Majority voting

Majority rule is a decision rule that selects alternatives which have a majority, that is, more than half the votes. It is the binary decision rule used most often in influential decision-making bodies, including the legislatures of democratic nations.

Though plurality (first-past-the post) is often mistaken for majority rule, they are not the same. Plurality makes the option with the most votes the winner, regardless of whether the fifty percent threshold is passed. This is equivalent to majority rule when there are only two alternatives. However, when there are more than two alternatives, it is possible for plurality to choose an alternative that has less than fifty percent of the votes cast in its favor.

Although it is a binary decision rule, majority rule has become the dominant factor in many modern western democracies, with most elections and referendums being decided by majority rule. It is frequently used in legislatures and other bodies in which alternatives can be considered and amended in a process of deliberation until the final version of a proposal is adopted or rejected by majority rule. It is one of the basic rules prescribed in books like Robert's Rules of Order. The rules in such books and those rules adopted by groups may additionally prescribe the use of a supermajoritarian rule under certain circumstances, such as a two-thirds rule to close debate.

According to May, majority rule is the only reasonable decision rule that is "fair", that is, that does not privilege voters by letting some votes count for more or privilege an alternative by requiring fewer votes for its passing. Stated more formally, majority rule is the only binary decision rule that has the following properties:

Strictly speaking, it has been shown that majority rule meets these criteria only if the number of voters is odd or infinite. If the number of voters is even, there is the chance that there will be a tie, and so the criterion of neutrality is not met. Many deliberative bodies reduce one participant's voting capacity—namely, they allow the chair to vote only to break ties. This substitutes a loss of total anonymity for the loss of neutrality.

In group decision-making it is possible for a voting paradox to form. That is, it is possible that there are alternatives a, b, and c such that a majority prefers a to b, another majority prefers b to c, and yet another majority prefers c to a. Because majority rule requires an alternative to have only majority support to pass, a majority under majority rule is especially vulnerable to having its decision overturned. (The minimum number of alternatives that can form such a cycle (voting paradox) is 3 if the number of voters is different from 4, because the Nakamura number of the majority rule is 3. For supermajority rules the minimum number is often greater, because the Nakamura number is often greater.)

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