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The Geometry of Efficient Fair Division

Author

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  • Barbanel,Julius B. Introduction by-Name:Taylor,Alan D.

Abstract

What is the best way to divide a 'cake' and allocate the pieces among some finite collection of players? In this book, the cake is a measure space, and each player uses a countably additive, non-atomic probability measure to evaluate the size of the pieces of cake, with different players generally using different measures. The author investigates efficiency properties (is there another partition that would make everyone at least as happy, and would make at least one player happier, than the present partition?) and fairness properties (do all players think that their piece is at least as large as every other player's piece?). He focuses exclusively on abstract existence results rather than algorithms, and on the geometric objects that arise naturally in this context. By examining the shape of these objects and the relationship between them, he demonstrates results concerning the existence of efficient and fair partitions.

Suggested Citation

  • Barbanel,Julius B. Introduction by-Name:Taylor,Alan D., 2005. "The Geometry of Efficient Fair Division," Cambridge Books, Cambridge University Press, number 9780521842488, January.
    • Handle: RePEc:cup:cbooks:9780521842488
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    Citations

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    Cited by:

    1. Edith Cohen & Michal Feldman & Amos Fiat & Haim Kaplan & Svetlana Olonetsky, 2010. "Envy-Free Makespan Approximation," Discussion Paper Series dp539, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    2. 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.
    3. 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.
    4. Barbanel, Julius B. & Brams, Steven J., 2011. "Two-person cake-cutting: the optimal number of cuts," MPRA Paper 34263, University Library of Munich, Germany.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. Barbanel, Julius B. & Brams, Steven J., 2010. "Two-person pie-cutting: The fairest cuts," MPRA Paper 22703, University Library of Munich, Germany.
    11. 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.
    12. Marco Dall’Aglio & Camilla Luca, 2014. "Finding maxmin allocations in cooperative and competitive fair division," Annals of Operations Research, Springer, vol. 223(1), pages 121-136, December.
    13. Maurice Salles, 2014. "‘Social choice and welfare’ at 30: its role in the development of social choice theory and welfare economics," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(1), pages 1-16, January.
    14. Steven Brams & Michael Jones & Christian Klamler, 2008. "Proportional pie-cutting," International Journal of Game Theory, Springer;Game Theory Society, vol. 36(3), pages 353-367, March.
    15. 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.
    16. Sagara, Nobusumi, 2008. "A characterization of [alpha]-maximin solutions of fair division problems," Mathematical Social Sciences, Elsevier, vol. 55(3), pages 273-280, May.
    17. Chèze, Guillaume, 2017. "Existence of a simple and equitable fair division: A short proof," Mathematical Social Sciences, Elsevier, vol. 87(C), pages 92-93.
    18. Taylor, Alan D., 2005. "A paradoxical Pareto frontier in the cake-cutting context," Mathematical Social Sciences, Elsevier, vol. 50(2), pages 227-233, September.
    19. 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.
    20. Marco Dall'Aglio & Camilla Di Luca & Lucia Milone, 2017. "Finding the Pareto optimal equitable allocation of homogeneous divisible goods among three players," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 27(3), pages 35-50.

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