The nucleolus offers a desirable payoff-sharing solution in cooperative games, thanks to its attractive properties—it always exists and lies in the core (if the core is non-empty), and it is unique. The nucleolus is considered as the most ‘stable’ solution in the sense that it lexicographically minimizes the dissatisfactions among all coalitions. Although computing the nucleolus is very challenging, the Kohlberg criterion offers a powerful method for verifying whether a solution is the nucleolus in relatively small games (i.e. with the number of players $$n \le 15$$ n ≤ 15 ). This approach, however, becomes more challenging for larger games because of the need to form and check a criterion involving possibly exponentially large collections of coalitions, with each collection potentially of an exponentially large size. The aim of this work is twofold. First, we develop an improved version of the Kohlberg criterion that involves checking the ‘balancedness’ of at most $$(n-1)$$ ( n - 1 ) sets of coalitions. Second, we exploit these results and introduce a novel descent-based constructive algorithm to find the nucleolus efficiently. We demonstrate the performance of the new algorithms by comparing them with existing methods over different types of games. Our contribution also includes the first open-source code for computing the nucleolus for games of moderately large sizes.

中文翻译：

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合作博弈核仁的发现与验证

由于其吸引人的特性，核仁在合作博弈中提供了一种理想的收益共享解决方案——它始终存在并位于核心中（如果核心非空），并且它是独一无二的。核仁被认为是最“稳定”的解决方案，因为它在字典上最大限度地减少了所有联盟之间的不满。尽管计算核仁非常具有挑战性，但科尔伯格准则提供了一种强大的方法来验证解决方案是否是相对较小游戏中的核仁（即玩家数量 $$n \le 15$$n ≤ 15 ）。然而，这种方法对于大型游戏变得更具挑战性，因为需要形成和检查一个标准，该标准涉及可能呈指数级大的联盟集合，每个集合的大小可能呈指数级大。这项工作的目的是双重的。首先，我们开发了 Kohlberg 准则的改进版本，该准则涉及检查最多 $$(n-1)$$ ( n - 1 ) 组联盟的“平衡性”。其次，我们利用这些结果并引入一种新的基于下降的构造算法来有效地找到核仁。我们通过将新算法与不同类型游戏的现有方法进行比较来展示新算法的性能。我们的贡献还包括第一个用于计算中等大小游戏核仁的开源代码。我们通过将新算法与不同类型游戏的现有方法进行比较来展示新算法的性能。我们的贡献还包括第一个用于计算中等大小游戏核仁的开源代码。我们通过将新算法与不同类型游戏的现有方法进行比较来展示新算法的性能。我们的贡献还包括第一个用于计算中等大小游戏核仁的开源代码。