Document Open Access Logo

Simple Envy-Free and Truthful Mechanisms for Cake Cutting with a Small Number of Cuts

Author Takao Asano

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2021.68.pdf
  • Filesize: 0.74 MB
  • 17 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory and mechanism design
Keywords
  • cake-cutting problem
  • envy-freeness
  • fairness
  • truthfulness
  • mechanism design

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

For the cake-cutting problem, Alijani, et al. [Reza Alijani et al., 2017; Masoud Seddighin et al., 2019] and Asano and Umeda [Takao Asano and Hiroyuki Umeda, 2020; Takao Asano and Hiroyuki Umeda, 2020] gave envy-free and truthful mechanisms with a small number of cuts, where the desired part of each player’s valuation function is a single interval on a given cake. In this paper, we give envy-free and truthful mechanisms with a small number of cuts, which are much simpler than those proposed by Alijani, et al. [Reza Alijani et al., 2017; Masoud Seddighin et al., 2019] and Asano and Umeda [Takao Asano and Hiroyuki Umeda, 2020; Takao Asano and Hiroyuki Umeda, 2020]. Furthermore, we show that this approach can be applied to the envy-free and truthful mechanism proposed by Chen, et al. [Yiling Chen et al., 2013], where the valuation function of each player is more general and piecewise uniform. Thus, we can obtain an envy-free and truthful mechanism with a small number of cuts even if the valuation function of each player is piecewise uniform, which solves the future problem posed by Alijani, et al. [Reza Alijani et al., 2017; Masoud Seddighin et al., 2019].

Cite As Get BibTex

Takao Asano. Simple Envy-Free and Truthful Mechanisms for Cake Cutting with a Small Number of Cuts. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 68:1-68:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ISAAC.2021.68

Author Details

Takao Asano
  • Chuo University, Tokyo, Japan

Acknowledgements

The author would like to thank Professor Shigeo Tsujii of Chuo University.

