We suggest new characterizations of the Banzhaf value without the

symmetry axiom, which reveal that the characterizations by Lehrer

(1988, International Journal of Game Theory 17, 89-99) and Nowak

(1997, International Journal of Game Theory 26, 127-141) as well

as most of the characterizations by Casajus (2010, Theory and Decision, forthcoming) are redundant. Further, we explore symmetry implications of Lehrer's 2-efficiency axiom.