If there are only two candidates in a race, electing the candidate with the most
votes is the obvious and fair way to select the winner. This is the notion of
majority rules. But if there are more than two candidates, no candidate may
have a majority, and there are several inconsistent ways to select a winner. We
shall illustrate this with an election involving four candidates Rasputin,
Sheherezade, Tamburlaine, and Ursula and 14 voters. The order in which each
voter prefers the candidates is given below, although ballots are only cast for
The easiest method is to elect the candidate with the most votes (as indicated
by the top row). In this case, Rasputin wins with five votes. However, five
out of 14 is not a majority it is what is called a plurality (the dictionary
defines plurality as a majority over each other candidate, but not over all
together; this definition is somewhat ambiguous as we shall discuss below under
the Condorcet winner).
If one insists on getting a majority winner, it may be necessary to restrict
the election to two candidates. An easy way to do this is to have a runoff
election between the top two vote getters, in this case Rasputin and
Sheherezade. In accordance with the given preference schedules, with only
Rasputin and Sheherezade in the race, those who voted for Tamburlaine or Ursula
would vote for Sheherezade and she would win with nine votes to five votes for
It may be possible to get a majority without restricting the election to two
candidates. This can be attemped by removing one candidate at a time until
someone has a majority. This is called the Hare method. In this case, Ursala
with only two votes would be eliminated first; shifting her votes to Tamburlaine
would not create a majority, but Sheherezade with only four votes would be the
bottom candidate at that point and be eliminated. This would leave only two
candidates, and with sheherezade's votes transferred to Tamburlaine, he would
have nine votes, which is a majority and Tamburlaine would be the winner.
Although this method assumes voting for only one candidate at a time and running
sequential elections, in practice preferential ballots are often collected so
that the tellers can determine the winner as we have here.
The Borda count is simiar to scoring a track meet. The entire voting preference
schedules must be submitted, and each candidate receives as many points as
there are candidates for each first place vote, one fewer points for each
second place vote, ... , and one point for each last place vote. In the above
example, each candidate receives 4 points for each first place vote, three
points for each second place vote, two points for each third place vote, and
one point for each fourth place vote. This yields Rasputin 29 points,
Sheherezade 31 points, Tamburlaine 36 points, and Ursula 44 points. Ursula
wins by the Borda count.
A Condorcet winner is a candidate who would win a two candidate election against
every other candidate (a majority over each other candidate in a sense other
thanthat used above). It is readily verified that with the above preference
schedules, Ursula would win a two candidate race against each other candidate,
hence is the Condorcet winner. Although this may seem like the best criterion
to use for determining an election, there is not always a Condorcet winner.
This example has been constructed to illustrate that the different procedures
for determining the winner of an election can indeed result in different
winners. Of course, if any candidate had a majority of the votes on the first
ballot, he would win under any of these procedures.
Exercise: Consider the election between Athelstan, Beowulf, Charles, Damon, end
Elsie with the preferences of 12 voters as indicated below.
Who is the plurality winner?
Who is the Run-off winner?
Who is the Hare winner?
Who is the Condorcet winner?
Sometimes a voter can obtain a more favorable outcome by not voting in
accordance with his preference schedule. Such voting is called strategic voting.
This is the concern when one wonders whether a vote for a third party candidate
is wasted in a national election. The four candidate 14 voter preference
schedule above illustrates this phenomenon.
If the Plurality criterion was used to determine the winner, those who voted for
Sheherezade would be better off voting for Tamurlaine, because that would make
Tamurlaine the winner, and they prefer Tamurlaine to Rasputin (of course, if
those who favor Tamurlaine voted for Sheherezade, that would cancel the shift in
votes and Rasputin would still win). In the case of the runoff procedure,
if those who voted for Tamurlaine voted for Ursula, she would be the
ultimate winner, and the prefer Ursula to Sheherezade. The same result
would ensue under the Hare method. No improvement over the Condorcet winner
by strategic voting is possible in this example.
CompetencyCan strategic voting produce a better outcome for any of the
voters under any of the voting procedures with the voting preferences of the
exercise above? (Thisis the apportionment for the methodofgreatest divisors.
Reflection: WHich of the voting procedures is most fair?
Challenge: Can strategic voting ever alter a Condorcet winner?