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Generalized Potentials, Value, and Core

Author

Listed:
  • Takaaki Abe

    (Graduate School of Economics, Waseda University)

  • Satoshi Nakada

    (Department of Business Economics, Tokyo University of Science)

Abstract

Our objective is to analyze the relationship between the Shapley value and the core from the perspective of the potential of a game. To this end, we introduce a new concept, generalized HM-potential, which is a generalization of the potential function defined by Hart and Mas-colell (1989). We show that the Shapley value lies in the core if and only if the maximum of the generalized HM-potential of a game is less than a cutoff value. Moreover, we show that this is equivalent to the minimum of the generalized HM-potential of a game being greater than another, different cutoff value. We also provide a geometric characterization of the class of games in which the Shapley value lies in the core, which also shows the relationship with convex games and average convex games as a corollary. Our results suggest a new approach to utilizing the potential function in cooperative game theory.

Suggested Citation

  • Takaaki Abe & Satoshi Nakada, 2018. "Generalized Potentials, Value, and Core," Discussion Paper Series DP2018-19, Research Institute for Economics & Business Administration, Kobe University.
  • Handle: RePEc:kob:dpaper:dp2018-19
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    File URL: https://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/DP2018-19.pdf
    File Function: First version, 2018
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    References listed on IDEAS

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    More about this item

    Keywords

    Shapley value; Core; Potential; Cooperative game;
    All these keywords.

    JEL classification:

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

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