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A new basis and the Shapley value

Author

Listed:
  • Koji Yokote

    (Graduate School of Economics, Waseda University)

  • Yukihiko Funaki

    (Faculty of Political Science and Economics, Waseda University)

  • Yoshio Kamijo

    (Kochi University of Technology, Department of Management)

Abstract

The purpose of this paper is to introduce a new basis of the set of all TU games. Shapley (1953) introduced the unanimity game inwhich cooperation of all players in a given coalition yields payoff. We introduce the commander game in which only one player in a given coalition yields payoff. The set of the commander games forms a basis and has two properties. First, when we express a game by a linear combination of the basis, the coefficients related to singletons coincide with the Shapley value. Second, the basis induces the null space of the Shapley value.

Suggested Citation

  • Koji Yokote & Yukihiko Funaki & Yoshio Kamijo, 2015. "A new basis and the Shapley value," Working Papers 1418, Waseda University, Faculty of Political Science and Economics.
    • Handle: RePEc:wap:wpaper:1418
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    References listed on IDEAS

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    1. Dragan, I. & Potters, J.A.M. & Tijs, S.H., 1989. "Superadditivity for solutions of coalitional games," Other publications TiSEM 283e2594-e3a0-418d-ae5e-2, Tilburg University, School of Economics and Management.
    2. René Brink & Gerard Laan & Valeri Vasil’ev, 2014. "Constrained core solutions for totally positive games with ordered players," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(2), pages 351-368, May.
    3. Rene van den Brink & Yukihiko Funaki, 2010. "Axiomatization and Implementation of Discounted Shapley Values," Tinbergen Institute Discussion Papers 10-065/1, Tinbergen Institute.
    4. René Brink & Yukihiko Funaki, 2015. "Implementation and axiomatization of discounted Shapley values," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 45(2), pages 329-344, September.
    5. O'Neill, Barry & Peleg, Bezalel, 2008. "Lexicographic composition of simple games," Games and Economic Behavior, Elsevier, vol. 62(2), pages 628-642, March.
    6. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
    7. Chun, Youngsub, 1989. "A new axiomatization of the shapley value," Games and Economic Behavior, Elsevier, vol. 1(2), pages 119-130, June.
    8. René van den Brink, 2002. "An axiomatization of the Shapley value using a fairness property," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(3), pages 309-319.
    9. Llerena, Francesc & Rafels, Carles, 2006. "The vector lattice structure of the n-person TU games," Games and Economic Behavior, Elsevier, vol. 54(2), pages 373-379, February.
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    2. Gasca, María-Victoria & Rigo-Mariani, Remy & Debusschere, Vincent & Sidqi, Yousra, 2025. "Fairness in energy communities: Centralized and decentralized frameworks," Renewable and Sustainable Energy Reviews, Elsevier, vol. 208(C).
    3. Takaaki Abe & Satoshi Nakada, 2023. "Core stability of the Shapley value for cooperative games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 60(4), pages 523-543, May.
    4. Qianqian Kong & Hao Sun & Genjiu Xu & Dongshuang Hou, 2019. "Associated Games to Optimize the Core of a Transferable Utility Game," Journal of Optimization Theory and Applications, Springer, vol. 182(2), pages 816-836, August.
    5. Yokote, Koji & Funaki, Yukihiko & Kamijo, Yoshio, 2017. "Coincidence of the Shapley value with other solutions satisfying covariance," Mathematical Social Sciences, Elsevier, vol. 89(C), pages 1-9.
    6. Sylvain Béal & Mihai Manea & Eric Rémila & Phillippe Solal, 2018. "Games With Identical Shapley Values," Working Papers 2018-03, CRESE.
    7. Takaaki Abe & Satoshi Nakada, 2018. "Generalized Potentials, Value, and Core," Discussion Paper Series DP2018-19, Research Institute for Economics & Business Administration, Kobe University.

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    JEL classification:

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

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