BUYING TIME

Cindy Boyd Anthony Ice Donna Osborn Margaret Sanders

Class: Pre-Algebra, Algebra, Statistics, Discrete Math

Materials: Poll sheets, tally sheets, calculator, background information, practice problem, student project sheet.

Goals: After gathering information on preferences, examine the different methods for determining a winner - Plurality, Borda Count, Run-Off, Sequential Run-Off, and Condorcet. To enable students to become familiar with oral and written presentations in Discrete Math.

Time Required: Approximately 2-4 days. One day to teach methods, if needed. One day to tally the data, then time either in or out of class to apply the decision making methods, make the decisions, and write the report. One day for oral presentations.

Background: Knowledge of methods of voting, frequency tables and survey techniques. Following this page are a series of pages illustrating these different methods with different soft drinks.

Setting: A client has come into your advertising agency with a new theme park for fun-loving young adults. They want to advertise during one prime time situation comedy.

Problem: Take a preferential survey of the top three situation comedies. Use Plurality, Borda Count, Run-Off, Sequential Run-Off, and Condorcet methods to determine winners. Then make a final choice. Each advertising agency should submit a data sheet and a written recommendation which defends and explains their method for determining when to advertise the new product. Also, each agency will orally present their report to the client.

Evaluation: One grade will be given to the entire group. Point values will be assigned for the written report and presentation. Points for the written report should be divided among the description of the problem, the list of assumptions, the data sheet with the results of the survey, the results of the decision-making processes, the recommendation for the client, and the explanation of the decision.

Extension:

1. For a more challenging project, increase the survey to include four TV shows.

2. Call local TV stations to find the cost for advertising during the shows surveyed.

3. Choose a specific night or time slot.

4. Change the product and/or target audience.

Teacher Notes:

1. To choose the shows for the survey: Have each student put his/her favorite prime time situation comedy on a slip of paper. Place slips in a container or a box. Draw out three slips (with different shows); these will be the three shows used in the survey. If two classes are doing the project simultaneously, use the same shows for both classes.

2. Survey at least 2 classes (50 students).

3. Depending on the level of the class, the background concept sheets could be run for all of the students. The background concept sheets are explanations of the different decision making methods.


4. The project can be scaled up or down to match the level of the class by increasing/decreasing the number of methods used to choose a winner.

5. The oral presentation should present the written report to the class. It should not be read.

6. Enclosed is a student tally sheet with six places for tallies; these are for the six possible rank orders. If four choices are given, then there are 24 possible rank orders.


PREFERENTIAL VOTING

Consider this set of preferences for soft drinks. Each column represents the rank order, while the number below indicates the number of people voting that preferential order.

Pepsi

Diet Coke

Dr. Pepper

Coke

Sprite

Sprite

Coke

Sprite

Diet Coke

Dr. Pepper

Diet Coke

Dr. Pepper

Sprite

Coke

Sprite

Sprite

Coke

Coke

Dr. Pepper

Dr. Pepper

Coke

Diet Coke

Dr. Pepper

Diet Coke

Diet Coke

Pepsi

Pepsi

Pepsi

Pepsi

Pepsi

18

12

10

9

4

2

Some methods of determining the winner:

1. Plurality

2. Borda Count

3. Run-Off

4. Sequential Run-Off

5. Condorcet

When a tie occurs for any of the methods, use the other methods to make a decision.

The Plurality Method

The winner is the choice that receives the most first place votes.

Pepsi

Diet Coke

Dr. Pepper

Coke

Sprite

Sprite

Coke

Sprite

Diet Coke

Dr. Pepper

Diet Coke

Dr. Pepper

Sprite

Coke

Sprite

Sprite

Coke

Coke

Dr. Pepper

Dr. Pepper

Coke

Diet Coke

Dr. Pepper

Diet Coke

Diet Coke

Pepsi

Pepsi

Pepsi

Pepsi

Pepsi

18

12

10

9

4

2

Diet Coke = 12 Pepsi = 18 Coke = 9 Dr. Pepper = 10 Sprite = 4 + 2 = 6

The plurality winner is Pepsi. The plurality winner is not always the majority winner. Note that the plurality winner is ranked last by a majority of the voters.


The Borda Method

If there are N choices on the ballot, the Borda method awards N points for a first-place ranking, N-1 points for a second-place ranking, etc.

