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The undercut procedure: an algorithm for the envy-free division of indivisible items

Author

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

Abstract

We propose a procedure for dividing indivisible items between two players in which each player ranks the items from best to worst and has no information about the other player’s ranking. It ensures that each player receives a subset of items that it values more than the other player’s complementary subset, given that such an envy-free division is possible. We show that the possibility of one player’s undercutting the other’s proposal, and implementing the reduced subset for himself or herself, makes the proposer “reasonable” and generally leads to an envy-free division, even when the players rank items exactly the same. Although the undercut procedure is manipulable, each player’s maximin strategy is to be truthful. Applications of the undercut procedure are briefly discussed.

Suggested Citation

  • Brams, Steven J. & Kilgour, D. Marc & Klamler, Christian, 2009. "The undercut procedure: an algorithm for the envy-free division of indivisible items," MPRA Paper 12774, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:12774
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    File URL: https://mpra.ub.uni-muenchen.de/12774/1/MPRA_paper_12774.pdf
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    References listed on IDEAS

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    1. Brams, S.J. & Kaplan, T.R., 2002. "Dividing the Indivisible: Procedures for Allocating Cabinet Ministries to Political Parties in a Parliamentary System," Working Papers 02-06, C.V. Starr Center for Applied Economics, New York University.
    2. 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.
    3. Weingast, Barry R. & Wittman, Donald, 2008. "The Oxford Handbook of Political Economy," OUP Catalogue, Oxford University Press, number 9780199548477.
    4. Steven J. Brams & Paul H. Edelman & Peter C. Fishburn, 2003. "Fair Division Of Indivisible Items," Theory and Decision, Springer, vol. 55(2), pages 147-180, September.
    5. 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.
    6. I. D. Hill, 2008. "Mathematics and Democracy: Designing Better Voting and Fair-division Procedures," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 171(4), pages 1032-1033.
    7. Steven J. Brams & D. Marc Kilgour, 2001. "Competitive Fair Division," Journal of Political Economy, University of Chicago Press, vol. 109(2), pages 418-443, April.
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. Haris Aziz, 2015. "A note on the undercut procedure," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 45(4), pages 723-728, December.
    4. repec:eee:matsoc:v:89:y:2017:i:c:p:100-108 is not listed on IDEAS
    5. 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.
    6. 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.
    7. 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.
    8. repec:spr:grdene:v:22:y:2013:i:2:d:10.1007_s10726-011-9263-5 is not listed on IDEAS
    9. 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.
    10. 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.

    More about this item

    Keywords

    Fair division; allocation of indivisible items; envy-freeness; ultimatum game;

    JEL classification:

    • 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
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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