Mathematics
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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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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.


Mathematics

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


Mathematics

Example 1.1: The Math Club Election (Page 4)


Mathematics

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


Mathematics

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


Mathematics

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.


Mathematics

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.


Mathematics

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.


Mathematics

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.


Mathematics

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


Mathematics

Example 1.2: The Math Club Election (Page 6)


Mathematics

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.


Mathematics

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.


Mathematics

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.


Mathematics

Which methods satisfy which criterion?

Y

N


Homework

Homework

  • Read pages 1 – 11

  • Page 30: 1, 2, 3, 6, 11, 12, 13, 14,

    18a


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