References

  1. Hassene Aissi, S. Thomas McCormick, and Maurice Queyranne. Faster algorithms for next breakpoint and max value for parametric global minimum cuts. In 21st International Conference on Integer Programming and Combinatorial Optimization, pages 27-39, 2020. Google Scholar
  2. Reza Alijani, Majid Farhadi, Mohammad Ghodsi, Masoud Seddighin, and Ahmad S. Tajik. Envy-free mechanisms with minimum number of cuts. In 31st AAAI Conference on Artificial Intelligence, 2017, pages 312-318, 2017. Google Scholar
  3. Takao Asano. Envy-free and truthful cake-cuttings based on parametric flows. In RIMS Kôkyûroku 2182, Kyoto University, April, 2021, pages 1-39, 2021. Google Scholar
  4. Takao Asano and Hiroyuki Umeda. Cake cutting: An envy-free and truthful mechanism with a small number of cuts. In 31st International Symposium on Algorithms and Computation, pages 15.1-15.16, 2020. Google Scholar
  5. Takao Asano and Hiroyuki Umeda. An envy-free and truthful mechanism for the cake-cutting problem. In RIMS Kôkyûroku 2154, Kyoto University, April, 2020, pages 54-91, 2020. Google Scholar
  6. Haris Aziz and Simon Mackenzie. A discrete and bounded envy-free cake cutting protocol for any number of agents. In 57th Annual Symposium on Foundations of Computer Science, 2016, pages 416-427, 2016. Google Scholar
  7. Haris Aziz and Chun Ye. Cake cutting algorithms for piecewise constant and piecewise uniform valuations. In 10th International Conference on Web and Internet Economics, 2014, pages 1-14, 2014. Google Scholar
  8. Xiaohui Bei, Ning Chen, Xia Hua, Biaoshuai Tao, and Endong Yang. Optimal proportional cake cutting with connected pieces. In 26th AAAI Conference on Artificial Intelligence, 2012, pages 1263-1269, 2012. Google Scholar
  9. Xiaohui Bei, Ning Chen, Guangda Huzhang, Biaoshuai Tao, and Jiajun Wu. Cake cutting: envy and truth. In 26th International Joint Conference on Artificial Intelligence, pages 3625-3631, 2017. Google Scholar
  10. Steven J. Brams, Michal Feldman, John K. Lai, Jamie Morgenstern, and Ariel D. Procaccia. On maxsum fair cake divisions. In 26th AAAI Conference on Artificial Intelligence, 2012, pages 1285-1291, 2012. Google Scholar
  11. Steven J. Brams and Alan D. Taylor. Fair Division: From Cake Cutting to Dispute Resolution. Cambridge University Press, 1996. Google Scholar
  12. Ioannis Caragiannis, John K. Lai, and Ariel D. Procaccia. Towards more expressive cake cutting. In 22nd International Joint Conference on Artificial Intelligence, 2011, pages 127-132, 2011. Google Scholar
  13. Yiling Chen, John K. Lai, David C. Parkes, and Ariel D. Procaccia. Truth, justice, and cake cutting. Games and Economic Behavior, 77:284-297, 2013. Google Scholar
  14. Xiaotie Deng, Qi Qi, and Amin Saberi. Algorithmic solutions for envy-free cake cutting. Operations Research, 60:1461-1476, 2012. Google Scholar
  15. Jeff Edmonds and Kirk Pruhs. Balanced allocations of cake. In 47th Annual Symposium on Foundations of Computer Science, 2006, pages 623-634, 2006. Google Scholar
  16. Jeff Edmonds and Kirk Pruhs. Cake cutting really is not a piece of cake. In 17th ACM-SIAM Symposium on Discrete Algorithms, pages 271-278, 2006. Google Scholar
  17. Giorgio Gallo, Michael D. Grigoriadis, and Robert E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM Journal on Computing, 18:30-55, 1989. Google Scholar
  18. Paul W. Goldberg, Alexandros Hollender, and Warut Suksompong. Contiguous cake cutting: hardness results and approximation algorithms. In 34th AAAI Conference on Artificial Intelligence, 2020, pages 1990-1997, 2020. Google Scholar
  19. David Kurokawa, John K. Lai, and Ariel D. Procaccia. How to cut a cake before the party ends. In 27th AAAI Conference on Artificial Intelligence, 2013, pages 555-561, 2013. Google Scholar
  20. Avishay Maya and Noam Nisan. Incentive compatible two player cake cutting. In 8th International Conference on Web and Internet Economics, 2012, pages 170-183, 2012. Google Scholar
  21. Ariel D. Procaccia. Thou shalt covet thy neighbor’s cake. In 21st International Joint Conference on Artificial Intelligence, pages 239-244, 2009. Google Scholar
  22. Ariel D. Procaccia. Cake cutting: not just child’s play. Commun. ACM, 56:78-87, 2013. Google Scholar
  23. Ariel D. Procaccia. Cake cutting algorithms. Handbook of Computational Social Choice (F. Brandt , V. Conitzer, U. Endriss, J. Lang, and A.D. Procaccia, Eds., Cambridge University Press), Chapter 13:313-329, 2016. Google Scholar
  24. Jack M. Robertson and William A. Webb. Cake Cutting Algorithms: Be Fair If you Can. A.K. Peters, 1998. Google Scholar
  25. Masoud Seddighin, Majid Farhadi, Mohammad Ghodsi, Reza Alijani, and Ahmad S. Tajik. Expand the shares together: envy-free mechanisms with a small number of cuts. Algorithmica, 81:1728-1755, 2019. Google Scholar
  26. Katerina Sherstyuk. How to gerrymander: A formal analysis. Public Choice, 95:27-49, 1998. Google Scholar
  27. Hugo Steinhaus. The problem of fair division. Econometrica, 16:101-104, 1948. Google Scholar
  28. Hugo Steinhaus. Sur la division pragmatique. Econometrica (suppliment), 17:315-319, 1949. Google Scholar
  29. Walter Stromquist. How to cut a cake fairly. American Mathematical Monthly, 87:640-644, 1980. Google Scholar
  30. Walter Stromquist. Envy-free cake divisions cannot be found by finite protocols. The Electronic Journal of Combinatorics, 15:#R11 (pp.1-10), 2008. Google Scholar
  31. Francis E. Su. Rental harmony: Sperner’s lemma in fair division. American Mathematical Monthly, 106:930-942, 1999. Google Scholar
  32. William Thomson. Children crying at birthday parties. why? Economic Theory, 31:501-521, 2007. Google Scholar
  33. Takeshi Tokuyama. Minimax parametric optimization problems and multi-dimensional parametric searching. In 33rd ACM Symposium on Theory of Computing, pages 75-83, 2001. Google Scholar
  34. David P. Williamson. Network Flow Algorithms. Cambridge University Press, 2019. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact