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.

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

Example 1.1: The Math Club Election (Page 4)

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

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

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.

1st 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

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.

2nd criteria for a fair election

POLL Preference Schedule of 100 votes for favorite restaurant

Votes 40 35 25

1st place McD. BK Subw.

2nd place Subw. Subw. BK.

3rd place BK McD. McD

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.

3rd criteria for a fair election:

MOCK POLL Preference Schedule for the 2004 Presidential Election on October 1

Percentage of voters

48% 47% 3%

1st choice Bush Kerry Nadar

2nd choice Kerry Bush Bush

3rd choice Nadar Nadar Kerry

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.

4th criteria for a fair election:

MOCK POLL Preference Schedule for actual election in November 2004

Percentage of voters

48% 47% 3%

1st choice Bush Kerry Nadar

2nd choice Kerry Bush Bush

3rd choice Nadar Nadar Kerry

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 Method2. Borda Count Method3. Plurality-with-Elimination Method4. Method of Pairwise Comparison

Example 1.2: The Math Club Election (Page 6)

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.

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.

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.

2nd criteria for a fair election

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PluralityBorda CountPlurality w eliminationPairwise comparison

Which methods satisfy which criterion?

Y N

Homework

• Read pages 1 – 11

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

18a