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Finding the Nucleoli of Large Cooperative Games: A Disproof with Counter-Example

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  • Meinhardt, Holger Ingmar

Abstract

Nguyen and Thomas (2016) claimed that they have found a method to compute the nucleoli of games with more than 50 players using nested linear programs (LP). Unfortunately, this claim is false. They incorrectly applied the indirect proof by ``A and not B implies A and not A'' to conclude that ``if A then B''is valid. In fact, they prove that a truth implies a falsehood. As established by Meinhardt (2015a), this is a wrong statement. Therefore, instead of giving a proof of their main Theorem 4b, they give a disproof. It comes as no surprise to us that the flow game example presented by these authors to support their arguments is obviously a counter-example of their algorithm. We show that the computed solution by this algorithm is neither the nucleolus nor a core element of the flow game. Moreover, the stopping criterion of all proposed methods is wrong, since it does not satisfy one of Kohlberg's properties (cf. Kohlberg (1971)). As a consequence, none of these algorithms is robust.

Suggested Citation

  • Meinhardt, Holger Ingmar, 2016. "Finding the Nucleoli of Large Cooperative Games: A Disproof with Counter-Example," MPRA Paper 69789, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:69789
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    File URL: https://mpra.ub.uni-muenchen.de/69789/1/MPRA_paper_69789.pdf
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    References listed on IDEAS

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    1. Nguyen, Tri-Dung & Thomas, Lyn, 2016. "Finding the nucleoli of large cooperative games," European Journal of Operational Research, Elsevier, vol. 248(3), pages 1078-1092.
    2. Meinhardt, Holger Ingmar, 2015. "The Incorrect Usage of Propositional Logic in Game Theory: The Case of Disproving Oneself," MPRA Paper 66637, University Library of Munich, Germany.
    3. repec:spr:thdchp:978-3-642-39549-9_1 is not listed on IDEAS
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    Cited by:

    1. Meinhardt, Holger Ingmar, 2017. "Simplifying the Kohlberg Criterion on the Nucleolus: A Disproof by Oneself," MPRA Paper 77143, University Library of Munich, Germany.

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    More about this item

    Keywords

    Transferable Utility Game; Nucleolus; Flow Problem; Propositional Logic; Circular Reasoning (circulus in probando); Indirect Proof; Proof by Contradiction; Proof by Contraposition.;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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