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Two-Person Fair Division of Indivisible Items: An Efficient, Envy-Free Algorithm

Author

Listed:
  • Brams, Steven J.
  • Kilgour, D. Marc
  • Klamler, Christian

Abstract

Many procedures have been suggested for the venerable problem of dividing a set of indivisible items between two players. We propose a new algorithm (AL), related to one proposed by Brams and Taylor (BT), which requires only that the players strictly rank items from best to worst. Unlike BT, in which any item named by both players in the same round goes into a “contested pile,” AL may reduce, or even eliminate, the contested pile, allocating additional or more preferred items to the players. The allocation(s) that AL yields are Pareto-optimal, envy-free, and maximal; as the number of items (assumed even) increases, the probability that AL allocates all the items appears to approach infinity if all possible rankings are equiprobable. Although AL is potentially manipulable, strategizing under it would be difficult in practice.

Suggested Citation

  • 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.
    • Handle: RePEc:pra:mprapa:47400
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    File URL: https://mpra.ub.uni-muenchen.de/47400/1/MPRA_paper_47400.pdf
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    References listed on IDEAS

    as
    1. Steven J. Brams & Todd R. Kaplan, 2002. "Dividing the Indivisible: Procedures for Allocating Cabinet Ministries to Political Parties in a Parliamentary System," Discussion Papers 0202, University of Exeter, Department of Economics.
    2. 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.
    3. Rudolf Vetschera & D. Kilgour, 2014. "Fair division of indivisible items between two players: design parameters for Contested Pile methods," Theory and Decision, Springer, vol. 76(4), pages 547-572, April.
    4. Rudolf Vetschera & D. Marc Kilgour, 2013. "Strategic Behavior in Contested-Pile Methods for Fair Division of Indivisible Items," Group Decision and Negotiation, Springer, vol. 22(2), pages 299-319, March.
    5. Dorothea Herreiner & Clemens Puppe, 2002. "A simple procedure for finding equitable allocations of indivisible goods," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 19(2), pages 415-430.
    6. Steven J. Brams & Daniel L. King, 2005. "Efficient Fair Division," Rationality and Society, , vol. 17(4), pages 387-421, November.
    7. Steven J. Brams & Todd R. Kaplan, 2004. "Dividing the Indivisible," Journal of Theoretical Politics, , vol. 16(2), pages 143-173, April.
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    Citations

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    Cited by:

    1. Manurangsi, Pasin & Suksompong, Warut, 2017. "Asymptotic existence of fair divisions for groups," Mathematical Social Sciences, Elsevier, vol. 89(C), pages 100-108.
    2. Brams, Steven J. & Kilgour, D. Marc & Klamler, Christian, 2014. "An algorithm for the proportional division of indivisible items," MPRA Paper 56587, University Library of Munich, Germany.
    3. Steven J. Brams & D. Marc Kilgour & Christian Klamler, 2017. "Maximin Envy-Free Division of Indivisible Items," Group Decision and Negotiation, Springer, vol. 26(1), pages 115-131, January.
    4. Fedor Sandomirskiy & Erel Segal-Halevi, 2019. "Efficient Fair Division with Minimal Sharing," Papers 1908.01669, arXiv.org, revised Apr 2022.
    5. Andreas Darmann & Christian Klamler, 2016. "Proportional Borda allocations," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 47(3), pages 543-558, October.
    6. Brams, Steven & Kilgour, D. Marc & Klamler, Christian, 2014. "How to divide things fairly," MPRA Paper 58370, University Library of Munich, Germany.
    7. Haris Aziz, 2016. "A generalization of the AL method for fair allocation of indivisible objects," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(2), pages 307-324, October.

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    More about this item

    Keywords

    Two-person fair division; indivisible items; envy-freeness; efficiency; algorithm;
    All these keywords.

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory
    • D6 - Microeconomics - - Welfare Economics
    • D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
    • D7 - Microeconomics - - Analysis of Collective Decision-Making
    • D74 - Microeconomics - - Analysis of Collective Decision-Making - - - Conflict; Conflict Resolution; Alliances; Revolutions

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