- Published 1997 in Int. J. Game Theory

We prove that the Banzhaf value is a unique symmetric solution having the dummy player property, the marginal contributions property introduced by Young (1985) and satisfying a very natural reduction axiom of Lehrer (1988). This note is devoted to the value concept introduced by Banzhaf (1965). It is wellknown that the Banzhaf value is not an efficient solution. Many authors tried to find a substitute of the efficiency axiom in the derivation of the Shapley value (Shapley, 1953) that would help to determine the Banzhaf value uniquely; see Roth (1977), Dubey and Shapley (1979) or Feltkamp (1995). The approach in Owen (1982) is based on a different idea but does not determine the Banzhafvalue uniquely. A very interesting "reduction" property was introduced by Lehrer (1988) who gave two axiomatic characterizations of the Banzhaf value. His first characterization is based on a "transfer" property of the value restricted to simple games. Such a property was introduced by Dubey (1975) and then applied to studying the Banzhaf value by Dubey and Shapley (1979) and Feltkamp (1995). The second axiomatization of Lehrer (1988) is (in a very essential way) based on the linearity assumption. Haller (1994) characterized the Banzhaf value by using some collusion neutrality properties, but his approach is also heavily based on the linearity assumption. In this paper, we employ equal treatment, dummy player axioms, a version of the reduction property due to Lehrer (1988) and the well-known postulate of Young (1985) saying that the value is (in some sense) determined by the marginal contributions of the players. Our axiomatic characterization of the Banzhaf value is a counterpart of the theorem of Young (1985) on the Shapley value. Let N be a finite set of players. Subsets of N are called coalitions. To simplify the notation, one membered coalitions {i} will sometimes be denoted by i. The cardinality of any coalition S is denoted by ISt. A transferable utility game (a TU-game) is a function v that assigns to each coalition S a real number v(S) and, in particular, v(O) = 0. For each coalition S, v(S) represents the worth or the power of S. For each nonempty coalition T, the unanimity game ur of the coalition T is defined by ur(S) = 1 if T c S, and ur(S) = 0 otherwise. A value is a mapping ~ which associates with each n-person TU-game v a vector ~ ( v ) = (~l(v), . . . ,~n(V)). The real number ~i(v) represents the individual value for player i in the game v. 0020-7276/97/1/137-14152.50 9 1997 Physica-Verlag, Heidelberg

@article{Nowak1997OnAA,
title={On an axiomatization of the banzhaf value without the additivity axiom},
author={Andrzej S. Nowak},
journal={Int. J. Game Theory},
year={1997},
volume={26},
pages={137-141}
}