G. Raith, and F. E. Su. Bidding for envy-freeness: A procedural approach to n-player fair-division problems. Social Choice and Welfare, To appear.Google Scholar
  9. 9.
    Jerzy Legut and Wilczýnski. Optimal partitioning of a measuarble space. Proceedings of the American Mathematical Society, 104(1):262–264, September 1988.Google Scholar
  10. 10.
    Malik Magdon-Ismail, Costas Busch, and Mukkai Krishnamoorthy. Cake-cutting is not a piece of cake. Technical Report 02-12, Rensselaer Polytechnic Institute, Troy, NY 12180, USA, 2002.Google Scholar
  11. 11.
    Elisa Peterson and F. E. Su. Four-person envy-free chore division. Mathematics Magazine, April 2002.Google Scholar
  12. 12.
    K. Rebman. How to get (at least) a fair share of the cake. in Mathematical Plums (Edited by R. Honsberger), The Mathematical Association of America, pages 22–37, 1979.Google Scholar
  13. 13.
    Jack Robertson and William Webb. Approximating fair division with a limited number of cuts. J. Comp. Theory, 72(2):340–344, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jack Robertson and William Webb. Cake-Cutting Algorithms: Be Fair If You Can. A. K. Peters, Nattick, MA, 1998.zbMATHGoogle Scholar
  15. 15.
    H. Steinhaus. The problem of fair division. Econometrica, 16:101–104, 1948.Google Scholar
  16. 16.
    F. E. Su. Rental harmony: Sperner’s lemma in fair division. American Mathematical Monthly, 106:930–942, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gerhard J. Woeginger. An approximation scheme for cake division with a linear number of cuts. In European Symposium on Algorithms (ESA), pages 896–901, 2002.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Malik Magdon-Ismail
    • 1
  • Costas Busch
    • 1
  • Mukkai S. Krishnamoorthy
    • 1
  1. 1.Department of Computer ScienceRensselaer Polytechnic InstituteTroy

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