Holger I. Meinhardt | Searcharxiv

arXiv2026

Polynomial-Time Algorithms for Computing the Nucleolus: An Assessment

Recently, Maggiorano et al. (2025) claimed that they have developed a strongly polynomial-time combinatorial algorithm for the nucleolus in convex games that is based on the reduced game approach and submodular function minimization method. Thereby, avoiding the ellipsoid method with its negative side effects in numerical computation completely. However, we shall argue that this is a fallacy based on an incorrect application of the Davis/Maschler reduced game property (RGP). Ignoring the fact that despite the pre-nucleolus, other solutions like the core, pre-kernel, and semi-reactive pre-bargaining set possess this property as well. This causes a severe selection issue, leading to the failure to compute the nucleolus of convex games using the reduced games approach. In order to assess this finding in its context, the ellipsoid method of Faigle et al. (2001) and the Fenchel-Moreau conjugation-based approach from convex analysis of Meinhardt (2013) to compute a pre-kernel element were resumed. In the latter case, it was exploited that for TU games with a single-valued pre-kernel, both solution concepts coincide. Implying that one has computed the pre-nucleolus if one has found the sole pre-kernel element of the game. Though it is a specialized and highly optimized algorithm for the pre-kernel, it assures runtime complexity of O(n^3) for computing the pre-nucleolus whenever the pre-kernel is a single point, which indicates a polynomial-time algorithm for this class of games.

cs.GT↗

arXiv2021

Deduction Theorem: The Problematic Nature of Common Practice in Game Theory

We consider the Deduction Theorem used in the literature of game theory to run a purported proof by contradiction. In the context of game theory, it is stated that if we have a proof of $ϕ\vdash φ$, then we also have a proof of $ϕ\Rightarrow φ$. Hence, the proof of $ϕ\Rightarrow φ$ is deduced from a previously known statement. However, we argue that one has to manage to establish that a proof exists for the clauses $ϕ$ and $φ$, i.e., they are known true statements in order to show that $ϕ\vdash φ$ is provable, and that therefore $ϕ\Rightarrow φ$ is provable as well. Thus, we are not allowed to assume that the clause $ϕ$ or $φ$ is a true statement. This leads immediately to a wrong conclusion. Apart from this, we stress to other facts why the Deduction Theorem is not applicable to run a proof by contradiction. Finally, we present an example from industrial cooperation where the Deduction Theorem is not correctly applied with the consequence that the obtained result contradicts the well-known aggregation issue.

cs.AI↗