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Core Stability of the Shapley Value for 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

Our objective is to analyze the relationship between the Shapley value and the core of cooperative games with transferable utility. We first characterize balanced games, namely, the set of games with a nonempty core, by means of geometric properties. We show that the set of balanced games generates a polyhedral cone and that a game is balanced if and only if it is a nonnegative linear combination of some simple games. Moreover, we show that the set of games whose Shapley value is in the core also yields a polyhedral cone and that a game obeys this property if and only if it is a nonnegative linear combination of some “easy” games. In addition, we also show that the number of games that correspond to the extreme rays of the polyhedron coincides with the number of minimal balanced collections.

Suggested Citation

  • Takaaki Abe & Satoshi Nakada, 2021. "Core Stability of the Shapley Value for Cooperative Games," Working Papers 2112, Waseda University, Faculty of Political Science and Economics.
    • Handle: RePEc:wap:wpaper:2112
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    More about this item

    Keywords

    Cooperative games; Shapley value; Core; Minkowski-Weyl’s Theorem;
    All these keywords.

    JEL classification:

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

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