Tournament solutions based on cooperative game theory

  • Aleksei Y. KondratevEmail author
  • Vladimir V. Mazalov
Original Paper


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.


Tournament solution Simple game Shapley–Shubik index Penrose–Banzhaf index Desirability relation Uncovered set 

Mathematics Subject Classification

MSC 91A12 MSC 91B14 

JEL Classification

C71 D71 C44 



We are grateful to Jennifer Rontganger (WZB Berlin Social Science Center) and Alexander Mazurov for help with language editing. We thank Elena Yanovskaya, Alexander Nesterov, Artem Sedakov and Nadezhda Smirnova for helpful comments on the manuscript.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsSt. PetersburgRussia
  2. 2.Institute for Problems of Regional Economics RASSt. PetersburgRussia
  3. 3.Institute of Applied Mathematical ResearchKarelian Research Center of Russian Academy of SciencesPetrozavodskRussia
  4. 4.School of Mathematics and Statistics and Institute of Applied MathematicsQingdao UniversityQingdaoPeople’s Republic of China

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