There are many situations that call for a group decision. At one extreme, three of us might be trying to decide where to go for dinner. At the other end of things, millions of people often need to decide which individual, or which political party, will lead their country. Very often we decide by voting. But what is the best—fairest, most representative—voting system? This is more complicated, and less obvious, than you might think.

When there are just two candidates for a post, it is all very simple: just vote for the person you prefer. But, as soon as there are more, confusion arises. Remember the 2000 Presidential election. Many people believe the presence of Ralph Nader on the 2000 ballot affected the results.

And this is a very straightforward case. Only one person was to be elected— what we will call a simple ballot. One can also have a complex election, where more than one candidate is to be elected. For example, if a club is to have a president,  a secretary and three committee members, there may be two simple elections—for president and secretary—plus a complex election for the other three.

Other sorts of election are possible; for example, the number to be elected might not be fixed. Some experts have proposed elections where points are allocated to the candidates and the high scorer wins. And sporting contests, where points are awarded not by electors (the judges) but based on the players’ performance, are very similar to elections of that kind.

In this chapter we shall look at some of the systems that have been suggested for simple ballots and are used in various places. And in the following chapter we shall cover complex ballots and discuss some of the complications that arise in voting.

Simple Voting

Suppose two people are running for an office. After each person makes his or her one vote, the person who gets more than half the votes wins. This is called the majority or absolute majority method. Ties are possible—ties are possible in any electoral system—but apart from this the absolute majority method always produces a result.

If there are three or more candidates, the majority method is not so good;  there may quite easily be no winner. Several schemes have been devised that allow a candidate with an absolute majority to be elected, and try to find a good approximation when there is no “absolute” winner. These are called majoritarian or plurality systems.

The first generalization is the plurality method, often called “first-past-the-post”  voting. (It is also called the simple majority method, although a majority is not always involved.) Each voter makes one vote, and the person who receives the most votes wins. For example, if there were three candidates, A, B and C, and 100 voters,  the absolute majority method requires 51 votes for a winner. If A received 40 votes and B and C each got 30, there would be no winner under absolute majority; under plurality, A would be elected.

The problem with the plurality method is that the winner might be very unpopular with a majority of voters. In our example, suppose all the supporters of B and C thought that both these candidates were better-qualified than A. Then the plurality method results in the election of the candidate that the majority thought was the worst possible choice. This problem is magnified if there are more candidates; even if there are only four or five candidates, people often think the plurality method elects the wrong person.

To overcome this difficulty in countries with only two major political parties, it is common for each party to endorse only one candidate. For example, in the United States, if there are two or more members of the Republican party who wish to run for some office, a preliminary election, called a primary election, is held, and party members vote on the proposed candidates; the one that receives the most votes is nominated by the party, and usually the others do not stand for election. This election would be called the Republican primary. There usually will also be a Democratic primary, and sometimes other parties run primaries. However, this method will not solve the problems if there are several major parties, or if the post for which the election is held is not a political one.