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Asymptotic existence of fair divisions for groups

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  • Manurangsi, Pasin
  • Suksompong, Warut

Abstract

The problem of dividing resources fairly occurs in many practical situations and is therefore an important topic of study in economics. In this paper, we investigate envy-free divisions in the setting where there are multiple players in each interested party. While all players in a party share the same set of resources, each player has her own preferences. Under additive valuations drawn randomly from probability distributions, we show that when all groups contain an equal number of players, a welfare-maximizing allocation is likely to be envy-free if the number of items exceeds the total number of players by a logarithmic factor. On the other hand, an envy-free allocation is unlikely to exist if the number of items is less than the total number of players. In addition, we show that a simple truthful mechanism, namely the random assignment mechanism, yields an allocation that satisfies the weaker notion of approximate envy-freeness with high probability.

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  • Manurangsi, Pasin & Suksompong, Warut, 2017. "Asymptotic existence of fair divisions for groups," Mathematical Social Sciences, Elsevier, vol. 89(C), pages 100-108.
    • Handle: RePEc:eee:matsoc:v:89:y:2017:i:c:p:100-108
    DOI: 10.1016/j.mathsocsci.2017.05.006
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    References listed on IDEAS

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    1. Steven Brams & D. Kilgour & Christian Klamler, 2012. "The undercut procedure: an algorithm for the envy-free division of indivisible items," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(2), pages 615-631, July.
    2. Brams, Steven J. & Kilgour, D. Marc & Klamler, Christian, 2013. "Two-Person Fair Division of Indivisible Items: An Efficient, Envy-Free Algorithm," MPRA Paper 47400, University Library of Munich, Germany.
    3. Eric Budish, 2011. "The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes," Journal of Political Economy, University of Chicago Press, vol. 119(6), pages 1061-1103.
    4. Suksompong, Warut, 2016. "Asymptotic existence of proportionally fair allocations," Mathematical Social Sciences, Elsevier, vol. 81(C), pages 62-65.
    5. Steven J. Brams & Jeffrey M. Togman, 1996. "Camp David: Was The Agreement Fair?," Conflict Management and Peace Science, Peace Science Society (International), vol. 15(1), pages 99-112, February.
    6. Varian, Hal R., 1974. "Equity, envy, and efficiency," Journal of Economic Theory, Elsevier, vol. 9(1), pages 63-91, September.
    7. Claus-Jochen Haake & Matthias G. Raith & Francis Edward Su, 2002. "Bidding for envy-freeness: A procedural approach to n-player fair-division problems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 19(4), pages 723-749.
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    Cited by:

    1. Erel Segal-Halevi & Warut Suksompong, 2023. "Cutting a Cake Fairly for Groups Revisited," Papers 2301.09061, arXiv.org.
    2. Pasin Manurangsi & Warut Suksompong, 2020. "Closing Gaps in Asymptotic Fair Division," Papers 2004.05563, arXiv.org.
    3. Suksompong, Warut, 2018. "Approximate maximin shares for groups of agents," Mathematical Social Sciences, Elsevier, vol. 92(C), pages 40-47.
    4. SEGAL-HALEVI, Erel & NITZAN, Shmuel, 2018. "Fair Cake-Cutting among Families," Discussion paper series HIAS-E-79, Hitotsubashi Institute for Advanced Study, Hitotsubashi University.
    5. Uriel Feige & Yehonatan Tahan, 2022. "On allocations that give intersecting groups their fair share," Papers 2204.06820, arXiv.org.
    6. Erel Segal-Halevi & Shmuel Nitzan, 2019. "Fair cake-cutting among families," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 53(4), pages 709-740, December.
    7. Bade, Sophie & Segal-Halevi, Erel, 2023. "Fairness for multi-self agents," Games and Economic Behavior, Elsevier, vol. 141(C), pages 321-336.
    8. Sophie Bade & Erel Segal-Halevi, 2018. "Fairness for Multi-Self Agents," Papers 1811.06684, arXiv.org, revised Apr 2022.
    9. Fedor Sandomirskiy & Erel Segal-Halevi, 2019. "Efficient Fair Division with Minimal Sharing," Papers 1908.01669, arXiv.org, revised Apr 2022.

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