Excursions in Modern Mathematics, 7e: 1.1 - 2Copyright © 2010 Pearson Education, Inc.

1 The Mathematics of Voting

The Paradoxes of Democracy

• Vote! In a democracy, the rights and duties

of citizenship are captured in that simple

one-word mantra.

• We vote in presidential elections,

gubernatorial elections, local elections,

school bonds, stadium bonds, American Idol

selections, and initiatives large and small.

Excursions in Modern Mathematics, 7e: 1.1 - 3Copyright © 2010 Pearson Education, Inc.

1 The Mathematics of Voting

The Paradoxes of Democracy

• The paradox is that the more opportunities

we have to vote, the less we seem to

appreciate and understand the meaning of

voting.

• Why should we vote?

• Does our vote really count?

• How does it count?

Excursions in Modern Mathematics, 7e: 1.1 - 4Copyright © 2010 Pearson Education, Inc.

1 The Mathematics of Voting

Voting Theory

• First half is voting;

• Second half is counting.

• Arrow’s impossibility theorem:

A method for determining election results that is democratic and always fair is a mathematical impossibility.

Excursions in Modern Mathematics, 7e: 1.1 - 5Copyright © 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: 1.1 - 6Copyright © 2010 Pearson Education, Inc.

The Math Appreciation Society (MAS) is a student

organization dedicated to an unsung but worthy cause, that

of fostering the enjoyment and appreciation of mathematics

among college students. The Tasmania State University

chapter of MAS is holding its annual election for president.

There are four candidates running for president: Alisha,

Boris, Carmen, and Dave (A, B, C, and D for short). Each of

the 37 members of the club votes by means of a ballot

indicating his or her first, second, third, and fourth choice.

The 37 ballots submitted are shown on the next slide. Once

the ballots are in, it’s decision time. Who should be the

winner of the election? Why?

Example 1.1 The Math Club Election

Excursions in Modern Mathematics, 7e: 1.1 - 7Copyright © 2010 Pearson Education, Inc.

Example 1.1 The Math Club Election

Excursions in Modern Mathematics, 7e: 1.1 - 8Copyright © 2010 Pearson Education, Inc.

Essential ingredients of every election:

Example 1.1 The Math Club Election

• Voters

• Candidates (electing people); Choice

(nonhuman alternatives-cities, colleges,

pizza toppings, etc.)

• Ballots:

Preference Ballot: rank in order of

preference

Linear Ballot: ties are not allowed

Excursions in Modern Mathematics, 7e: 1.1 - 9Copyright © 2010 Pearson Education, Inc.

Preference Schedule

Example 1.1 The Math Club Election

• Only a few different ways to rank results:

organize in a preference schedule

Excursions in Modern Mathematics, 7e: 1.1 - 10Copyright © 2010 Pearson Education, Inc.

Transitivity and Elimination of Candidates

Example 1.1 The Math Club Election

• Transitive: Voter prefers A over B and B

over C then automatically prefers A over C

• Elimination: Relative preferences are not

affected by the elimination of one or more

candidates

Excursions in Modern Mathematics, 7e: 1.1 - 11Copyright © 2010 Pearson Education, Inc.

In the next section we return to the

business of deciding the outcome of

elections in general and the Math Club

election in particular.

Example 1.1 The Math Club Election