arXiv subjects

Holger Ingmar Meinhardt

Publications and source records attributed to Holger Ingmar Meinhardt.

4 recordsLinked to original sources

arXiv2017

Simplifying the Kohlberg Criterion on the Nucleolus: A Correct Approach

Nguyen (2016, 2017) 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 by $(ϕ\Rightarrow \bot) \Leftrightarrow \neg ϕ$. 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,2017)). 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 $(\cup_{j=1}^{k}\,T_{j})$ within his proposed methods cannot be appropriate. 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. Therefore, our objections from (Meinhardt,2017) are still valid with some modifications. In particular, he failed to work out modified necessary and sufficient conditions of the nucleolus.

math.OC↗

arXiv2016

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

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 "$A \land \neg B$ implies $A \land \neg A$" 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.

cs.GT↗

arXiv2015

On the Single-Valuedness of the Pre-Kernel

Based on results given in the recent book by Meinhardt (2013), which presents a dual characterization of the pre-kernel by a finite union of solution sets of a family of quadratic and convex objective functions, we could derive some results related to the uniqueness of the pre-kernel. Rather than extending the knowledge of game classes for which the pre-kernel consists of a single point, we apply a different approach. We select a game from an arbitrary game class with a single pre-kernel element satisfying the non-empty interior condition of a payoff equivalence class, and then establish that the set of related and linear independent games which are derived from this pre-kernel point of the default game replicates this point also as its sole pre-kernel element. In the proof we apply results and techniques employed in the above work. Namely, we prove in a first step that the linear mapping of a pre-kernel element into a specific vector subspace of balanced excesses is a singleton. Secondly, that there cannot exist a different and non-transversal vector subspace of balanced excesses in which a linear transformation of a pre-kernel element can be mapped. Furthermore, we establish that on the restricted subset on the game space that is constituted by the convex hull of the default and the set of related games, the pre-kernel correspondence is single-valued, and therefore continuous. Finally, we provide sufficient conditions that preserve the pre-nucleolus property for related games even when the default game has not a single pre-kernel point.

math.OC↗

arXiv2015

The Incorrect Usage of Propositional Logic in Game Theory: The Case of Disproving Oneself

Recently, we had to realize that more and more game theoretical articles have been published in peer-reviewed journals with severe logical deficiencies. In particular, we observed that the indirect proof was not applied correctly. These authors confuse between statements of propositional logic. They apply an indirect proof while assuming a prerequisite in order to get a contradiction. For instance, to find out that "if A then B" is valid, they suppose that the assumptions "A and not B" are valid to derive a contradiction in order to deduce "if A then B". Hence, they want to establish the equivalent proposition "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, which is a wrong statement. As a consequence, "if A then B" is invalid, disproving their own results. We present and discuss some selected cases from the literature with severe logical flaws, invalidating the articles.

cs.GT↗