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Potentials and Solutions of Cooperative Games

Author

Listed:
  • Takaaki Abe

    (: School of Political Science and Economics, Waseda University, 1-6-1, Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050, Japan.)

  • Satoshi Nakada

    (School of Management, Department of Business Economics, Tokyo University of Science, 1-11-2, Fujimi, Chiyoda-ku, Tokyo, 102-0071, Japan)

Abstract

This paper considers the solution concepts of cooperative games that admit a potential function. We say that a solution admits a potential function if the solution is given as the gradient vector of the potential function at the player set. Hart and Mas-Collel (1989) show that the Shapley value is the only solution that is efficient and admits the potential function for games with variable player sets. In this paper, first, we argue that if we remove efficiency, various solutions admit a potential function. Second, we characterize the class of the solutions that admit a potential function and provide their general functional form. Third, we define a potential function for games with a fixed player set and associate a potential function with the axioms that the Shapley value obeys. Finally, we discuss how the efficiency requirement induces the uniqueness of the Shapley value through a potential function.

Suggested Citation

  • Takaaki Abe & Satoshi Nakada, 2021. "Potentials and Solutions of Cooperative Games," Working Papers 2114, Waseda University, Faculty of Political Science and Economics.
    • Handle: RePEc:wap:wpaper:2114
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    More about this item

    Keywords

    Cooperative games; Efficiency; Potential function; Shapley value;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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