1 The Mathematics of Voting
1.1 Preference Ballots and Preference
Schedules
1.2 The Plurality Method
1.3 The Borda Count Method
1.4 The Plurality-with-Elimination Method
(Instant Runoff Voting)
1.5 The Method of Pairwise Comparisons
1.6 Rankings
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Excursions in Modern Mathematics, 7e: 1.3 - 2
The Borda Count Method
• Each place on a ballot is assigned points
• With N candidates, 1 point for last place, 2
points for second from last, and so on
• First-place vote is worth N points
• Tally points for each candidate separately
• Candidate with highest total is winner
• Candidate is called the Borda winner
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Example 1.5 The Math Club Election
(Borda Method)
Let’s use the Borda count method to choose
the winner of the Math Appreciation Society
election first introduced in Example 1.1. Table
1-4 shows the point values under each
column based on first place worth 4 points,
second place worth 3 points, third place worth
2 points, and fourth place worth 1 point.
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Example 1.5 The Math Club Election
(Borda Method)
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Example 1.5 The Math Club Election
(Borda Method)
Tally the points:
A gets: 56 + 10 + 8 + 4 + 1 = 79 points
B gets: 42 + 30 + 16 + 16 + 2 = 106 points
C gets: 28 + 40 + 24 + 8 + 4 = 104 points
D gets: 14 + 20 + 32 + 12 + 3 = 81 points
The Borda winner of this election is Boris!
(Wasn’t Alisha the winner of this election
under the plurality method?)
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What’s wrong with the Borda Method?
In contrast to the plurality method, the Borda
count method takes into account all the
information provided in the voters’ preference
ballots, and the Borda winner is the candidate
with the best average ranking - the best
compromise candidate if you will. On its face, the
Borda count method seems like an excellent way
to take full consideration of the voter’s
preferences, so indeed, what’s wrong with it?
The next example illustrates some of the
problems with the Borda count method.
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Example 1.6 The School Principal
Election
The last principal at Washington Elementary
School has just retired and the School Board
must hire a new principal. The four finalists for
the job are Mrs. Amaro, Mr. Burr, Mr. Castro,
and Mrs. Dunbar (A, B, C, and D, respectively).
After interviewing the four finalists, each of the
11 school board members gets to rank the
candidates by means of a preference ballot, and
the Borda winner gets the job. Table 1-5 shows
the preference schedule for this election.
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Example 1.6 The School Principal
Election
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Example 1.6 The School Principal
Election
A simple count tells us that Mr. B is the Borda
winner with 32 points.
What about Mrs. A? Majority candidate, 6 of
11 first-place votes, and Condorcet candidate,
with 6 first-place votes beats each candidate
in head-to-head comparison
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What’s wrong with the Borda Method?
The Borda Method violates two basic criteria
of fairness:
• Majority criterion
• Condorcet criterion
Despite its flaws, experts in voting theory
consider the Borda count method one of the
best, if not the very best, method for deciding
elections with many candidates.
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In Defense of the Borda count method?
(1) Although violations of the majority
criterion can happen, they do not happen
very often, and when there are many
candidates such violations are rare; and
(2) violations of the Condorcet criterion
automatically follow violations of the
majority criterion, since a majority
candidate is automatically a Condorcet
candidate.
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Borda count method in Real Life
• individual sports awards (Heisman Trophy
winner, NBA Rookie of the Year, NFL MVP,
etc.)
• college football polls
• music industry awards
• hiring of school principals, university
presidents, and corporate executives
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