Generating Semivalues via Unanimity Games

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.