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Tournament solutions based on cooperative game theory

Author

Listed:
  • Aleksei Y. Kondratev

    (National Research University Higher School of Economics
    Institute for Problems of Regional Economics RAS)

  • Vladimir V. Mazalov

    (Karelian Research Center of Russian Academy of Sciences
    Qingdao University)

Abstract

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.

Suggested Citation

  • Aleksei Y. Kondratev & Vladimir V. Mazalov, 2020. "Tournament solutions based on cooperative game theory," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(1), pages 119-145, March.
    • Handle: RePEc:spr:jogath:v:49:y:2020:i:1:d:10.1007_s00182-019-00681-5
    DOI: 10.1007/s00182-019-00681-5
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    More about this item

    Keywords

    Tournament solution; Simple game; Shapley–Shubik index; Penrose–Banzhaf index; Desirability relation; Uncovered set;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory

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