Simplifying the Kohlberg Criterion on the Nucleolus: A Disproof by Oneself

Meinhardt, Holger Ingmar (2017): Simplifying the Kohlberg Criterion on the Nucleolus: A Disproof by Oneself.

Abstract

(Nguyen 2016) claimed that he has developed a simplifying set of the Kohlberg criteria that involves checking the balancedness of at most (n-1) sets of coalitions. This claim is not true. Analogous to Nguyen and Thomas (2016), he has incorrectly applied the indirect proof. He established in his purported proofs of the main results that a truth implies a falsehood. This is a wrong statement and such a hypotheses must be rejected (cf. Meinhardt (2015,2016a,2016b)). Executing a logical correct interpretation ought immediately lead him to the conclusion that his proposed algorithms are deficient. In particular, he had to detect that the imposed balancedness requirement on the test condition within his proposed methods cannot be appropriate. As a consequence, either a nucleolus with a weakly balanced set will be dismissed by the implemented algorithms or a solution which is not a nucleolus will be selected as a nucleolus. Hence, one cannot expect that one of these algorithms makes a correct selection. The supposed algorithms are wrongly designed and cannot be set in any relation with Kohlberg.

• Item Type: MPRA Paper
• Language: English
• Keywords: Transferable Utility Game, Nucleolus, Balancedness, Kohlberg Criteria; Convexity; Affine Hull; Propositional Logic; Circular Reasoning (circulus in probando); Indirect Proof; Proof by Contradiction
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C71 - Cooperative Games

References

E. Kohlberg. On the Nucleolus of a Characteristic Function Game. SIAM Journal of Applied Mathematics, 20:62–66, 1971.

H. I. Meinhardt. The Incorrect Usage of Propositional Logic in Game Theory: The Case of Disproving Oneself. ArXiv e-prints, abs/1509.05883, 2015. URL http://arxiv.org/abs/1509.05883.

H. I. Meinhardt. Finding the nucleoli of large cooperative games: A disproof with counter-example. CoRR,abs/1603.00226, 2016a. URL http://arxiv.org/abs/1603.00226.

H. I. Meinhardt. The Incorrect Usage of Propositional Logic in Game Theory: The Case of Disproving Oneself. MPRA, 75876, 2016b. URL https://mpra.ub.uni-muenchen.de/75876/. Revised Version.

Tri-Dung Nguyen. Simplifying the Kohlberg Criterion on the Nucleolus. ArXiv e-prints, June 2016. URL http://arxiv. org/abs/1606.05987.

Tri-Dung Nguyen and Lyn Thomas. Finding the nucleoli of large cooperative games. European Journal of Operational Research, 248(3):1078 – 1092, 2016. ISSN 0377-2217. doi: http://dx.doi.org/10.1016/j.ejor.2015.08.017. URL http: //www.sciencedirect.com/science/article/pii/S0377221715007547.

B. Peleg and P. Sudhölter. Introduction to the Theory of Cooperative Games, volume 34 of Theory and Decision Library: Series C. Springer-Verlag, Heidelberg, 2 edition, 2007.