In an ideal world, elections should be two things: free and fair. Every adult, with a few sensible exceptions, should be able to vote for a candidate of their choice, and each single vote should be worth the same.
Ensuring a free vote is a matter for the law. Making elections fair is more a matter for mathematicians. They have been studying voting systems for hundreds of years, looking for sources of bias that distort the value of individual votes, and ways to avoid them. Along the way, they have turned up many paradoxes and surprises. What they have not done is come up with the answer. With good reason: it probably doesn’t exist.
The many democratic electoral systems in use around the world attempt to strike a balance between mathematical fairness and political considerations such as accountability and the need for strong, stable government. Take first-past-the-post or “plurality” voting, which used for national elections in the US, Canada, India – and the UK, which goes to the polls next week. Its principle is simple: each electoral division elects one representative, the candidate who gained the most votes.
This system scores well on stability and accountability, but in terms of mathematical fairness it is a dud. Votes for anyone other than the winning candidate are disregarded. If more than two parties with substantial support contest a constituency, as is typical in Canada, India and the UK, a candidate does not have to get anything like 50 per cent of the votes to win, so a majority of votes are “lost”.
Dividing a nation or city into bite-sized chunks for an election is itself a fraught business (see “Borderline case”) that invites other distortions, too. A party can win outright by being only marginally ahead of its competitors in most electoral divisions. In the UK general election in 2005, the ruling Labour party won 55 per cent of the seats on just 35 per cent of the total votes. If a candidate or party is slightly ahead in a bare majority of electoral divisions but a long way behind in others, they can win even if a competitor gets more votes overall – as happened most notoriously in recent history in the US presidential election of 2000, when George W. Bush narrowly defeated Al Gore.
The anomalies of a plurality voting system can be more subtle, though, as mathematician Donald Saari at the University of California, Irvine, showed. Suppose 15 people are asked to rank their liking for milk (M), beer (B), or wine (W). Six rank them M-W-B, five B-W-M, and four W-B-M. In a plurality system where only first preferences count, the outcome is simple: milk wins with 40 per cent of the vote, followed by beer, with wine trailing in last.
So do voters actually prefer milk? Not a bit of it. Nine voters prefer beer to milk, and nine prefer wine to milk – clear majorities in both cases. Meanwhile, 10 people prefer wine to beer. By pairing off all these preferences, we see the truly preferred order to be W-B-M – the exact reverse of what the voting system produced. In fact Saari showed that given a set of voter preferences you can design a system that produces any result you desire.
In the example above, simple plurality voting produced an anomalous outcome because the alcohol drinkers stuck together: wine and beer drinkers both nominated the other as their second preference and gave milk a big thumbs-down. Similar things happen in politics when two parties appeal to the same kind of voters, splitting their votes between them and allowing a third party unpopular with the majority to win the election.
Can we avoid that kind of unfairness while keeping the advantages of a first-past-the-post system? Only to an extent. One possibility is a second “run-off” election between the two top-ranked candidates, as happens in France and in many presidential elections elsewhere. But there is no guarantee that the two candidates with the widest potential support even make the run-off. In the 2002 French presidential election, for example, so many left-wing candidates stood in the first round that all of them were eliminated, leaving two right-wing candidates, Jacques Chirac and Jean-Marie Le Pen, to contest the run-off.
Another strategy allows voters to place candidates in order of preference, with a 1, 2, 3 and so on. After the first-preference votes have been counted, the candidate with the lowest score is eliminated and the votes reapportioned to the next-choice candidates on those ballot papers. This process goes on until one candidate has the support of over 50 per cent of the voters. This system, called the instant run-off or alternative or preferential vote, is used in elections to the Australian House of Representatives, as well as in several US cities. It has also been suggested for the UK.
Preferential voting comes closer to being fair than plurality voting, but it does not eliminate ordering paradoxes. The Marquis de Condorcet, a French mathematician, noted this as early as 1785. Suppose we have three candidates, A, B and C, and three voters who rank them A-B-C, B-C-A and C-A-B. Voters prefer A to B by 2 to 1. But B is preferred to C and C preferred to A by the same margin of 2 to 1. To quote the Dodo in Alice in Wonderland: “Everybody has won and all must have prizes.”
One type of voting system avoids such circular paradoxes entirely: proportional representation. Here a party is awarded a number of parliamentary seats in direct proportion to the number of people who voted for it. Such a system is undoubtedly fairer in a mathematical sense than either plurality or preferential voting, but it has political drawbacks. It implies large, multi-representative constituencies; the best shot at truly proportional representation comes with just one constituency, the system used in Israel. But large constituencies weaken the link between voters and their representatives. Candidates are often chosen from a centrally determined list, so voters have little or no control over who represents them. What’s more, proportional systems tend to produce coalitions of two or more parties, potentially leading to unstable and ineffectual government – although plurality systems are not immune to such problems, either (see “Power in the balance”).
Proportional representation has its own mathematical wrinkles. There is no way, for example, to allocate a whole number of seats in exact proportion to a larger population. This can lead to an odd situation in which increasing the total number of seats available reduces the representation of an individual constituency, even if its population stays the same (see “Proportional paradox”).
Such imperfections led the American economist Kenneth Arrow to list in 1963 the general attributes of an idealised fair voting system. He suggested that voters should be able to express a complete set of their preferences; no single voter should be allowed to dictate the outcome of the election; if every voter prefers one candidate to another, the final ranking should reflect that; and if a voter prefers one candidate to a second, introducing a third candidate should not reverse that preference.
All very sensible. There’s just one problem: Arrow and others went on to prove that no conceivable voting system could satisfy all four conditions. In particular, there will always be the possibility that one voter, simply by changing their vote, can change the overall preference of the whole electorate.
So we are left to make the best of a bad job. Some less fair systems produce governments with enough power to actually do things, though most voters may disapprove; some fairer systems spread power so thinly that any attempt at government descends into partisan infighting. Crunching the numbers can help, but deciding which is the lesser of the two evils is ultimately a matter not for mathematics, but for human judgement.