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
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.
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
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.
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.
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.