Lecture on Mathematics of Voting and Apportionment

[Pages:81]Math Circle

Lecture on Mathematics of Voting

and Apportionment

Ernesto Diaz

Assistant Professor of Mathematics Department of Natural Sciences and

Mathematics

Slide 1.1-1

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.

Copyright ? 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 1.1 - 2

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?

Copyright ? 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 1.1 - 3

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.

Copyright ? 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 1.1 - 4

1 The Mathematics of Voting

1.1 Preference Ballots and Preference Schedules

1.2 The Plurality Method 1.3 The Borda Count Method

Copyright ? 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 1.1 - 5

Example 1.1 The Math Club Election

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?

Copyright ? 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 1.1 - 6

Example 1.1 The Math Club Election

Copyright ? 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 1.1 - 7

1 The Mathematics of Voting

1.1 Preference Ballots and Preference Schedules

1.2 The Plurality Method 1.3 The Borda Count Method

Copyright ? 2010 Pearson Education, Inc.

Excursions in Modern Mathematics, 7e: 1.2 - 8

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