ABSTRACT
Majority Rule and its Weighted Analog Introduction “Majority” means “more than half.” Majority rule is well defined only for cases where there are just two options. For these cases, majority rule is the voteprocessing rule that chooses the option that has the greater number of supporters (the majority of those voting). If there are more than two options, it is possible that no option has the support of a majority (more than half) of the voters. Voteprocessing rules for these cases are considered in later chapters. In terms of the dimensions of voting developed in Chapter 7, majority rule is characterized by the fact that it involves two options, with one to be selected. Traditionally, it also involves equally weighted votes. This standard version of majority rule is treated in the next section, where separate subsections consider majority rule historically, as a procedure for estimating statistically which option is likely to be better, and as an axiomatic system. The statistical subsection accords separate treatment to majority rule for voters who are self-interested advocates and for voters who are disinterested judges. The next section discusses the variation on majority rule generated by the assignment of predetermined, unequal weights to the votes. Here again, separate treatment is accorded to the use of this rule when voters are disinterested judges and when they are self-interested advocates. Equally Weighted Votes Majority rule is so much a part of our cultural heritage that it is difficult to address the question of what is attractive about it. Should we have minority rule instead? Absurd. Historical Approaches For a systematic explanation of why majority rule is attractive, it is interesting to examine theories of its origin. J.A.O. Larsen argues that a comparison of decisionmaking procedures described in the Homeric epics with what is known about the origins of democracy in Greece suggests that voting originated in the seventh century B.C.1 He says that the earlier practice was simply to talk until it seemed
that someone had come up with the right idea. But this has a potential for leading to conflict when not all persons perceive the discussion to have come to the same conclusion. Larsen offers such an example from the second book of the Iliad. Agamemnon planned to have others oppose his insincere suggestion that they all go home. The men responded by rushing to the boats, and Odysseus had to drive them back to the meeting.2 An orderly discussion followed by a vote can help avoid such disorder. Another theory holds that voting originated independently in Germanic tribes of Northern Europe, in an environment where it was understood that there should be unanimous agreement on all decisions.3 If consensus did not emerge through discussion, it was acceptable to employ coercion to reach a decision to which all agreed, with those who did not wish to be coerced allowed to flee. Majority rule emerged when it became customary to allow the majority (who could be expected to prevail in the event of a clash) to have their way without a physical clash. Statistical Justifications Self-Interested Advocates There is a statistical justification of majority rule in terms of the “principle of insufficient reason.” If one is trying to estimate which of two options is likely to provide greater total utility to the members of a collectivity, and if the only information that is available is the number of voters who favor each option, then there is no reason to suppose that those who favor one option will feel any more strongly about the matter than those who favor the other. Therefore it is more likely that greater total utility is obtained if the option favored by the majority is adopted than if the option favored by the minority is adopted. Disinterested Judges If one views voters as disinterested, then a justification of majority rule first offered by Condorcet is possible.4 If every voter is regarded as equally skillful in discerning which of two options is better, and if prior to the vote one has no reason for believing that one option is more likely to be better than the other, then after the vote it is reasonable to believe that the option favored by the majority is more likely to be the better one. May’s Axiomatic Justification A different way of justifying majority rule is to look for a set of formal, mathematical conditions that describe an attractive vote-processing rule and show that they are uniquely satisfied by majority rule. This approach was taken by K.O. May.5 May characterized the object of a vote as a “motion” and used the numbers “1,” “0” and “–1” to symbolize a vote in favor, an abstention, and a vote against the motion respectively. In describing the outcome of the vote, he used the same numbers to symbolize “the motion is passed,” “there is a tie,” and “the motion is defeated” respectively. In May’s framework, a “voting rule” (what I have called a
vote-processing rule) is a mathematical function that maps a sequence of numbers, each of which is 1, 0, or –1, to one of the numbers 1, 0, and –1. Thus if S is a sequence of 1’s, 0’s and –1’s, majority rule may be described as the following function:
f (S ) = °¯ °®
negative. is of components theof sum theif 1 0; is of components theof sum theif 0 positive; is of components theof sum theif 1
S S S
May showed that the following four conditions are necessary and sufficient for majority rule:
1. Universal domain: The function yields exactly one outcome for every sequence of 1’s, 0’s and –1’s interpreted as votes.
