The banzhaf value and general semivalues for differentiable mixed games

We consider semivalues on pM_∞—a vector space of games with a continuum of players (among which there may be atoms) that possess a robust differentiability feature. We introduce the notion of a derivative semivalue on pM_∞ and extend the standard Banzhaf value from the domain of finite games onto pM_∞ as a certain particularly simple derivative semivalue. Our main result shows that any semivalue on pM_∞ is a derivative semivalue. It is also shown that the Banzhaf value is the only semivalue on pM_∞ that satisfies a version of the composition property of Owen and that, in addition, is nonzero for all nonzero monotonic finite games.