The recent axiomatic study of probabilistic values of games has clariﬁed the relationship between various valuation methods and the players’ subjective perceptions of the coalition-formation process. This has important bearing upon the increasingly-common use of the Banzhaf value in measuring the apportionment of power among the players in voting games. The incompatibility of the players’ hypothesized subjective beliefs (under the Banzhaf valuation scheme) leads to the strange phenomenon of “pitfall” points (points of value discontinuity) in weighted majority games with several major players and an ocean of minor players. Such results argue against the use of the Banzhaf value (or indeed, of any value other than the Shapley-Shubik index) in the measurement of power in weighted voting systems.
Weber, Robert J., "Subjectivity in the Valuation of Games" (1979). Cowles Foundation Discussion Papers. 750.