Values, nullifiers and dummifiers

Abstract

[Abstract]: In this chapter, we provide new axiomatic characterizations of the Shapley and Banzhaf values using properties concerning nullifying and dummifying players. Then, we introduce a novel value for TU-games using an axiomatic approach: the e-Banzhaf value. Specifically, we take the properties of efficiency, symmetry, additivity, and combine them with the property for nullifying players that we consider most natural: the nullifying players pay for the mean property. We prove that these four properties characterize the e-Banzhaf value. This novel value possesses some properties of the Shapley value and some of the Banzhaf value, and can be seen as a kind of efficient variation of the Banzhaf value. However, it is not invariant to S-equivalence and it does not satisfy the dummifying players pay for the mean property. Finally, we introduce an alternative novel value for TU-games: the ie-Banzhaf value. It is the unique value for TU-games that satisfies efficiency, symmetry, additivity and the dummifying players pay for the mean property. We prove that the ie-Banzhaf value is also a kind of efficient variation of the Banzhaf value and that it is invariant to S-equivalence.