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.

翻译：最近，Maggiorano等人（2025年）声称他们基于简化博弈方法和子模函数最小化技术，为凸博弈的核仁设计了一种强多项式时间的组合算法。从而完全避免了数值计算中具有负面影响的椭球法。然而，我们认为这一论断是基于对Davis/Maschler简化博弈性质（RGP）的错误应用而产生的谬误。他们忽略了这样一个事实：除了预核仁之外，其他解概念如核心、预核与半反应预谈判集同样具备该性质。这导致了严重的选择问题，使得通过简化博弈方法计算凸博弈的核仁无法实现。为了在具体背景下评估这一发现，我们重新审视了Faigle等人（2001年）的椭球法以及Meinhardt（2013年）基于凸分析中Fenchel-Moreau共轭理论的预核元素计算方法。在后一种方法中，我们利用了以下性质：对于具有单值预核的TU博弈，预核仁与预核这两个解概念是重合的。这意味着当找到博弈中唯一的预核元素时，实际上就计算出了预核仁。尽管该算法是针对预核的专用高度优化算法，但当预核为单点集时，它能确保以O(n^3)的时间复杂度计算预核仁，这表明了针对此类博弈的多项式时间算法的存在性。