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Chapter 1

Chapter 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.

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Chapter 1

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  1. Chapter 1 The Mathematics of Voting

  2. The Paradoxes of Democracy • Vote! In a democracy, the rights and duties of citizenship are captured in that simple one-word mantra. • The paradox is that the more opportunities we have to vote, the less we seem to appreciate and understand the meaning of voting.

  3. The Paradoxes of Democracy • Why should we vote? • Does our vote really count? • How does it count?

  4. The Paradoxes of Democracy • Arrow’s impossibility theorem: A method for determining election results that is democratic and always fair is a mathematical impossibility.

  5. Preference Ballots and Preference Schedules Notes 1 – Section 1.1

  6. Essential Learnings • Students will understand and be able to construct and interpret a preference schedule for an election involving preference ballots.

  7. Essential Ingredients of every 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

  8. 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?

  9. Preference Schedule • A way to organize ballot information From now on, all election examples will be given as a preference schedule.

  10. Transitivity and Elimination of Candidates • 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

  11. Practice Example 1 • Use the ballots below to create a preference schedule.

  12. Practice Example 1 cont. Preference Schedule: Total voters = 5 + 6 + 7 + 9 + 3 = 30 voters

  13. Example 1.1: The Math Club Election • We will use this example multiple times using different counting methods to determine the winner of the election.

  14. Assignments Signed Syllabus Slip p. 29: 1, 2, 4, 8, 9 (completed on engineering paper) Bring Covered Textbook to Class Every Day!

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