A tournament can be represented as a set of candidates and the results from pairwise comparisons of the candidates. In our setting, candidates may form coalitions. The candidates can choose to fix who wins the pairwise comparisons within their coalition. A coalition is winning if it can guarantee that a candidate from this coalition will win each pairwise comparison. This approach divides all coalitions into two groups and is, hence, a simple game. We show that each minimal winning coalition consists of a certain uncovered candidate and its dominators. We then apply solution concepts developed for simple games and consider the desirability relation and the power indices which preserve this relation. The tournament solution, defined as the maximal elements of the desirability relation, is a good way to select the strongest candidates. The Shapley–Shubik index, the Penrose–Banzhaf index, and the nucleolus are used to measure the power of the candidates. We also extend this approach to the case of weak tournaments.

中文翻译：

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基于合作博弈论的锦标赛解决方案

一场比赛可以表示为一组候选者和候选者成对比较的结果。在我们的环境中，候选人可以组成联盟。候选人可以选择确定谁在他们的联盟内赢得了成对比较。如果一个联盟能够保证来自该联盟的候选人将赢得每次成对比较，那么它就是获胜。这种方法将所有联盟分为两组，因此是一个简单的游戏。我们表明，每个最小获胜联盟都由某个未被发现的候选人及其统治者组成。然后我们应用为简单游戏开发的解决方案概念，并考虑可取性关系和保持这种关系的权力指数。锦标赛解，定义为合意性关系的最大元素，是选择最强候选者的好方法。Shapley-Shubik 指数、Penrose-Banzhaf 指数和核仁用于衡量候选者的能力。我们还将这种方法扩展到弱锦标赛的情况。