I posted this a couple of years back, and though it would be nice to repost since it is favorite of mine, and PR/IR seems to be on the front burner of late.
SCIENCE AND TECHNOLOGY
The mathematics of voting
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.
paper just published in the journal Economic
Theory, Donald Saari, a mathematician at
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.
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
Admittedly, this is more complex than plurality voting and cannot be
used with current American voting machines (though it is used in