MA 110: Finite MathDr. Maria Byrne

Instructional Laboratory 0345

Lecture 10/31/2008

1. Plurality Method

2. Pairwise Comparison Method

We’ll Study:Two Voting Systems

Plurality Method

• Only considers voter’s top choice.

• The candidate with the most votes wins.

• A tie can be handled in any way the voting system agrees upon.

• If there is an absolute majority vote for one candidate, that candidate wins.

Plurality Method

• Only considers voter’s top choice.

• The candidate with the most votes wins.

• A tie can be handled in any way the voting system agrees upon.

• If there is an absolute majority vote for one candidate, that candidate wins.

Simple and easy voting method.

Pairwise Comparison MethodAlso known as condorcet method.

• Considers all ranked choices of each voter.

• If one candidate beats every other candidate in a head-to-head match, that candidate is the winner.

• If one candidate beats one candidate but is beaten by another candidate, a point-system is used to find the winner that will satisfy people the most.

Plurality and Pairwise Comparison Methods

• Both satisfy the majority wins rule.

• Only pairwise comparison satisfies the head-to-head winning rule.

• Both fail the irrelevant loser rule.

Comparing Plurality and Pairwise Comparison

Fairness Rules Plurality Pairwise Comparison

Majority Wins Rule X X

Head-to-head Winning Rule X

Irrelevant Loser Rule

Comparing Plurality and Pairwise Comparison Methods

Fairness Rules Plurality Pairwise Comparison

Majority Wins Rule X X

Head-to-head Winning Rule X

Irrelevant Loser Rule

The Head-to-Head Criterion

If a candidate wins when compared head to head (individually) with each

of the other candidates, then that candidate should be declared the

winner.

Plurality can fail

the head-to-head criterion:

Example 14 from book

Page 429

Comparing Plurality and Pairwise Comparison

Fairness Rules Plurality Pairwise Comparison

Majority Wins Rule X X

Head-to-head Winning Rule X

Irrelevant Loser Rule

Failing Irrelevant Losers Rulewith Plurality Voting

What kind of pie to have? 9 votes:

Apple Blueberry Cherry

3 4 2

Failing Irrelevant Losers Rulewith Plurality Voting

What kind of pie to have?

Apple Blueberry Cherry

3 4 2

With cherry pie, Blueberry pie wins!

Failing Irrelevant Losers Rulewith Plurality Voting

What kind of pie to have?

Apple Blueberry Cherry

3 4 2

With cherry pie, Blueberry pie wins!

If there’s no cherry pie, 2 people revote:

Failing Irrelevant Losers Rulewith Plurality Voting

What kind of pie to have?

Apple Blueberry Cherry

3 4 2

With cherry pie, Blueberry pie wins!

Apple Blueberry Cherry

3+2=5 4 0

If there’s no cherry pie, 2 people revote:

Failing Irrelevant Losers Rulewith Plurality Voting

What kind of pie to have?

Apple Blueberry Cherry

3 4 2

With cherry pie, Blueberry pie wins!

Apple Blueberry Cherry

3+2=5 4 0

Without cherry pie, apple pie wins!

If there’s no cherry pie, 2 people revote:

On the independence of irrelevant alternatives:

After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says "In that case I'll have the blueberry pie."

Failing Irrelevant Losers Rulewith Plurality Voting

What kind of pie to have?

Apple Blueberry Cherry

3 4 2

With cherry pie, Blueberry pie wins!

Apple Blueberry Cherry

3+2=5 4 0

Without cherry pie, apple pie wins!

If there’s no cherry pie, 2 people revote:

Voting Systems And Us

I. Who would have been president

if other voting systems were used?

Often, details of the voting system don’t matter:

• 1984, 1988, 1992, 1996

• Different voting schemes (popular vote, the Electoral College, the pairwise comparison), all selected the same candidate.

Yoshie Furuhashi , 2004

What happened in 2000?

2000 Presidential Election

George W. Bush Al Gore

Did Nader spoil the election?

Impossible to say but…

Did Nader spoil the election?

Impossible to say but…

In the 2000 presidential election in Florida. Bush beat Gore by only 537 votes. Nader received 97,421 votes.

A Voting Game

A Voting Game

• Each student is an elf, a hobbit or a dwarf living in Middle Earth.

• Middle Earth is a democracy, but the inhabitants haven’t decided how to vote.

• Running for president:• Gandalf (elf)

• Gimli (dwarf)

• Bilbo Baggins (hobbit)

• Aragorn (man)

Winning

• If Gandalf gets elected, elves and men win.

• If Gimli gets elected, dwarves and hobbits win.

• If Bilbo gets elected, elves and hobbits win.

• If Aragorn gets elected, man and dwarves win.

Plurality Voting

• Vote #1

• Vote #2

• Vote #3

Pairwise Comparison Voting

• Vote #1