Election procedures

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 a single candidate.



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 Rasputin.

Hare method

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.

Borda Count

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.

Condorcet winner

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?

Strategic voting

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?