This isn't an article about Brexit in any way. But, whatever Brexit has done for good or bad, it has certainly raised all of our awareness of the parliamentary process. We live in a democracy and the basis for our parliamentary system is the voting process.

The purpose of voting is to allow the general public to express their views, either directly as in a referendum, or through electing representatives, as in a general election. What could be simpler? However, the process of translating the "will of the people" into a fair representation of that will in parliament, is very far from perfect. And, as we shall see, involves a substantial amount of mathematics.

Why no voting system can ever be perfect

Perhaps the key contribution to our understanding of voting was made in the 1950s by the economist Kenneth Arrow, who won the 1972 Nobel Prize in Economics for his work in this field. Arrow's seminal contribution was to show that there is no perfect voting system.

Arrow did this by considering what might be reasonable expectations of any voting system, and then showing that there was no one system which could satisfy all of them. He assumed that there was an election in which there were multiple candidates, and that each voter could rank those candidates in order of preference. For example, if there were candidates A, B, C and D, then one voter might rank them in the order A, B, C, D, while another might rank them as B, D, C, A. These preferences would then be expressed anonymously at a ballot box in an election. The purpose of the voting system is to elect a single candidate as the output of the voting process. The question is, can this be done fairly?

Arrow's original conditions that he thought any fair voting system should fullfill were as follows:

Kenneth Arrow showed that no voting system with three or more candidates can satisfy all the four conditions he set out. Image: Linda A. Cicero / Stanford News Service, CC BY 3.0

1. Dictator: The system should reflect the wishes of more than one voter, so there is no dictator.
2. Unanimity: If all voters prefer candidate A to candidate B then A should come out ahead of B in the final result.
3. Universality: The voting system should always return one clear result.
4. Independence of irrelevant alternatives: In the final result, the ranking of A above B should only depend on how individual votes ranked A compared to B and not how they ranked them when compared to a third alternative C.

Other conditions can also be added, indeed many have been since Arrow's work.

One is the majority condition, which stipulates that if a candidate is top choice for a majority of the voters then this candidate should get elected.

Finally there is the anonymity condition, which says that it should not be possible to tell from the vote who voted for which candidate.

All of these conditions seem pretty reasonable. But here is the bad news, which is Arrow's key conclusion and for which he won the Nobel Prize: it is mathematically impossible for any voting system involving three or more candidates to satisfy all his four conditions above.

Oh dear! Does this imply the end of democracy? Well, not quite. What it shows is that the conditions laid down above are really very demanding, and that in a realistic voting method we have to make certain compromises, balancing different views of what it means to be fair. These inevitably lead to anomalies in the voting process, and we have to work these out and then be aware of them. Therein lies the science (and mathematics) of voting.

An example

To illustrate Arrow's result, let's see how well a simple voting system fairs against his conditions.

Even when there are only three candidates, the Borda count fails some of the conditions mentioned above.

A system that is often used when a committee ranks candidates for a job is the Borda system. The system is named after the 18th-century French mathematician Jean-Charles de Borda who devised it in 1770, and it has been (re)discovered many times both before and since.

In the simple Borda system each voter ranks the candidates in order of preference. If there are N candidates, then a candidate gets N-1 points for every time he or she is ranked first, N-2 votes for every time he or she is ranked second, and so on, and 0 points for every time they are ranked last.

The points are then added up and the candidate with the largest total score wins. Variants of the Borda system use different numbers for the rankings such as 1, 1/2, 1/3 etc. In Strictly come dancing the judges use a version of the Borda method where they give a mark between 1 and 10 for each dance pair.

Let's see how the Borda system works with three candidates A, B, C and three voters V₁, V₂, and V₃, who assign scores as follows:

Adding these up we see that

A gets a total score of 5
B gets a total score of 1
C gets a total score of 3.

So in this system A is the clear winner. We also note that A is the first choice for two out of the three voters.

How well does the Borda system measure up against Arrow's conditions?

Dictator is clearly satisfied. The cap on the maximum score that any voter can give to a candidate prevents the emergence of a dictator. In fact the Borda system does very well in this case. It is a consensus system in that it tends to elect candidates who are broadly supported by the electorate.

Unanimity is also clearly satisfied. If all voters prefer A to B then A will always get a higher ranking than B. It follows the sum of the scores of A will be higher than the sum of the scores of B.

Universality however fails. Suppose the outcome of our election above had been:

It's hard to translate the choice of millions of individuals into a fair election result.

In this case A and B tie for first place with 4 votes each (even though A is the first choice for two out of the three voters). In this case the election is repeated, in the hope that in the second run a winner emerges. Of course this is easily done in a committee (which is one reason that this system is used in this case) but is much harder to do in a real election. (Another problem with a real election is that it requires votes to be added up. Again this is easy for a small election such as Strictly come dancing, but much harder if you are considering tens of thousands of voters. )

The majority condition and independence of irrelevant alternatives also fail. This is more subtle, and is a weakness of the Borda system which can be exploited in tactical voting and has meant that it is not widely used. As an example, consider an election with five voters and four candidates. The following table shows the scores the candidates get from each voter, based on the voters' rankings:

Here B wins with a score of 10, with A and C tying in second place. Note that A does not win despite being the favourite of a majority of three voters, and B wins it even though he or she is favourite of none. B could be described as a safe candidate, who is never going to be exceptional: a safe pair of hands, but no superstar.

Now imagine that, after the voters have given their scores, candidate D is exposed of corruption and drops out of the race. Assuming that the relative rankings of the other candidates by each voter remain the same (e.g. if a voter preferred candidate A over candidate B before D was eliminated, then they still prefer A to B after D got eliminated), the scores corresponding to the rankings are now:

Now candidate A is the winner, so independence of irrelevant alternatives fails. The outcome of the election hinges on whether the largely unpopular candidate D runs or not. One could imagine that the two voters who put A last, V₄ and V₅, did so not because they really thought A worse than the outsider D, but because they wanted to make sure A didn't win over B or C: that's tactical voting.

Given these problems we can see that it is hard to find a voting method which even comes close to satisfying reasonable assumptions. In the next article we will look at three voting methods that are widely used in real elections and see how they fare in terms of fairness.

About this article

This article is based on a talk in Budd's ongoing Gresham College lecture series (see video above). You can see other articles based on the talk here.

Chris Budd.

Chris Budd OBE is Professor of Applied Mathematics at the University of Bath, Vice President of the Institute of Mathematics and its Applications, Chair of Mathematics for the Royal Institution and an honorary fellow of the British Science Association. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.

He has co-written the popular mathematics book Mathematics Galore!, published by Oxford University Press, with C. Sangwin, and features in the book 50 Visions of Mathematics ed. Sam Parc.