1 the mathematics of voting

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


• 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


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)


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?)


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?


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


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


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.


(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?


• 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

1 the mathematics of voting

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