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Algorithmica

, Volume 81, Issue 4, pp 1728–1755 | Cite as

Expand the Shares Together: Envy-Free Mechanisms with a Small Number of Cuts

  • Masoud SeddighinEmail author
  • Majid Farhadi
  • Mohammad Ghodsi
  • Reza Alijani
  • Ahmad S. Tajik
Article
  • 71 Downloads

Abstract

We study the problem of fair division of a heterogeneous resource among strategic players. Given a divisible heterogeneous cake, we wish to divide the cake among n players to meet these conditions: (I) every player (weakly) prefers his allocated cake to any other player’s share (such notion is known as envy-freeness), (II) the allocation is dominant strategy-proof (truthful) (III) the number of cuts made on the cake is small. We provide methods for dividing the cake under different assumptions on the valuation functions of the players. First, we suppose that the valuation function of every player is a single interval with a special property, namely ordering property. For this case, we propose a process called expansion process and show that it results in an envy-free and truthful allocation that cuts the cake into exactly n pieces. Next, we remove the ordering restriction and show that for this case, an extended form of the expansion process, called expansion process with unlocking yields an envy-free allocation of the cake with at most \(2(n-1)\) cuts. Furthermore, we show that in the expansion process with unlocking, the players may misrepresent their valuations to earn more share. In addition, we use a more complex form of the expansion process with unlocking to obtain an envy-free and truthful allocation that cuts the cake in at most \(2(n-1)\) locations. We also, evaluate our expansion method in practice. In the empirical results, we compare the number of cuts made by our method to the number of cuts in the optimal solution (\(n-1\)). The experiments reveal that the number of cuts made by the expansion and unlocking process for envy-free division of the cake is very close to the optimal solution. Finally, we study piecewise constant and piecewise uniform valuation functions with m pieces and present the conditions, under which a generalized form of expansion process can allocate the cake via O(nm) cuts.

Keywords

Cake cutting Envy-free Mechanism design Approximation Fairness 

Notes

References

  1. 1.
    Alijani, R., Farhadi, M., Ghodsi, M., Seddighin, M., Tajik, A.S.: Envy-free mechanisms with minimum number of cuts. In: Thirty-First AAAI Conference on Artificial Intelligence (2017)Google Scholar
  2. 2.
    Aziz, H., Mackenzie, S.: A discrete and bounded envy-free cake cutting protocol for any number of agents. In: 2016 IEEE 57th Annual Symposium onFoundations of Computer Science (FOCS), pp. 416–427. IEEE (2016)Google Scholar
  3. 3.
    Aziz, H., Ye, C.: Cake cutting algorithms for piecewise constant and piecewise uniform valuations. In: International Conference on Web and Internet Economics, pp. 1–14. Springer (2014)Google Scholar
  4. 4.
    Barbanel, J.B., Brams, S.J.: Cake division with minimal cuts: envy-free procedures for three persons, four persons, and beyond. Math. Soc. Sci. 48, 251–269 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bei, X., Chen, N., Hua, X., Tao, B., Yang, E.: Optimal proportional cake cutting with connected pieces. In: Twenty-sixth AAAI Conference on Artificial Intelligence (2012)Google Scholar
  6. 6.
    Bei, X., Chen, N., Huzhang, G., Tao, B., Wu, J.: Cake cutting: envy and truth. In: Twenty-sixth International Joint Conference on Artificial Intelligence, pp. 3625–3631. AAAI Press (2017)Google Scholar
  7. 7.
    Brams, S.J., Feldman, M., Lai, J.K., Morgenstern, J., Procaccia, A.D.: On maxsum fair cake divisions. In: Twenty-sixth AAAI Conference on Artificial Intelligence (2012)Google Scholar
  8. 8.
    Brams, S.J., Jones, M.A., Klamler, C.: Better ways to cut a cake. Not. AMS 53, 1314–1321 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Brams, S.J., Taylor, A.D.: An envy-free cake division protocol. Am. Math. Mon. 102, 9–18 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Caragiannis, I., Lai, J.K., Procaccia, A.D.: Towards more expressive cake cutting. IJCAI Proc. Int. Jt. Conf. Artif. Intell. 22, 127 (2011)Google Scholar
  11. 11.
    Chen, Y., Lai, J.K., Parkes, D.C., Procaccia, A.D.: Truth, justice, and cake cutting. Games Econ. Beh. 77, 284–297 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Deng, X., Qi, Q., Saberi, A.: Algorithmic solutions for envy-free cake cutting. Oper. Res. 60, 1461–1476 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kurokawa, D., Lai, J.K., Procaccia, A.D.: How to cut a cake before the party ends. In: Twenty-Seventh AAAI Conference on Artificial Intelligence (2013)Google Scholar
  14. 14.
    Maya, A., Nisan, N.: Incentive compatible two player cake cutting. In: International Workshop on Internet and Network Economics, pp. 170–183. Springer (2012)Google Scholar
  15. 15.
    Procaccia, A.D.: Cake cutting: not just child’s play. Commun. ACM 56, 78–87 (2013)CrossRefGoogle Scholar
  16. 16.
    Procaccia, A.D.: Cake cutting algorithms. In: Handbook of Computational Social Choice, chapter 13, Citeseer (2015)Google Scholar
  17. 17.
    Steinhaus, H.: The problem of fair division. Econometrica 16, 101–104 (1948)Google Scholar
  18. 18.
    Stromquist, W.: How to cut a cake fairly. Am. Math. Mon. 87, 640–644 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Stromquist, W.: Envy-free cake divisions cannot be found by finite protocols. Electr. J. Comb. 15, 11 (2008)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Masoud Seddighin
    • 1
    Email author
  • Majid Farhadi
    • 1
  • Mohammad Ghodsi
    • 1
    • 2
  • Reza Alijani
    • 3
  • Ahmad S. Tajik
    • 4
  1. 1.Sharif University of TechnologyTehranIran
  2. 2.IPM - Institute for Research in Fundamental SciencesTehranIran
  3. 3.Duke universityDurhamUSA
  4. 4.University of MichiganAnn ArborUSA

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