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Computing solutions for matching games

Author

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  • Péter Biró
  • Walter Kern
  • Daniël Paulusma

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.
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  • Péter Biró & Walter Kern & Daniël Paulusma, 2012. "Computing solutions for matching games," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(1), pages 75-90, February.
    • Handle: RePEc:spr:jogath:v:41:y:2012:i:1:p:75-90
    DOI: 10.1007/s00182-011-0273-y
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    1. 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.
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    4. Solymosi, Tamas & Raghavan, Tirukkannamangai E S, 1994. "An Algorithm for Finding the Nucleolus of Asignment Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(2), pages 119-143.
    5. 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.
    6. 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.
    7. M. L. Balinski, 1965. "Integer Programming: Methods, Uses, Computations," Management Science, INFORMS, vol. 12(3), pages 253-313, November.
    8. 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.
    9. 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.
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    Cited by:

    1. Frits Hof & Walter Kern & Sascha Kurz & Kanstantsin Pashkovich & Daniël Paulusma, 2020. "Simple games versus weighted voting games: bounding the critical threshold value," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 54(4), pages 609-621, April.
    2. Zhuan Khye Koh & Laura Sanità, 2020. "Stabilizing Weighted Graphs," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1318-1341, November.
    3. Biró, Péter & Kern, Walter & Paulusma, Daniël & Wojuteczky, Péter, 2018. "The stable fixtures problem with payments," Games and Economic Behavior, Elsevier, vol. 108(C), pages 245-268.
    4. Vijay V. Vazirani, 2022. "Cores of Games via Total Dual Integrality, with Applications to Perfect Graphs and Polymatroids," Papers 2209.04903, arXiv.org, revised Nov 2022.
    5. Akiyoshi Shioura, 2017. "On the Partnership formation problem," The Journal of Mechanism and Institution Design, Society for the Promotion of Mechanism and Institution Design, University of York, vol. 2(1), pages 105-140, December.
    6. Vazirani, Vijay V., 2022. "The general graph matching game: Approximate core," Games and Economic Behavior, Elsevier, vol. 132(C), pages 478-486.
    7. F.Javier Martínez-de-Albéniz & Carles Rafels & Neus Ybern, 2015. "Insights into the nucleolus of the assignment game," UB School of Economics Working Papers 2015/333, University of Barcelona School of Economics.
    8. Vijay V. Vazirani, 2021. "The General Graph Matching Game: Approximate Core," Papers 2101.07390, arXiv.org, revised Jul 2021.
    9. Han Xiao & Tianhang Lu & Qizhi Fang, 2021. "Approximate Core Allocations for Multiple Partners Matching Games," Papers 2107.01442, arXiv.org, revised Oct 2021.
    10. Peter Biro & Matthijs Bomhoff & Walter Kern & Petr A. Golovach & Daniel Paulusma, 2012. "Solutions for the Stable Roommates Problem with Payments," CERS-IE WORKING PAPERS 1211, Institute of Economics, Centre for Economic and Regional Studies.
    11. Vijay V. Vazirani, 2022. "New Characterizations of Core Imputations of Matching and $b$-Matching Games," Papers 2202.00619, arXiv.org, revised Dec 2022.
    12. Han Xiao & Qizhi Fang, 2022. "Population monotonicity in matching games," Journal of Combinatorial Optimization, Springer, vol. 43(4), pages 699-709, May.
    13. Vijay V. Vazirani, 2023. "The Investment Management Game: Extending the Scope of the Notion of Core," Papers 2302.00608, arXiv.org, revised Sep 2023.
    14. Vijay V. Vazirani, 2023. "LP-Duality Theory and the Cores of Games," Papers 2302.07627, arXiv.org, revised Mar 2023.

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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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