Pepsi

Diet Coke

Dr. Pepper

Coke

Sprite

Sprite

Coke

Sprite

Diet Coke

Dr. Pepper

Diet Coke

Dr. Pepper

Sprite

Coke

Sprite

Sprite

Coke

Coke

Dr. Pepper

Dr. Pepper

Coke

Diet Coke

Dr. Pepper

Diet Coke

Diet Coke

Pepsi

Pepsi

Pepsi

Pepsi

Pepsi

18

12

10

9

4

2

In order, the totals for Diet Coke, Pepsi, Coke, Dr. Pepper, and Sprite:

Diet Coke = 18(1) + 12(5) + 10(4) + 9(2) + 4(4) +2(2) = 156

Pepsi = 18(5) + 12(1) + 10(1) + 9(1) + 4(1) +2(1) = 127

Coke = 18(4) + 12(3) + 10(2) + 9(5) + 4(3) +2(3) = 191

Dr. Pepper = 18(2) + 12(2) + 10(5) + 9(4) + 4(2) +2(4) = 162

Sprite = 18(3) + 12(4) + 10(3) + 9(3) + 4(5) +2(5) = 189

Coke wins.


The Run-Off Method

The Run-Off method conducts a new election between the two choices with the most first-place rankings.

Pepsi

Diet Coke

Dr. Pepper

Coke

Sprite

Sprite

Coke

Sprite

Diet Coke

Dr. Pepper

Diet Coke

Dr. Pepper

Sprite

Coke

Sprite

Sprite

Coke

Coke

Dr. Pepper

Dr. Pepper

Coke

Diet Coke

Dr. Pepper

Diet Coke

Diet Coke

Pepsi

Pepsi

Pepsi

Pepsi

Pepsi

18

12

10

9

4

2

Diet Coke = 12; Pepsi = 18; Coke = 9; Dr. Pepper = 10; Sprite = 4 + 2 = 6

The Run-Off is between Diet Coke and Pepsi. Eliminate all but Pepsi and Diet Coke. Next consider each of the preference schedules which were eliminated. Dr. Pepper (10) should be awarded to Diet Coke because Diet Coke was rated above Pepsi in that preferential list. Coke (9) will be awarded to Diet Coke. Sprite (4) will be awarded to Diet Coke. Sprite (2) will be awarded to Diet Coke.

Diet Coke = 12 + 10 + 9 + 4 + 2 = 37

Pepsi = 18

Diet Coke wins.


The Sequential Run-Off Method

The sequential Run-Off method eliminates only one choice and then conducts a new election. This process is repeated until one choice remains.

Pepsi

Diet Coke

Dr. Pepper

Coke

Sprite

Sprite

Coke

Sprite

Diet Coke

Dr. Pepper

Diet Coke

Dr. Pepper

Sprite

Coke

Sprite

Sprite

Coke

Coke

Dr. Pepper

Dr. Pepper

Coke

Diet Coke

Dr. Pepper

Diet Coke

Diet Coke

Pepsi

Pepsi

Pepsi

Pepsi

Pepsi

18

12

10

9

4

2

Diet Coke = 12; Pepsi = 18; Coke = 9; Dr. Pepper = 10; Sprite = 4 + 2 = 6

Sprite is eliminated first. Sprite=s four votes in column five are awarded to Diet Coke because Diet Coke was the second choice of those four people. Sprite=s two votes in column six are awarded to Dr. Pepper because Dr. Pepper was the second choice of those two people. Now the vote totals are changed as shown below.

Diet Coke = 12 + 4 = 16

Pepsi = 18

Coke = 9

Dr. Pepper = 10 + 2 = 12

Now the new low, Coke, is eliminated and Coke=s nine votes are awarded to Dr. Pepper because Dr. Pepper was the second choice of those nine people. Now the vote totals are changed as shown below.

Diet Coke = 16

Pepsi = 18

Dr. Pepper = 12 + 9 = 21

Now the new low, Diet Coke, is eliminated and Diet Coke=s 16 votes are given to Dr. Pepper because the second choice, Sprite, and the third choice, Coke, have already been eliminated. Now the vote totals are changed as shown below.

Pepsi = 18

Dr. Pepper = 21 + 12 = 33

Now the new low is eliminated and Dr. Pepper wins. But it is interesting to note that 43 people ranked Coke above Dr. Pepper originally and 12 people ranked Dr. Pepper above Coke.


The Condorcet Method

The Condorcet method awards the election to the choice that can beat each of the other choices in one-on-one contests.

