Banzhaf Voting Power Measure

The Banzhaf voting power measure (β') is one of the best known and most widely applied measures of a priori voting power. Although introduced in 1946 by Lionel Sharples Penrose, it was unknowingly and more famously reinvented by John Banzhaf in 1965. Moreover, James S. Coleman reinvented it once again in 1971. Thus, it is known variously as the Banzhaf, Penrose, Penrose-Banzhaf, or Banzhaf-Coleman-Penrose measure (or index). It is also known as the absolute Banzhaf measure to distinguish it from the normalized Banzhaf index (β) introduced by subsequent authors.

The definitions of the absolute and normalized versions of the Banzhaf measure will be presented, links with the previously mentioned indices will be illustrated, and some axiomatizations and algorithms will be noted. An example will be presented in conclusion.

The essential element in the construction of both the absolute and the normalized Banzhaf measures is the concept of “swing.” Let N = {1, 2,…, n} be the set of members of a collectivity. A swing for member i of N is a pair of coalitions (S, S/{i}) such that S is a winning coalition and S/{i} is not winning. The number of swings for each member i is called his Banzhaf score and is denoted by ci. For each voter i, the absolute Banzhaf measure (β') is the quotient between his Banzhaf score and the number of pairs of complementary coalitions (or bipartitions), that is, .

In 1979, Pradeep Dubey and Lloyd Stowell Shapley developed an axiomatization of the Banzhaf index using four axioms. Guillermo Owen and others have proposed successive axiomatizations.

As noted at the outset, the absolute Banzhaf measure β' is essentially identical to the measure r previously proposed by Penrose. (Specifically, β'=2r.) Furthermore, the measures of the power to prevent action γ and the power to initiate action γ* subsequently introduced by J. S. Coleman are both rescalings of β' (and r). In particular, β' and both Coleman measures are equal if every coalition or its complement is winning (i.e., there are no blocking coalitions), otherwise β is the harmonic mean of γ and γ*.

The normalized Banzhaf index β is the rescaling of β' such that the power of all voters sums to 1. That is, for each voter i, this index is the ratio between his Banzhaf score and the sum of Banzhaf scores across all voters.

Dan S. Felsenthal and Moshé Machover have classified β' and β, as well as γ and γ*, in the group of the I-power indices (power as influence).

A key problem with the Banzhaf index regards monotonicity. Both β' and β are locally monotonic—that is, if a member has greater weight than another, the member's index is not less. But only β' is globally monotonic, that is, if the weight of the i-th member increases and the weights of all other members decrease or remain unchanged, then β' of [Page 55]the member i does not decrease, but this is not generally true for β.

Numerous algorithms have been introduced to calculate Banzhaf values given a large number of voters. In 1962, Irwin Mann and L. S. Shapley proposed a method of calculating the Shapley—Shubik Index that uses generating functions. This method was extended to the Banzhaf index by Steven J. Brams and Paul J. Affuso in 1976 and was subsequently implemented by Jesús Mario Bilbao, and others. Another algorithm, developed by Gianfranco Gambarelli in 1996, can be applied to weighted majority games and supplies a method for the direct calculation of the measure's variations in response to variations of weight distribution among the players.

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