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Semivalues: power,potential and multilinear extensions


  • Carreras, Francesc
  • Giménez, José Miguel


The notions of power and potential, both defined for any semivalue, give rise to two endomorphisms of the vector space of all cooperative games on a given player set. Several properties of these linear mappings are stated and their action on unanimity games is emphasized. We also relate in both cases the multilinear extension of the image game to the multilinear extension of the original game.

Suggested Citation

  • Carreras, Francesc & Giménez, José Miguel, 2010. "Semivalues: power,potential and multilinear extensions," MPRA Paper 27620, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:27620

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    References listed on IDEAS

    1. Pradeep Dubey & Abraham Neyman & Robert James Weber, 1981. "Value Theory Without Efficiency," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 122-128, February.
    2. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
    3. Pradeep Dubey & Lloyd S. Shapley, 1979. "Mathematical Properties of the Banzhaf Power Index," Mathematics of Operations Research, INFORMS, vol. 4(2), pages 99-131, May.
    4. Robert J. Weber, 1977. "Probabilistic Values for Games," Cowles Foundation Discussion Papers 471R, Cowles Foundation for Research in Economics, Yale University.
    5. Dragan, Irinel, 1996. "New mathematical properties of the Banzhaf value," European Journal of Operational Research, Elsevier, vol. 95(2), pages 451-463, December.
    6. Sergiu Hart, 2006. "Shapley Value," Discussion Paper Series dp421, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    7. Robert J. Weber, 1979. "Subjectivity in the Valuation of Games," Cowles Foundation Discussion Papers 515, Cowles Foundation for Research in Economics, Yale University.
    8. Rafael Amer & José Miguel giménez, 2003. "Modification of Semivalues for Games with Coalition Structures," Theory and Decision, Springer, vol. 54(3), pages 185-205, May.
    9. Feltkamp, Vincent, 1995. "Alternative Axiomatic Characterizations of the Shapley and Banzhaf Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 24(2), pages 179-186.
    10. Calvo, Emilio & Santos, Juan Carlos, 1997. "Potentials in cooperative TU-games," Mathematical Social Sciences, Elsevier, vol. 34(2), pages 175-190, October.
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    More about this item


    Cooperative game; Semivalue; Power; Potential; Multilinear extension;

    JEL classification:

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

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