Alternative axiomatic characterizations of the Shapley and Banzhaf values

Dubey characterized the Shapley-Shubik index axiomatically on the class of monotonic simple games. Dubey and Shapley characterized the Banzhaf index in a similar way. This paper extends these characterizations to axiomatic characterizations of the Shapley and Banzhaf values on the class of control games, on the class of simple games and on the class of all transferable utility games. In particular, it is shown that the additivity axiom which is usually used to characterize these values on the class of all transferable utility games can be weakened without changing the result.