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

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 "$A \land \neg B$ 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.