FIFTEEN mathematicians went out to buy drink for a party. They decided to buy a single beverage in bulk to save money, but they wanted to choose which one in as logical and fair a way as possible. So each listed the three drinks on offer (beer, wine and milk) in order of preference. Six preferred milk, followed by wine and then beer; five liked beer the most, followed by wine and then milk; and four were wine-lovers whose second choice was beer, followed by milk.

The question was how to decide the outcome from these preferences. One milk-lover proposed a plurality vote, in which each person casts a single vote for their first choice. This would give milk six votes, beer five, and wine four, ensuring that his own favourite would prevail. Not so fast, said a beer-drinker. Given that wine was the least popular first choice, why not stage a run-off between milk and beer? Since the four whose first choice was wine said that they preferred beer to milk, this would mean that beer would win, by nine votes to six.

Humbug, said a wine buff. She suggested a more elaborate approach: pairwise comparison. Taking all stated preferences into account, it was clear that, given a choice between wine and beer, a majority (ten of the 15) would choose wine; given the choice between wine and milk, a majority (nine of the 15) would also choose wine. Although it had the smallest number of first choices, in other words, wine had the broadest appeal.

This sorry tale has a serious point: that the outcome of an election is a reflection of voting procedure as much as voters' wishes. In 70% of three-candidate elections, changing the procedure changes the final ranking. So the results of real-world elections can seem paradoxical, or downright unfair.

In a paper just published in the journal *Economic Theory*, Donald Saari, a mathematician at Northwestern University in Evanston, Illinois, claims to have got to the root of the problem. It is, he says, all to do with symmetry (technically, with something called the wreath product of symmetry groups). Essentially, says Dr Saari, voting paradoxes arise when the system fails to respect natural cancellations of votes. In a two-candidate contest, for example, nobody would deny that the candidate with the most first-preference votes should win. One way to explain this is that votes of the form AB (ie, candidate A is preferred to candidate B) should cancel out votes of the form BA. If this leaves a surplus of AB, then A wins.

These cancellations are a form of reflectional symmetry. But votes in a three-candidate election should cancel out, too. Consider three votes in such a contest: ABC, BCA and CAB. Each candidate is placed first, second and third once, so it is clear that these three votes should cancel each other out. This is a form of rotational symmetry, since the three votes form a rotating cycle.

Taking these two symmetries into account, it is possible to characterise all paradoxes for a three-candidate election under any voting procedure. Dr Saari's results can also be generalised for elections with more than three candidates using more complicated, but closely related symmetries. It is thus possible to evaluate the “fairness” of different voting systems.

Plurality voting, one of the most common democratic systems, fails to respect reflectional symmetry. Since it is only each voter's first choice that counts, a voter with preference ABC fails to cancel out an equal-and-opposite voter with preference CBA; instead, the result is one vote for A, and one for C. As a result, paradoxical results are possible under plurality voting. Similarly, pairwise comparison does not respect rotational symmetry, so it can lead to paradoxes too.

The fairest voting system, says Dr Saari, would respect both symmetries. The only system that fits the bill is the Borda count, proposed by Jean-Charles de Borda in 1770 to elect members to the Academy of Sciences in Paris. In an election with X candidates, each voter awards X points to his first choice, X-1 to his second choice, and so on. The results are added up and the candidate with the most points wins.

Admittedly, this is more complex than plurality voting and cannot be used with current American voting machines (though it is used in Australia). Also, if voters are not familiar with all candidates, and do not rank them all, the unassigned points must be divided up evenly between the unranked candidates. But for small elections, the system is ideal. And our thirsty mathematicians? Having read Dr Saari's results, they should now be merrily quaffing wine.

This article appeared in the Science & technology section of the print edition under the headline "Democratic symmetry"