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 \(N\) 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.
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Notes
The condition \(v(N)=1\) implies that there must be at least one winning coalition, due to the monotonicity condition. Sometimes this is omitted in the definition of simple game.
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Acknowledgments
The second author is indebted to H. Sendov for bringing his attention to completely monotonic sequences, and to provide some bibliographical information, and to Agata Caserta for some useful discussion on the topic of the paper. Research partially supported by Ministero dell’Istruzione, dell’Università e della Ricerca Scientifica (COFIN 2009).
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Bernardi, G., Lucchetti, R. Generating Semivalues via Unanimity Games. J Optim Theory Appl 166, 1051–1062 (2015). https://doi.org/10.1007/s10957-014-0660-1
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DOI: https://doi.org/10.1007/s10957-014-0660-1