Warm Up: Factorial

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# Warm Up: Factorial - PowerPoint PPT Presentation

Warm Up: Factorial. 5! is read five factorial and means start with 5 and count down to one multiplying as you go. 5! = 5*4*3*2*1 = 120 Find: 7!. Election Theory. How do you pick a winner and is your method “fair”?. Warm Up. Pick your top three lunch spots.

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Warm Up: Factorial

5! is read five factorial and means start with 5 and count down to one multiplying as you go.

5! = 5*4*3*2*1 = 120

Find: 7!

### Election Theory

How do you pick a winner and is your method “fair”?

Warm Up
• Pick your top three lunch spots.

McD Chick Cook Out Arby’s KFC Sub

BK Taco B Wendy’s

• Rank them 1st, 2nd, and 3rd.

Chocolates Flowers Jewelry Dinner Balloons

Stuffed Animal Poem Card

Chapter 1Election Theory

We will study:

• various voting systems & how winners are chosen
• the fairness of these systems
• the curious quirks built into each system
• weighted voting systems
Plurality
• Each person votes for their favorite option.
• The option with the most votes wins!

Example:

The students in the class voted on the format of the next test.

Multiple Choice 17

Essay 3

And the winner is……

Majority
• Each person votes for their favorite option.
• The option with greater than 50% of the votes wins!
• It is possible for there to be no winner.

Example 1:

The group voted on the lunch menu.

Hot dogs 10%

Hamburgers 25%

Steaks 60%

Majority

Example 2:

Hot dogs 20%

Hamburgers 30%

Steaks 40%

Skill Review: Percents

Percent means out of 100.

To find the percent of approval for each option divide the number of votes by the total number of votes cast.

Example:

If 4 votes were cast for steaks and 12 people voted, find the percent of votes for steaks.

What percent of the voters did not want steak?

Majority

Example 3:

How many votes are required to be a majority winner?

Let’s think
• If an option is a majority winner, would it automatically be a plurality winner?
• If an option is a plurality winner, would it automatically be a majority winner?
• Is there always a plurality winner?
Problem 1

How many first place votes did each option get?

Which option is the plurality winner?

What is the percentage associated with those votes?

Which option is the majority winner?

Problem 2

How many first place votes did each option get?

Which option is the plurality winner?

What is the percentage associated with those votes?

Which option is the majority winner?

HOMEWORK

Option 1: Take a survey with 3 to 5 fixed responses of at least 20 people and calculate the winner by each method.

(Which do you prefer hot dogs, tacos or wraps?)

Option 2: Find an example of these methods being used “in the real world”. Be prepared to discuss it with the class.

Warm Up Day 2
• A B C D
• B C B B
• C D D C
• D A AA
• 8 5 6 7
• 1) How many ordered their preference CBDA?
• 2) How many people were surveyed?
• 3) How many votes did “B” get for first or second place?
• 4) Who got the most votes for first place? Do they have a majority?
• 5) What percent of those surveyed ranked “A” last?
Borda Count
• Each voter ranks all options from first to last.
• Points are given based on rank. The point values need to be set.
• The option with the most total point wins.

Example:

Bob: first A then B then C.

May: first B then A then C.

Sue: first A then B then C.

Fred: first C then A then B.

Award 3 points for first place, 2 points for second place and 1 point for third place.

A gets ___ points.

B gets ___ points.

C gets ___ points.

Activities with Kids
• Nikeyta decided to take her 9 nephew nieces “on a lark”. She asks each of them to create a list that ranks his preferences for the following activities: swimming (S), ice skating (I), bowling (B), and hiking (H). Here are the preference lists:
• 1 2 3 4 5 6 7 8 9
• S S B BBB I I H
• I I S SS I H H I
• B B I II H S SS
• H HHHH S B BB
• A. Create a preference schedule summarizing the information.
• B. Is there a majority winner?
• C. Find the winner using Plurality.
• D. Find the winner using a Borda count.
Approval Voting
• Each voter ranks the options.
• The group decides on how many ranks to include in the approval vote.
• Eliminate the other ranks.
• The option with the most votes at any of the approval ranks wins.
Approval

Example

A

B

C

D

E

D

B

E

C

A

E

C

B

D

A

20

22

18

Runoff
• Find the top two vote getters for first place.
• Knock everyone else off. Give votes to the next candidate on the tree.
• Winner takes all!
Sequential Runoff
• Calculate the first place votes.
• Eliminate the weakest link one at a time.
• Redo the count until a majority winner is found.
• This method is also called Majority with Elimination
Arrow’s Conditions
• Non-dictatorship
• One person does not control the vote of the group.
• Individual Sovereignty
• Each person is allowed to vote their preferences.
• Unanimity
• The group ranking should be consistent with the votes of the individuals.
• Freedom from Irrelevant Alternatives
• The winning choice should still win if a lower ranked choice is eliminated.
• Uniqueness of the Group Rankings
• If the method is done correctly, you should get the same answer every time.
Pairwise Comparison Method
• Take every combination of two candidates.
• Assign a point if the voters prefer a candidate over another. Assign a half point for a tie.
• Candidate with the most points win.
Condorcet
• Set up the same as Pairwise but Condorcet states you must win over every other contender.
• Establish the condorcet matrix.

Borda count

In a Borda count election, 5 voters rank 5 alternatives [A, B, C, D, E].

3 voters rank [A>B>C>D>E]. 1 voter ranks [C>D>E>B>A]. 1 voter ranks [E>C>D>B>A].

Borda count (a=0, b=1): C=13, A=12, B=11, D=8, E=6. C wins.

Now, the voter who ranks [C>D>E>B>A] instead ranks [C>B>E>D>A]; and the voter who ranks [E>C>D>B>A] instead ranks [E>C>B>D>A]. Note that they change their preferences only over the pairs [B, D] and [B, E].

