Penrose Voting Power Measure

Lionel Sharples Penrose (1898–1972) was a British mathematician, statistician, psychiatrist, and geneticist. Although his main research work was in genetics and psychiatry, he also investigated the principles on which an ideal two-tier representative system should be based. In 1946 Penrose published an article in the Journal of the Royal Statistical Society that, as far as we know, is the first properly scientific discussion of voting power, and in which he proposed a probabilistic measure of absolute voting power. He slightly modifies its definition in his 1952 booklet.

In defining the voting power of an individual voter, Penrose assumes the Bernoullian model, in which, when voting on an arbitrary resolution, all 2n possible divisions of the set of n voters into yes and no subsets are a priori equally probable. (All probabilities mentioned below are the a priori probabilities of this model.) A resolution passes if some specified majority of the voters support it. No abstention is allowed.

Penrose defines the voting power of voter a to be ra-1/2, where ra is the probability of the event that the outcome of the division goes the way a votes. He then goes on to state (without proof) an identity—a theorem according to which the voting power of a, so defined, equals half the probability ψa of a situation in which a's vote is decisive, that is, that the other n-1 voters are divided in such a way that a's vote determines the outcome.

Penrose's 1952 modified definition is double what it was in his 1946 paper. Thus a's voting power is, by Penrose's identity:

Penrose's measure, in the form denoted here by ψa, was independently reinvented in 1965 by John F. Banzhaf III, and is often referred to as the absolute (or unnormalized) Banzhaf measure (or index).

If the decision rule is interpreted as a simple game, then Penrose's measure can be defined equivalently as

where ηa is the number of winning coalitions in which a's membership is critical—that is, which are no longer winning if a defects from them—and n (as before) is the number of players-voters. For any voter a the following useful identity holds:

where ωa is the number of winning coalitions to which a belongs and ω is the total number of winning coalitions.