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Use of Linear Programming to Find an Envy-Free Solution Closest to the Brams–Kilgour Gap Solution for the Housemates Problem

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  • Richard F. Potthoff

    (Duke University)

Abstract

Although the Gap Procedure that Brams and Kilgour (2001) proposed for determining the price of each room in the housemates problem has many favorable properties, it also has one drawback: Its solution is not always envy-free. Described herein is an approach that uses linear programming to find an envy-free solution closest (in a certain sense) to the Gap solution when the latter is not envy-free. If negative prices are allowed, such a solution always exists. If not, it sometimes exists, in which case linear programming can find it by disallowing negative prices. Several examples are presented.

Suggested Citation

  • Richard F. Potthoff, 2002. "Use of Linear Programming to Find an Envy-Free Solution Closest to the Brams–Kilgour Gap Solution for the Housemates Problem," Group Decision and Negotiation, Springer, vol. 11(5), pages 405-414, September.
  • Handle: RePEc:spr:grdene:v:11:y:2002:i:5:d:10.1023_a:1020485018300
    DOI: 10.1023/A:1020485018300
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    References listed on IDEAS

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    1. 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.
    2. Flip Klijn, 2000. "An algorithm for envy-free allocations in an economy with indivisible objects and money," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 17(2), pages 201-215.
    3. 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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    Cited by:

    1. Carraro, Carlo & Marchiori, Carmen & Sgobbi, Alessandra, 2005. "Advances in negotiation theory : bargaining, coalitions, and fairness," Policy Research Working Paper Series 3642, The World Bank.

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