Collusion, symmetry, and the Banzhaf value

We resolve redundancies in the characterizations of the Banzhaf value suggested by Haller (1994, Int J Game Theory 23, 261-281) and Malawski (2002, Int J Game Theory 31:47-67). In particular, we show that the collusion properties employed by them are equivalent. Combined with the dummy player axiom, any of the collusion properties has strong symmetry implications whenever the cardinality of the player set exceeds two. Finally, we establish that the Banzhaf value is non-redundantly characterized by the dummy player axiom and any of the collusion properties, provided that the player set is as above.