# Simplifying the Kohlberg Criterion on the Nucleolus: A Correct Approach

## TL;DR

This paper critically examines Nguyen's claimed simplification of the Kohlberg criterion for the nucleolus, identifying logical errors and demonstrating that the proposed algorithms are flawed and cannot correctly determine the nucleolus.

## Contribution

The paper provides a rigorous correction to Nguyen's approach, clarifying the proper logical framework and establishing accurate necessary and sufficient conditions for the nucleolus.

## Key findings

- Nguyen's algorithms are logically flawed and cannot correctly identify the nucleolus.
- The proposed simplified criteria are invalid due to incorrect application of logical implications.
- Corrected conditions for the nucleolus are necessary for accurate computation.

## Abstract

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 $(\phi \Rightarrow \bot) \Leftrightarrow \neg \phi$. 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.