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Voting often is used to decide yes or no questions. Legislatures vote on bills, stockholders vote on resolutions presented by the board of directors of a corporation, and juries vote to acquit or convict a defendant. In this chapter, we shall concentrate on situations where there are just two alternatives, “yes” or “no.” The theorem of Kenneth May quoted in Chapter 9 says that majority rule is the only system with the following properties:

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There are systems in which the voters appear to be unequal in power, but actually have all the properties required by May’s theorem. Any student of politics will attest that not all legislators are equally powerful. (Think of the speaker of the U.S. House of Representatives versus a freshman member, or the prime minister versus a backbencher in Parliament.) Nevertheless, the voting system actually treats the legislators equally: Each is allowed one vote. Our interest is in the voting system itself and not in the influence that some voters might acquire as a result of experience or accomplishment.

In this chapter, we shall consider voting systems that do not treat the voters equally. For example, shareholders of corporations are asked to vote on motions presented by the corporation’s board of directors. Each shareholder is allotted one vote per share that he or she owns. If two shareholders own different numbers of shares, they are not treated equally as voters. The voter with the larger number of shares has the greater investment and is given at least the appearance of greater influence. In this case, the stockholders are treated unequally because they are actually unequal.

The Council of Ministers of the European Union consists of the prime ministers from each of the 28 member states. The populations of the member states range from Malta’s population of half a million to Germany’s, which is more than 80 million. Until the voting provisions of the Treaty of Lisbon¹ went into effect on November 1, 2014, the Council of Ministers gave the ministers from the larger states more votes as a way of enhancing their influence. Voting systems in which the voters have varying numbers of votes are called weighted voting systems.

We will see that the number of votes that a participant is allowed to cast does not always reflect that participant’s influence. To assess a voter’s power to affect the outcome of a vote, we will define and explore two mathematical models of the decision-making process, each of which leads to a numerical measure of voting power, called a power index.

Our mathematical models start with a set of assumptions designed to capture the essence of a voting system—how a system allocates decision-making power to the voters, without regard to political alliances, skillful manipulation of the system by the more able participants, etc. The first model, developed by a mathematician, Lloyd S. Shapley, and an economist, Martin Shubik, is based on the assumption that the content of a proposal to be voted upon is subject to negotiation in order to attract votes. The second, developed by an attorney, John F. Banzhaf III, assumes that there is no communication between the participants. Formally, the Banzhaf model best describes a vote in which each participant decides by a coin toss. This ignores political reality, but it does extract the essence of the way the voting system allocates power when the participants operate independently of each other.