Power Indices

Early History

As far as we know, the first published work on voting power was by Luther Martin, a Maryland delegate to the 1787 Constitutional Convention in Philadelphia. Martin feared that the voting power [Page 510]of the large states in the U.S. House of Representatives would be disproportionately large compared to that of the small states, assuming that the representatives of each of the 13 states would always vote as a bloc. In a pamphlet published the following year he exposed the fallacy of equating voting power with weight (in this case size of a voting bloc), and made an attempt—albeit unsystematic and somewhat crude—to measure voting power. Martin's approach is broadly based on the notion of I-power, namely the ability of a voter—merely by means of his vote—to affect the passage or defeat of a proposed act. Significantly, this first contribution to the measurement of voting power was made by a politician, thereby showing that voting power measures are not mere mathematicians' fancies but are of practical political and constitutional importance.

Comments on the Main Voting-Power Measures

As each of the main voting-power measures has a separate entry in this encyclopedia, we will only comment here briefly on them.

The Penrose Measure

As far as we know, the first properly scientific discussion of voting power is a 1946 paper by Lionel Sharples Penrose in which he proposes a probabilistic measure of absolute voting power. According to Penrose, the power of a voter “can be measured by the amount by which his chance of being on the winning side exceeds one half. The power, thus defined, is the same as half the likelihood of a situation in which [he] can be decisive,” that is, in which the remaining voters are so divided that the given voter can tip the balance (p. 53). Thus, the power of voter v is defined as rv-1/2, where rv is the probability that the outcome of the division goes the way v votes. Penrose posits random voting on the part of the other voters, who are assumed to be indifferent. Although he does not define this explicitly, he is clearly using the a priori Bernoullian model, in which all 2n possible divisions of the set of n voters into yes and no camps are equiprobable.

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