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Computing Solutions for Matching Games

Author

Listed:
  • Peter Biro

    (Institute of Economics - Hungarian Academy of Sciences)

  • Walter Kern

    (Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, P.O.Box 217, NL-7500 AE Enschede)

  • Daniel Paulusma

    (Department of Computer Science, University of Durham Science Laboratories, South Road, Durham DH1 3EY, England)

Abstract

A matching game is a cooperative game (N; v) defined on a graph G = (N;E) with an edge weighting w : E ! R+. The player set is N and the value of a coalition S N is de ned as the maximum weight of a matching in the subgraph induced by S. First we present an O(nm+n2 log n) algorithm that tests if the core of a matching game defined on a weighted graph with n vertices and m edges is nonempty and that computes a core member if the core is nonempty. This algorithm improves previous work based on the ellipsoid method and can also be used to compute stable solutions for instances of the stable roommates problem with payments. Second we show that the nucleolus of an n-player matching game with a nonempty core can be computed in O(n4) time. This generalizes the corresponding result of Solymosi and Raghavan for assignment games. Third we prove that is NP-hard to determine an imputation with minimum number of blocking pairs, even for matching games with unit edge weights, whereas the problem of determining an imputation with minimum total blocking value is shown to be polynomial-time solvable for general matching games.

Suggested Citation

  • Peter Biro & Walter Kern & Daniel Paulusma, 2011. "Computing Solutions for Matching Games," CERS-IE WORKING PAPERS 1142, Institute of Economics, Centre for Economic and Regional Studies.
    • Handle: RePEc:has:discpr:1142
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    File URL: http://econ.core.hu/file/download/mtdp/MTDP1142.pdf
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    References listed on IDEAS

    as
    1. M. L. Balinski, 1965. "Integer Programming: Methods, Uses, Computations," Management Science, INFORMS, vol. 12(3), pages 253-313, November.
    2. Jeroen Kuipers & Ulrich Faigle & Walter Kern, 2001. "On the computation of the nucleolus of a cooperative game," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(1), pages 79-98.
    3. Xiaotie Deng & Toshihide Ibaraki & Hiroshi Nagamochi, 1999. "Algorithmic Aspects of the Core of Combinatorial Optimization Games," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 751-766, August.
    4. Johan Karlander & Kimmo Eriksson, 2001. "Stable outcomes of the roommate game with transferable utility," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(4), pages 555-569.
    5. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Walter Kern & Daniël Paulusma, 2003. "Matching Games: The Least Core and the Nucleolus," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 294-308, May.
    7. M. Maschler & B. Peleg & L. S. Shapley, 1979. "Geometric Properties of the Kernel, Nucleolus, and Related Solution Concepts," Mathematics of Operations Research, INFORMS, vol. 4(4), pages 303-338, November.
    8. Bettina Klaus & Alexandru Nichifor, 2010. "Consistency in one-sided assignment problems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 35(3), pages 415-433, September.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    matching game; nucleolus; cooperative game theory;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • 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

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