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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”, below) 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.

Borderline case In first-past-the-post or “plurality” systems, borders matter. To ensure that each vote has roughly the same weight, each constituency should have roughly the same number of voters. Threading boundaries between and through centres of population on the pretext of ensuring fairness is also a great way to cheat for your own benefit – a practice known as gerrymandering, after a 19th-century governor of Massachusetts, Elbridge Gerry, who created an electoral division whose shape reminded a local newspaper editor of a salamander. Suppose a city controlled by the Liberal Republican (LR) party has a voting population of 900,000 divided into three constituencies. Polls show that at the next election LR is heading for defeat – 400,000 people intend to vote for it but the 500,000 others will opt for the Democratic Conservative (DC) party. If the boundaries were to keep the proportions the same, each constituency would contain roughly 130,000 LR voters and 170,000 DC voters, and DC would take all three seats – the usual inequity of a plurality voting system. In reality, voters inclined to vote for one party or the other will probably clump together in the same neighbourhoods of the city, so LR might well retain one seat. However, it could be all too easy for LR to redraw the boundaries to reverse the result and secure itself a majority – as these two dividing strategies show.

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.

Order, order

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”).

Editorial: Giving democracy a shot in the arm

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.

Proportional paradox Although elections to the US House of Representatives use a first-past-the-post voting system, the constitution requires that seats be “apportioned among the several states according to their respective numbers” – that is, divvied up proportionally. In 1880, the chief clerk of the US Census Bureau, Charles Seaton, discovered that Alabama would get eight seats in a 299-seat House, but only seven in a 300-seat House. This “Alabama paradox” was caused by an algorithm known as the largest remainder method, which was used to round the number of seats a state would receive under strict proportionality to a whole number. Suppose for simplicity’s sake that a nation of 39 million voters has a parliament with four seats – giving a quota of 9.75 million voters per seat. The seats must, however, be shared among three states, Alabaska, Bolorado and Carofornia, with voting populations of 21, 13 and 5 million, respectively. Dividing these numbers by the quota gives each state’s fair proportion of seats. Rounded down to an integer, this number of seats is given to the states. Any seats left over go to the state or states with the highest remainders. See the results The rounded-down integers allocate three seats. The fourth goes to Carofornia, the state with the largest remainder. Suppose now the number of seats increases from four to five. The quota is 39 million divided by 5, or 7.8 million, and so our table becomes this: The rounded-down integers account for three seats as before. The two spare go to Alabaska and Bolorado, which have the two largest remainders, and Carofornia loses its only seat. (The US Constitution stipulates that each state must have at least one representative, which would protect Carofornia in this case – the size of the House would have to be increased by one seat.) The precise conditions that lead to the Alabama paradox are mathematically complex. For three states they can be portrayed graphically, as in this diagram. The left-hand diagram shows the populations (as a fraction of the country’s total) and fair proportions of three states in the case of four seats; the right-hand side superimposes the diagram for five seats. The Alabama paradox occurs for the shaded population combinations: our example lies in the leftmost orange-shaded region. Such quirks mean that seats in proportional systems are now generally apportioned using algorithms known as divisor methods. These work by dividing voting populations by a common factor so that when the fair proportions are rounded to a whole number they add up to the number of available seats. But this method is not foolproof: it sometimes gives a constituency more seats than the whole number closest to its fair proportion.

Power in the balance One criticism of proportional voting systems is that they make it less likely that one party wins a majority of the seats available, thus increasing the power of smaller parties as “king-makers” who can swing the balance between rival parties as they see fit. The same can happen in a plurality system if the electoral arithmetic delivers a hung parliament, in which no party has an overall majority – as might happen in the UK after its election next week. Where does the power reside in such situations? One way to quantify that question is the Banzhaf power index. First, list all combinations of parties that could form a majority coalition, and in all of those coalitions count how many times a party is a “swing” partner that could destroy the majority if it dropped out. Dividing this number by the total number of swing partners in all possible majority coalitions gives a party’s power index. For example, imagine a parliament of six seats in which party A has three seats, party B has two and party C has one. There are three ways to make a coalition with a majority of at least four votes: AB, AC and ABC. In the first two instances, both partners are swing partners. In the third instance, only A is – if either B or C dropped out, the remaining coalition would still have a majority. Among the total of five swing partners in the three coalitions, A crops up three times and B and C once each. So A has a power index of 3 ÷ 5, or 0.6, or 60 per cent – more than the 50 per cent of the seats it holds – and B and C are each “worth” just 20 per cent. In a realistic situation, the calculations are more involved. This diagram shows how the power shifts dramatically when there is no majority in a hypothetical parliament of 650 seats in which five voting blocs are represented.