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Mathematics The study of symbols, shapes, algorithms, sets, and patterns, using logical reasoning and quantitative calculation. Quantitative Reasoning:

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Presentation on theme: "Mathematics The study of symbols, shapes, algorithms, sets, and patterns, using logical reasoning and quantitative calculation. Quantitative Reasoning:"— Presentation transcript:

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

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

7 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. 1 st 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. CRITERIA FOR A FAIR ELECTION

8 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. 2 nd criteria for a fair election POLLPreference Schedule of 100 votes for favorite restaurant Votes st placeMcD.BKSubw. 2 nd placeSubw. BK. 3 rd placeBKMcD.McD

9 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. 3 rd criteria for a fair election: MOCK POLLPreference Schedule for the 2004 Presidential Election on October 1 Percentage of voters 48%47%3% 1 st choiceBushKerryNadar 2 nd choiceKerryBush 3 rd choiceNadar Kerry

10 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. 4 th criteria for a fair election: MOCK POLLPreference Schedule for actual election in November 2004 Percentage of voters 48%47%3% 1 st choiceBushKerryNadar 2 nd choiceKerryBush 3 rd choiceNadar Kerry

11 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: 1.Plurality Method 2.Borda Count Method 3.Plurality-with-Elimination Method 4.Method of Pairwise Comparison

12 Example 1.2: The Math Club Election (Page 6)

13 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.

14 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.

15 What’s wrong with the plurality method? 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. 2 nd criteria for a fair election

16 Which methods satisfy which criterion? YN

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


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