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The proportional random allocation of indivisible units

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  • Hervé Moulin

    (Rice University, Department of Economics, MS 22, P.O. Box 1892, Houston, TX 77251-1892, USA)

Abstract

Indivisible units are randomly allocated among agents with a claim/demand on the resources. The available resources fall short of the sum of individual claims. The proportional method distributes units sequentially, and the probability of receiving a unit at any step is proportional to the unsatisfied claims. We characterize the family of probabilistic rationing methods meeting the three axioms Consistency, Lower and Upper Composition. It contains the proportional method, all deterministic fixed priority methods, and the priority compositions of proportional methods. The proportional method is the only fair method in the family.

Suggested Citation

  • Hervé Moulin, 2002. "The proportional random allocation of indivisible units," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 19(2), pages 381-413.
    • Handle: RePEc:spr:sochwe:v:19:y:2002:i:2:p:381-413
    Note: Received: 30 November 1999/Accepted: 15 November 2000
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    Cited by:

    1. Bergantinos, Gustavo & Vidal-Puga, Juan J., 2006. "Additive rules in discrete allocation problems," European Journal of Operational Research, Elsevier, vol. 172(3), pages 971-978, August.
    2. Ricardo Martínez, 2020. "On how to divide a budget according to population and wealth," ThE Papers 20/10, Department of Economic Theory and Economic History of the University of Granada..
    3. Moulin, Herve & Stong, Richard, 2003. "Filling a multicolor urn: an axiomatic analysis," Games and Economic Behavior, Elsevier, vol. 45(1), pages 242-269, October.
    4. Martínez, Ricardo & Moreno-Ternero, Juan D., 2022. "Compensation and sacrifice in the probabilistic rationing of indivisible units," European Journal of Operational Research, Elsevier, vol. 302(2), pages 740-751.
    5. Moulin, Herve, 2005. "Split-Proof Probabilistic Scheduling," Working Papers 2004-06, Rice University, Department of Economics.
    6. Toulis, Panos & Parkes, David C., 2015. "Design and analysis of multi-hospital kidney exchange mechanisms using random graphs," Games and Economic Behavior, Elsevier, vol. 91(C), pages 360-382.
    7. Yan Dai & Moise Blanchard & Patrick Jaillet, 2025. "Non-Monetary Mechanism Design without Distributional Information: Using Scarce Audits Wisely," Papers 2502.08412, arXiv.org, revised Jun 2025.
    8. Chambers, Christopher P., 2006. "Asymmetric rules for claims problems without homogeneity," Games and Economic Behavior, Elsevier, vol. 54(2), pages 241-260, February.
    9. Lars Ehlers & Bettina Klaus, 2003. "Probabilistic assignments of identical indivisible objects and uniform probabilistic rules," Review of Economic Design, Springer;Society for Economic Design, vol. 8(3), pages 249-268, October.
    10. Thomson, William, 2015. "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: An update," Mathematical Social Sciences, Elsevier, vol. 74(C), pages 41-59.
    11. Patrick Harless, 2017. "Endowment additivity and the weighted proportional rules for adjudicating conflicting claims," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 63(3), pages 755-781, March.
    12. Moulin, Hervé, 2008. "Proportional scheduling, split-proofness, and merge-proofness," Games and Economic Behavior, Elsevier, vol. 63(2), pages 567-587, July.
    13. Vincent Mak & Darryl A. Seale & Eyran J. Gisches & Amnon Rapoport & Meng Cheng & Myounghee Moon & Rui Yang, 2018. "A network ridesharing experiment with sequential choice of transportation mode," Theory and Decision, Springer, vol. 85(3), pages 407-433, October.
    14. Moulin, Herve & Stong, Richard, 2001. "Fair Queuing and Other Probabilistic Allocation Methods," Working Papers 2000-09, Rice University, Department of Economics.
    15. Feige, Uriel & Tennenholtz, Moshe, 2014. "On fair division of a homogeneous good," Games and Economic Behavior, Elsevier, vol. 87(C), pages 305-321.
    16. Moulin, Herve, 2002. "Axiomatic cost and surplus sharing," Handbook of Social Choice and Welfare, in: K. J. Arrow & A. K. Sen & K. Suzumura (ed.), Handbook of Social Choice and Welfare, edition 1, volume 1, chapter 6, pages 289-357, Elsevier.
    17. Chambers, Christopher P., 2004. "Consistency in the probabilistic assignment model," Journal of Mathematical Economics, Elsevier, vol. 40(8), pages 953-962, December.
    18. Tasnadi, Attila, 2002. "On probabilistic rationing methods," Mathematical Social Sciences, Elsevier, vol. 44(2), pages 211-221, November.

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