1 the mathematics of voting
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1 The Mathematics of Voting. 1.1Preference Ballots and Preference Schedules 1.2The Plurality Method 1.3 The Borda Count Method 1.4The Plurality-with-Elimination Method (Instant Runoff Voting) 1.5The Method of Pairwise Comparisons 1.6Rankings. The Borda Count Method. - PowerPoint PPT PresentationTRANSCRIPT
Excursions in Modern Mathematics, 7e: 1.3 - 2Copyright © 2010 Pearson Education, Inc.
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
Excursions in Modern Mathematics, 7e: 1.3 - 3Copyright © 2010 Pearson Education, Inc.
• 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
The Borda Count Method
Excursions in Modern Mathematics, 7e: 1.3 - 4Copyright © 2010 Pearson Education, Inc.
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.
Example 1.5 The Math Club Election (Borda Method)
Excursions in Modern Mathematics, 7e: 1.3 - 5Copyright © 2010 Pearson Education, Inc.
Example 1.5 The Math Club Election (Borda Method)
Excursions in Modern Mathematics, 7e: 1.3 - 6Copyright © 2010 Pearson Education, Inc.
Tally the points:
Example 1.5 The Math Club Election (Borda Method)
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?)
Excursions in Modern Mathematics, 7e: 1.3 - 7Copyright © 2010 Pearson Education, Inc.
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.
What’s wrong with the Borda Method?
Excursions in Modern Mathematics, 7e: 1.3 - 8Copyright © 2010 Pearson Education, Inc.
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.
Example 1.6 The School Principal Election
Excursions in Modern Mathematics, 7e: 1.3 - 9Copyright © 2010 Pearson Education, Inc.
Example 1.6 The School Principal Election
Excursions in Modern Mathematics, 7e: 1.3 - 10Copyright © 2010 Pearson Education, Inc.
A simple count tells us that Mr. B is the Borda winner with 32 points.
Example 1.6 The School Principal Election
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
Excursions in Modern Mathematics, 7e: 1.3 - 11Copyright © 2010 Pearson Education, Inc.
The Borda Method violates two basic criteria of fairness:
What’s wrong with the Borda Method?
• 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.
Excursions in Modern Mathematics, 7e: 1.3 - 12Copyright © 2010 Pearson Education, Inc.
(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.
In Defense of the Borda count method?
Excursions in Modern Mathematics, 7e: 1.3 - 13Copyright © 2010 Pearson Education, Inc.
• 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
Borda count method in Real Life