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

TL;DR

This paper discredits Nguyen and Thomas's claimed method for computing the nucleoli of large cooperative games, providing a counter-example that shows their algorithm is incorrect and not robust.

Contribution

The paper provides a counter-example disproving Nguyen and Thomas's method and highlights fundamental errors in their proof and algorithm.

Findings

Nguyen and Thomas's algorithm does not compute the nucleolus correctly

Their stopping criterion violates Kohlberg's properties

The presented flow game example is a counter-example to their claims

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.