Voting Power

Voting power, in contrast to the number of votes an actor possesses, is the ability of an actor to influence the outcome of voting in a collectivity.

While the idea of voting power can be traced at least as far back as a report by Luther Martin in 1787, the first systematic study of voting power was made by Lionel Sharples Penrose in 1946, in the context of a hypothetical distribution of votes in the UN General Assembly. He introduced the concept of a priori voting power, a measurement of the ability of a participant of the voting body to influence the outcome when votes are cast randomly.

The share of votes a member of the collectivity casts does not directly determine the power that this voter exercises; that also depends on the configuration of votes allocated to all other members and on the rules of the voting system. Various measures of a priori voting power have been proposed, some of which measure relative voting power and are expressed in normalized form (so that the total power of all voters adds up to 1) and others of which measure absolute voting power. Examples of the former include the Shapley—Shubik index, the normalized Banzhaf index, the Deegan—Packel index, and the Public Goods Index. The prime example of the latter is the Penrose voting power measure, including variants later developed by Banzhaf and by James Coleman.

Relative voting power is useful for making comparisons among members of a given collectivity with a fixed voting rule, while absolute voting power allows comparisons among different collectivities and different voting rules.

Measuring voting power may give rise to certain paradoxes, and some power measures may be suitable for measuring certain aspects of voting power but not others.

In their important work, Dan S. Felsenthal and Moshé Machover distinguish between two categories [Page 700]that they call I-power and P-power. I-power interprets voting power as the probability of influencing the decisions of a voting body under a specified decision rule, that is, whether a proposed bill is approved or rejected. The original Penrose voting power measure and its derivatives measure I-power. P-power interprets a decision rule as a simple cooperative game with transferable utility and conceptualizes voting power as a voter's ability to win some share of a fixed total payoff. The Shapley—Shubik index measures P-power (see related entry).

Table 1 The voting power assigned by various power measures under two different decision rules

Differences among various measures of voting power are illustrated by the following example. Consider a miniature parliament composed of three parties with two, one, and one seats respectively and no ideological or other inclination to form one winning coalition in preference to another. Consider also two decision rules: the simple majority rule and the unanimity rule. In the first case, an act can be approved with at least three favorable votes, while in the second case, an act can be approved only with four favorable votes. Table 1 (p. 699) shows the power of each party under each decision rule under each power measure.

...