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Finding and verifying the nucleolus of cooperative games

Author

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  • Márton Benedek

    (Institute of Economics, Centre for Economic and Regional Studies,Toth Kalman utca 4, 1097 Budapest, Hungary Corvinus University of Budapest, Fővám tér 8., Budapest, 1093, Hungary,Budapest University of Technology and Economics, Egry József u. 1., Budapest, 1111, Hungary)

  • Jörg Fliege

    (Mathematical Sciences, University of Southampton, University Road, Southampton, SO17 1BJ, United Kingdom)

  • Tri-Dung Nguyen

    (Mathematical Sciences, Business School and CORMSIS, University of Southampton, Southampton, SO17 1BJ, United Kingdom)

Abstract

The nucleolus offers a desirable payoff-sharing solution in cooperative games, thanks to its attractive properties. Although computing the nucleolus is very challenging, the Kohlberg criterion offers a method for verifying whether a solution is the nucleolus in relatively small games (number of players n at most 15). This approach becomes more challenging for larger games as the criterion involves possibly exponentially large collections of coalitions, with each collection being potentially exponentially large. The aim of this work is twofold. First, we develop an improved Kohlberg criterion that involves checking the `balancedness' of at most (n-1) sets of coalitions. Second, we exploit these results and introduce a novel descent-based constructive algorithm to find the nucleolus efficiently. We demonstrate the performance of the new algorithms by comparing them with existing methods over different types of games. Our contribution also includes the first open-source code for computing the nucleolus of moderately large games.

Suggested Citation

  • Márton Benedek & Jörg Fliege & Tri-Dung Nguyen, 2020. "Finding and verifying the nucleolus of cooperative games," CERS-IE WORKING PAPERS 2021, Institute of Economics, Centre for Economic and Regional Studies.
    • Handle: RePEc:has:discpr:2021
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    References listed on IDEAS

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    15. 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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    More about this item

    Keywords

    nucleolus; cooperative games; Kohlberg criterion; computation;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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