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Mathematics

Mathematics. The study of symbols, shapes, algorithms, sets, and patterns, using logical reasoning and quantitative calculation. Quantitative Reasoning : Interpreting, understanding, making judgments, and applying mathematical concepts to analyze and solve problems from various backgrounds.

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Mathematics

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  1. Mathematics • The study of symbols, shapes, algorithms, sets, and patterns, using logical reasoning and quantitative calculation. • Quantitative Reasoning: Interpreting, understanding, making judgments, and applying mathematical concepts to analyze and solve problems from various backgrounds.

  2. Preference Ballot: Ballots in which a voter is asked to rank all candidates in order of preference

  3. Example 1.1: The Math Club Election (Page 4)

  4. preference schedule (page 5) - When we organize preference ballots by grouping together like ballots we have a preference schedule.

  5. Arrow’s Impossibility Theorem (page 3) - A method for determining election results that is democratic and always fair is a mathematical impossibility.

  6. CRITERIA FOR A FAIR ELECTION majority rule - in a democratic election between two candidates, the one with the majority (more than half) of the votes wins. 1st criteria for a fair election: The Majority Criterion (page 6) - If a choice receives a majority of the first place votes in an election, then that choice should be the winner of the election.

  7. 2nd criteria for a fair election The Condorcet Criterion (page 8) - If there is a choice that in a head-to-head comparison is preferred by the voters over every other choice, then that choice should be the winner of the election. A candidate that wins every head-to-head comparison with the other candidates is called a Condorcet candidate.

  8. 3rd criteria for a fair election: The Monotonicity Criterion (page 15). If choice X is a winner of an election and, in a reelection, the only changes in the ballots are changes that only favor X, then X should remain a winner of the election.

  9. 4th criteria for a fair election: The Independence-of-Irrelevant-Alternatives Criterion (page 18). If choice X is a winner of an election and one (or more) of the other choices is removed and the ballots recounted, then X should still be a winner of the election.

  10. Arrow’s Impossibility Theorem (page 3) - A method for determining election results that is democratic and always fair is a mathematical impossibility. • Methods used to find the winner of an election: • Plurality Method • Borda Count Method • Plurality-with-Elimination Method • Method of Pairwise Comparison

  11. Example 1.2: The Math Club Election (Page 6)

  12. I. THE PLURALITY METHOD plurality method (page 6) - the candidate (or candidates) with the most first place votes wins. A plurality does not imply a majority but a majority does imply a plurality.

  13. Example 1.3. The Band Election (page 7) What’s wrong with the plurality method? If we compare the Hula Bowl to any other bowl on a head-to-head basis, the Hula Bowl is always the preferred choice.

  14. What’s wrong with the plurality method? 2nd criteria for a fair election The Condorcet Criterion (page 8) - If there is a choice that in a head-to-head comparison is preferred by the voters over every other choice, then that choice should be the winner of the election.

  15. Which methods satisfy which criterion? Y N

  16. Homework • Read pages 1 – 11 • Page 30: 1, 2, 3, 6, 11, 12, 13, 14, 18a

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