Simple majority voting
From Academic Kids

Simple majority voting is a straightforward form of voting whereby the option with a simple majority of votes wins. It seems intuitively the most obvious example of democratic procedures. It is informally used in small groups to make all kinds of practical decisions, by counting hands in a group or judging the loudness of the cheers in a crowd. However, Kenneth May states that any social choice rule that satisfies certain conditions, will turn out to be the simple majority rule. We will first define simple majority voting and its characteristics, secondly we will state May’s theorem and its proof and finally, look at the implications of the theorem.
Assumptions
Before starting on the formal definition of simple majority voting, it is important to point out that certain matters should be considered if we look at the practical use of democratic procedures.
For example, when a country holds a referendum to decide on whether or not to join the EU, the following matters are dealt with. First, there is a selection of which people will be allowed to vote. In most cases, this selection is not a question for debate but an already decided matter. In western Europe, for example, generally everybody older than eighteen is allowed to vote. In other countries this age limit may differ or some groups may be excluded.
Secondly, the country will do all the necessary campaigning to make sure that the voters know what they are voting about. Even though the campaign is often rooted in selfinterest, since both factions will try to convince as many people as possible, the result is that most people will be wellinformed. This informed choice is a conditio sine qua non of democracy: if people are misinformed, they are manipulated, their choices limited and their wills restricted. For example: after being informed that her unborn child will have the Down syndrome, a women has an abortion. However, suppose this test is not entirely accurate and further examination shows that the fetus would have been normal. Here the women should have been informed about the accuracy of the test and the probabilities involved before making her decision. The concept of informed choice is particularly important in bioethics and medical ethics, but as shown is also relevant in political theory.
Thirdly, for democracy to work in any way (even though a perfect way may not exist), it is necessary for all voters to vote honestly. This is not only a problem of manipulability by the voter, but also of manipulability of the voter by an oppressor. In many countries who have a history of oppression, observers are present during the elections, to see to what extent it was fair and free. Depending on their observations, other countries decide whether or not to recognize the new government as a democratic one and adjust their trade relationships accordingly. Observers look for signs like armed men guarding the voting houses, cameras observing the voters and the transparency of the counting.
May's Theorem
Manipulability by voters is as such unobservable, but doesn’t constitute a problem with simple majority voting, since in a two option case, it is impossible to manipulate the result by voting strategically. May states that, since group choice must depend only upon individual preferences concerning the alternatives in a set, a pattern of group choice may be built up if we know the group preference for each pair of alternatives. However, manipulability in a more options case is not as simple as it sounds.
Definition Simple majority voting is an example of a social choice rule: a mapping that associates a list of individual preferences with a resulting outcome. Formally speaking, simple majority voting assigns +1 if only if N+1(d1, d2, .., dn) › [N+1(d1, d2, .., dn) + N1(d1, d2, .., dn)]/2 Or in more normal words: the winning choice is the one whose number of votes is greater than half of the numbers of individuals who aren’t indifferent between the two choices. This is in contrast with absolute majority voting, where the winner is the option who gets more than half of the votes. Or formally speaking, absolute majority voting assigns +1 if and only if N+1(d1, d2, .., dn) › n/2
The difference is clarified by means of the following example. Suppose D = (+1, +1, +1, 0, 0, 1, 1), the distribution of votes. Applying simple majority voting yields +1 as the winning option, applying absolute majority voting gives 0, indifference between the options.
Properties
What are the properties of simple majority voting? We are looking for a set of sufficient and necessary conditions.
To start with, simple majority voting satisfies the property of universal domain: it assigns an unambiguous value to every logically possible list of individual preferences. In other words, whatever every individual chooses, the procedure always yields a result. This trait is sometimes referred to as decisiveness.
Secondly, simple majority voting satisfies anonymity: it assigns the same value to two lists that are permutations of one another. This means that when two people change their votes in such a way that the number of voters for each option remains the same, than the result remains the same. The procedure does not care about which voter votes for an option, only about how many voters vote for that option. If the result changed, that would mean that one of those two votes overrides all the other votes, which comes down to a dictatorship.
Simple majority voting also satisfies neutrality: if everyone reverses their vote, the result is reversed, a tie remains a tie. Formally speaking: f(d1, d2, .., dn) = f(d1, d2, .., dn) This shows that the method of decision does not favor either alternative.
As fourth and last property, simple majority voting satisfies positive responsiveness. Formally speaking: if for all i, whenever (d1, d2, .., dn) and (d1’, d2’, .., dn’) are ivariants with di’ > di, then f (d1, d2, .., dn) ≥ 0 implies f (d1’, d2’, .., dn’) = +1. In words, if one of the two options has more or equally many votes as the other option and one voter changes his mind in favor of that first option, then the result will also change in favor of that option: a tie becomes a win and a win stays a win. This method of comparing lists and deducing results can be successively applied, enabling to compare two lists that are not ivariants, as they are connected by a series of ivariants.
See also: Qualified Majority Voting, Voting, May’s theorem