Majority rule is a characteristic of nations in which the majority of a population plays a role in decision-making. Political systems with majority rule are majoritarian while those with minority rule are minoritarian. The UK adopted the principle of "No Independence Before Majority African Rule" in its decolonization of Africa.

If there are only two alternatives, the choice of the majority can be determined without any problems: That alternative is the collective choice, which gets more votes than the other one. If both alternatives get the same number of votes, an additional decision is necessary. In cases with more than two alternatives under consideration, it is more difficult to say which alternative is preferred by the majority, as the following example will show, where a group of five individuals (A, B, C, D, E) has to choose one out of four alternatives (w, x , y, z).

The following preferences of the individuals are assumed:

	A	B	C	D	E
1.	y	y	x	z	w
2.	z	x	z	x	x
3.	w	z	y	y	y
4.	x	w	w	w	z

The question is: Which of the alternatives is preferred by the majority?

The voting system, by which that alternative wins, which in a single act of voting gets more votes than any other alternative, is called “plurality voting”. If this method is applied and everyone votes “sincerely” for his preferred alternative, then y would represent the collective choice. It gets two votes whereas the other ones get only one.

But is y the alternative the majority prefers? Alternative y is wanted by A and B, but two individuals are no majority in a group of five.

If one looks at the rankings of the table above one will notice that there is an alternative x, which is preferred to y by a majority (C, D, and E). Consequently alternative y cannot be what the majority wants.

One could try to solve the problem by applying the method of majority voting. Then that alternative is the collective choice, which gets more than half of the votes.

Let us assume that by the votes of A, C and D the alternative z is chosen. A, C and D are a majority. But is alternative z really that what the majority wants?

Apparently it is not. One can derive from the rankings that for a majority (B, C and E) alternative x is better than z. Consequently alternative z, too, cannot be what the majority wants.

The French scholar Condorcet (1743-94) therefore proposed to compare each alternative with each other. If an alternative in all cases receives a majority of votes, this alternative represents the will of the majority. “That motion, if any, which is able to obtain a simple majority over all the other motions concerned, is the majority motion.” (Duncan Black, The Theory of Committees and Elections, Cambridge 1958, p. 18.)

Contents 1 Majority rule with coalitions 2 Voting on single issues and suboptimality 3 Majority rule and minority rights 4 Further Reading 5 References 6 See also

[edit] Majority rule with coalitions

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Up to now it was assumed that everyone votes “sincerely” for his favourite alternative. But in many cases voters are able to improve their position by adopting a certain strategy of voting. If one looks upon voters as persons maximizing their utilities, the assumption of sincere voting will not work.

If one assumes instead that voters know the preferences of the other voters and are able to make binding agreements on how to vote, the whole scene changes.

As the example above demonstrates, under such conditions w, y, and z become unstable results, for in each case there is a majority of individuals preferring x to that result. In terms of game theory one could say that the Condorcet winner is the only point of stable equilibrium in the cooperative game of voting.

If voters

• know the preferences of the other voters,
• make binding agreements on how to vote and
• vote rationally according to their own interest

then an existing Condorcet winner will win in all voting systems giving equal weights to the individual preferences.^{[citation needed]}

Therefore all these voting systems are in accordance with majority rule.

The proof of this theorem is rather easy. If for instance a candidate y rather than the Condorcet winner x is chosen, those individuals preferring x to y could have established a winning majority coalition on the basis of x, what would have been better for each member of the respective coalition. A voting system which gives equal weights to the voters will only then produce a result other than the Condorcet winner when one exists in the case of irrational behavior, imperfect information, or inability to enforce binding contracts.

There is, however, the possibility that no Condorcet winner exists because of circular majorities: x > y and y > z and z > x. In this case, the group is said to have intransitive preferences. There is no stable point of equilibrium in the theoretical model of the voting process. In real life this is no great problem because whenever the voting process delivers no result the status quo normally will be chosen. Under real-life conditions there are "frictions" not considered by the theoretical model, which will stop the circular movement. For example there may be costs of changing partners and establishing a new majority coalition.

[edit] Voting on single issues and suboptimality

Majority rule may lead to quite different results if one votes separately on several single issues or if one puts these issues together and votes once on the corresponding bundles of alternatives.

An example may demonstrate this.

Suppose there are 3 voters, A, B and C, who have to decide 3 issues each with 2 alternatives: s or t, v or w, and x or y.

When a certain alternative is collectively chosen, voters either get a certain additional quantity of hours of leisure or their hours of leisure are reduced by a certain quantity. It is further assumed that each voter prefers more hours of leisure to less.

The 6 alternatives and the corresponding outcomes for the voters are given in the tables below:

	A	B	C
s:	0	0	0
t:	1	1	-3

	A	B	C
v:	0	0	0
w:	1	-3	1

	A	B	C
x:	0	0	0
y:	-3	1	1

From the tables you can see that for A and B alternative t is better than s, that for A and C alternative w is better than v, and that for B and C alternative y is better than x. Therefore t, w and y are the majority alternatives and thus the collective choice.

Now we put the 3 issues together. We get bundles of 3 alternatives each, for instance t+w+y and s+v+x, on which to vote. The bundles correspond to the following outcomes for the voters, consisting in hours of leisure (or quantities of any other good):

	A	B	C
s+v+x:	0	0	0
t+w+y:	-1	-1	-1

The table shows that now a majority prefers s+v+x to t+w+y. This result is quite the opposite of the former results gained by voting separately on each issue.

The bundle s+v+x now is preferred not only by a majority of voters but is even unanimously preferred by all the voters.

This means that s+v+x is superior to t+w+y according to the Pareto criterion.

Voting on each issue separately may thus lead to suboptimal results.

This is a rather strong argument against “direct democracy” and the indiscriminate use of referenda on single issues.

[edit] Majority rule and minority rights

More broadly, the term majority rule is used in discussions regarding the principles of majority rule and the protection of individual and minority rights.

A common misconception of majority rule is that it can be soundly used to determine majority rights among a class of voters, or be used as a form of mob rule over a minority as an expression of majoritarianism. However, the class of voters and their equal rights must be decided beforehand as a separate act.^[1] The majority rule, utilized thereafter as a convincing method of democratic decision making, is then assumed to be universally binding among all voters as a function of equal rights. This logic prevents the use of voting as a majoritarian tyranny. Any decision that unfairly targets a minority's right could be said to be majoritarian, but would not be a logically sound example of a majority decision, which categorically assumes equal rights established by charter or constitution. Of course, all of the above assumes a constitutional democracy, which is not always the case with all democratic countries. It also assumes that all individuals vote and accept the charter. It also assumes that all those governed by the charter are voluntary members of the society.

[edit] Further Reading

• Black, D.: The Theory of Committees and Elections, Cambridge 1958
• Farquharson, R.: Theory of Voting, Oxford 1961

[edit] References

1. ^ A Przeworski, JM Maravall, I NetLibrary Democracy and the Rule of Law (2003) p.223

[edit] See also

• Majority criterion
• Voting paradox
• Voting system criterion
• Voting system
• NIBMARhu:Majority Rule

ja:多数決