The new Borda count: B=14, C=13, A=12, E=6, D=5. B wins.

Note that the social choice has changed the ranking of [B, A], [B, C] and [D, E]. The changes in the social choice ranking are dependent on irrelevant changes in the preference profile. In particular, B now wins instead of C, even though no voter changed their preference over [B, C].

Consider the situation:
• You are in charge of the agenda. The day’s plan is to vote and decide on one of three proposals for new senior privileges.
• Of the three proposals, you like C best.
• The plan is to vote on the first two proposals on the agenda and then vote on the winner between the first two and the third proposal.
• Your early polls say that the other members have these preferences. {ABC,40}{BCA, 30}{CAB,30}
• How would you place the proposals on the agenda and why?(Which method(s) compare two options at a time?)
Which of Arrow’s Conditions were violated?
• Uniqueness of group ranking
• “The same answer every time.”
• Unanimity
• “If everyone prefers A over B then the group should prefer A over B”
• Individual Sovereignty
• “Each person can make his own ranking.”
• Non-dictatorship
• “The preferences of one do not determine the group.”
• Freedom from Irrelevant Alternatives
• “The group ranking does not change with the introduction of another choice”

An anecdote which illustrates a violation of this property has been attributed to Sidney Morgenbesser:

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."

### IIA

An

Freedom from Irrelevant Alternatives

Which of Arrow’s Conditions were violated?
• Uniqueness of group ranking
• “The same answer every time.”
• Unanimity
• “If everyone prefers A over B then the group should prefer A over B”
• Individual Sovereignty
• “Each person can make his own ranking.”
• Non-dictatorship
• “The preferences of one do not determine the group.”
• Freedom from Irrelevant Alternatives
• “The group ranking does not change with the introduction of another choice”
Which of Arrow’s Conditions were violated?
• Uniqueness of group ranking
• “The same answer every time.”
• Unanimity
• “If everyone prefers A over B then the group should prefer A over B”
• Individual Sovereignty
• “Each person can make his own ranking.”
• Non-dictatorship
• “The preferences of one do not determine the group.”
• Freedom from Irrelevant Alternatives
• “The group ranking does not change with the introduction of another choice”
Which of Arrow’s Conditions were violated?
• Uniqueness of group ranking
• “The same answer every time.”
• Unanimity
• “If everyone prefers A over B then the group should prefer A over B”
• Individual Sovereignty
• “Each person can make his own ranking.”
• Non-dictatorship
• “The preferences of one do not determine the group.”
• Freedom from Irrelevant Alternatives
• “The group ranking does not change with the introduction of another choice”
Which of Arrow’s Conditions were violated?
• Uniqueness of group ranking
• “The same answer every time.”
• Unanimity
• “If everyone prefers A over B then the group should prefer A over B”
• Individual Sovereignty
• “Each person can make his own ranking.”
• Non-dictatorship
• “The preferences of one do not determine the group.”
• Freedom from Irrelevant Alternatives
• “The group ranking does not change with the introduction of another choice”
Which of Arrow’s Conditions were violated?
• Uniqueness of group ranking
• “The same answer every time.”
• Unanimity
• “If everyone prefers A over B then the group should prefer A over B”
• Individual Sovereignty
• “Each person can make his own ranking.”
• Non-dictatorship
• “The preferences of one do not determine the group.”
• Freedom from Irrelevant Alternatives
• “The group ranking does not change with the introduction of another choice”

Duncan Black, in 1958, proposed a new voting system for multi-candidate elections (called Black's System):

• Each voter submits their entire preference order.
• If a Condorcet winner exist, then election that candidate.
• If not, then use the Borda count to determine the societal preference order.

Problem Set 7 : Which of the following criteria (anonymity, neutrality, monotonicity, the majority criterion and the CWC) does Black's system satisfy? Justify your answers. Also, suppose Dale, Paul,and Wayne are the three finalist in the "World's Sexiest Man" contest held aboard a luxury cruise ship. The 15 judges will be using Black's method and the preference schedule they vote for is:

• Number of Voters
• Under Black's system, what societal preference order would be produced? Suppose that after votes are cast, but before the winner is announced, Wayne is kicked off the ship for disorderly conduct, thus he is ineligible to be named the "World's Sexiest Man". Given your answer to the previous question, should Wayne's exclusion from the contest change its outcome? Suppose that Wayne's name is removed from each of the 15 ballots shown above, with the remaining contestant moved up wherever is appropriate. What would be the outcome, under Black's system, with this new two candidate preference schedule? Is this odd to you?
Example
• 7 votes for A > B > C
• 6 votes for B > C > A
• 5 votes for C > A > B
• Determine the majority winner with 3 options then all combinations of 2 options.

Borda count

In a Borda count election, 5 voters rank 5 alternatives [A, B, C, D, E].

3 voters rank [A>B>C>D>E]. 1 voter ranks [C>D>E>B>A]. 1 voter ranks [E>C>D>B>A].

Borda count (a=0, b=1): C=13, A=12, B=11, D=8, E=6. C wins.

Now, the voter who ranks [C>D>E>B>A] instead ranks [C>B>E>D>A]; and the voter who ranks [E>C>D>B>A] instead ranks [E>C>B>D>A]. Note that they change their preferences only over the pairs [B, D] and [B, E].

The new Borda count: B=14, C=13, A=12, E=6, D=5. B wins.

Note that the social choice has changed the ranking of [B, A], [B, C] and [D, E]. The changes in the social choice ranking are dependent on irrelevant changes in the preference profile. In particular, B now wins instead of C, even though no voter changed their preference over [B, C].