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An algorithm for the proportional division of indivisible items

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

Abstract

An allocation of indivisible items among n ≥ 2 players is proportional if and only if each player receives a proportional subset—one that it thinks is worth at least 1/n of the total value of all the items. We show that a proportional allocation exists if and only if there is an allocation in which each player receives one of its minimal bundles, from which the subtraction of any item would make the bundle worth less than 1/n. We give a practicable algorithm, based on players’ rankings of minimal bundles, that finds a proportional allocation if one exists; if not, it gives as many players as possible minimal bundles. The resulting allocation is maximin, but it may be neither envy-free nor Pareto-optimal. However, there always exists a Pareto-optimal maximin allocation which, when n = 2, is also envy-free. We compare our algorithm with two other 2-person algorithms, and we discuss its applicability to real-world disputes among two or more players.

Suggested Citation

  • 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.
    • Handle: RePEc:pra:mprapa:56587
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    File URL: https://mpra.ub.uni-muenchen.de/56587/1/MPRA_paper_56585.pdf
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    References listed on IDEAS

    as
    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. Steven J. Brams & Peter C. Fishburn, 2000. "Fair division of indivisible items between two people with identical preferences: Envy-freeness, Pareto-optimality, and equity," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 17(2), pages 247-267.
    4. Eve Ramaekers, 2013. "Fair allocation of indivisible goods: the two-agent case," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(2), pages 359-380, July.
    5. Edelman, Paul & Fishburn, Peter, 2001. "Fair division of indivisible items among people with similar preferences," Mathematical Social Sciences, Elsevier, vol. 41(3), pages 327-347, May.
    6. 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.
    7. Steven J. Brams & Daniel L. King, 2005. "Efficient Fair Division," Rationality and Society, , vol. 17(4), pages 387-421, November.
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    Cited by:

    1. Brams, Steven & Kilgour, D. Marc & Klamler, Christian, 2014. "How to divide things fairly," MPRA Paper 58370, University Library of Munich, Germany.

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    More about this item

    Keywords

    Fair division; indivisible items; proportionality; envy-freeness;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory
    • D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
    • D74 - Microeconomics - - Analysis of Collective Decision-Making - - - Conflict; Conflict Resolution; Alliances; Revolutions
    • D78 - Microeconomics - - Analysis of Collective Decision-Making - - - Positive Analysis of Policy Formulation and Implementation

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