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Generating Semivalues via Unanimity Games

Author

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  • Giulia Bernardi

    (Politecnico di Milano)

  • Roberto Lucchetti

    (Politecnico di Milano)

Abstract

We provide a condition guaranteeing when a value defined on the base of the unanimity games and extended by linearity on the space of all games with a fixed, finite set $$N$$ N of players is a semivalue. Furthermore, we provide a characterization of the semivalues on the vector space of all finite games, by proving that the coefficients on the base of the unanimity games form a completely monotonic sequence. We also give a characterization of irregular semivalues. In the last part, we remind some results on completely monotonic sequences, which allow one to easily build regular semivalues, with the above procedure.

Suggested Citation

  • Giulia Bernardi & Roberto Lucchetti, 2015. "Generating Semivalues via Unanimity Games," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 1051-1062, September.
    • Handle: RePEc:spr:joptap:v:166:y:2015:i:3:d:10.1007_s10957-014-0660-1
    DOI: 10.1007/s10957-014-0660-1
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    References listed on IDEAS

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    1. Roberto Lucchetti & Paola Radrizzani & Emanuele Munarini, 2011. "A new family of regular semivalues and applications," International Journal of Game Theory, Springer;Game Theory Society, vol. 40(4), pages 655-675, November.
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    3. Carreras, Francesc & Freixas, Josep & Puente, Maria Albina, 2003. "Semivalues as power indices," European Journal of Operational Research, Elsevier, vol. 149(3), pages 676-687, September.
    4. Stefano Moretti & Fioravante Patrone, 2008. "Rejoinder on: Transversality of the Shapley value," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 16(1), pages 60-61, July.
    5. Stefano Moretti & Fioravante Patrone, 2008. "Transversality of the Shapley value," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 16(1), pages 1-41, July.
    6. Monderer, Dov & Samet, Dov, 2002. "Variations on the shapley value," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 54, pages 2055-2076, Elsevier.
    7. Stefano Moretti & Fioravante Patrone & Stefano Bonassi, 2007. "The class of microarray games and the relevance index for genes," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 15(2), pages 256-280, December.
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