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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. Ken SCIENCE AND TECHNOLOGY
The mathematics of voting Democratic symmetry 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 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 Admittedly, this is more complex than plurality voting and cannot be
used with current American voting machines (though it is used in |