Pepsi

Diet Coke

Dr. Pepper

Coke

Sprite

Sprite

Coke

Sprite

Diet Coke

Dr. Pepper

Diet Coke

Dr. Pepper

Sprite

Coke

Sprite

Sprite

Coke

Coke

Dr. Pepper

Dr. Pepper

Coke

Diet Coke

Dr. Pepper

Diet Coke

Diet Coke

Pepsi

Pepsi

Pepsi

Pepsi

Pepsi

18

12

10

9

4

2

To examine the data for a Condorcet winner, compare each choice with every other choice. First put an X in each box which compares a choice to itself. Then compare Diet Coke with Pepsi. Diet Coke is ranked higher than Pepsi on columns 2, 3, 4, 5, and 6; therefore Diet Coke gets the votes from those columns (12+10+9+4+2=37 votes). Pepsi is ranked higher than Diet Coke on column 1; therefore, Pepsi gets the votes from that column (18 votes). So Diet Coke wins and we place a W in the Diet Coke row, Pepsi column. Now we compare Diet Coke to Coke. Diet Coke is ranked higher than Coke in columns 2, 3, and 5; therefore Diet Coke get the votes from those columns (12 + 10 + 4 = 26 votes). Coke is ranked higher than Diet Coke in columns 1, 4, and 6; therefore, Coke gets the votes from those columns (18+ 9 + 2 = 29 votes). So Diet Coke loses and we place an L in the Diet Coke row, Coke column. Since Diet Coke does not beat all of the other choices, Diet Coke can not win. Now we would begin the same process with Pepsi.

Diet Coke

Pepsi

Coke

Dr. Pepper

Sprite

Diet Coke

X

W

L

L

L

Pepsi

L

X

L

L

L

Coke

W

W

X

W

L

Dr. Pepper

W

W

L

X

L

Sprite

W

W

W

W

X

The winner is Sprite because it obtains a majority over each of the other choices. If no one wins against all of the other choices, then there is no winner. This is called a paradox.


Practice Problem

Student Worksheet

Here are some survey results for you to analyze using the different methods. The school cafeteria is considering three different possibilities - pizza, hamburger, or hot dog - for lunch. Only one of the three will be served. Here are the results of the survey. Use all of the different methods to determine the lunch choice.

Pizza

Pizza

Hot Dog

Hot Dog

Hamburger

Hamburger

Hamburger

Hot Dog

Pizza

Hamburger

Pizza

Hot Dog

Hot Dog

Hamburger

Hamburger

Pizza

Hot Dog

Pizza

13

18

12

17

13

17


ANSWERS TO PRACTICE PROBLEM

PLURALITY: Pizza

BORDA COUNT: Hot Dog

RUN-OFF: Hamburger

SEQUENTIAL RUN-OFF: Hamburger

CONDORCET: Hot Dog


VECTOR AD AGENCY

You are a member of a creative team at the Vector Ad Agency, Athe agency with magnitude and direction@. SumWare Company is offering your firm a chance to market their newest theme park, CalcLand. SumWare wants to purchase ad-time during the most popular prime time situation comedy. It is important to Vector Ad Agency to snag the SumWare account from a rival company, Shakespeare Ltd. The SumWare account would generate over $2.1 million for your agency. Your boss has assigned the account to your creative group and to another group in the firm. Your boss will use the written and oral presentations to decide which group gets the $25,000 bonus.

Your group=s plan of action is:

a) To conduct a preferential survey of the top three situation comedies

_______________ , ________________ , and _____________________ .

b) To use Plurality, Borda Count, Run-Off, Sequential Run-Off, and Condorcet to determine

winners.

c) To make a final choice using the results of your decision-making process.

Your creative team will

* submit a written report which contains

* a description of the problem

* a list of assumptions

* a data sheet with the results of the survey

* results of the decision-making processes

* a recommendation for the client

* an explanation of your decision

* give an oral presentation which covers everything in the written report


Rank the 3 shows in order with 1 as your favorite.

Rank the 3 shows in order with 1 as your favorite.

_____________________

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_____________________

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Rank the 3 shows in order with 1 as your favorite.

Rank the 3 shows in order with 1 as your favorite.

_____________________

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_____________________

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_____________________

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_____________________

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_____________________

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_____________________

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Rank the 3 shows in order with 1 as your favorite.

Rank the 3 shows in order with 1 as your favorite.

_____________________

__

_____________________

__

_____________________

__

_____________________

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_____________________

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_____________________

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Rank the 3 shows in order with 1 as your favorite.

Rank the 3 shows in order with 1 as your favorite.

_____________________

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_____________________

__

_____________________

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_____________________

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_____________________

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_____________________

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STUDENT TALLY SHEET

_______________________________

____________________________

_______________________________

____________________________

_______________________________

____________________________

TALLY TOTAL ______

TALLY TOTAL ______

_______________________________

____________________________

_______________________________

____________________________

_______________________________

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TALLY TOTAL ______

TALLY TOTAL ______

_______________________________

____________________________

_______________________________

____________________________

_______________________________

____________________________

TALLY TOTAL ______

TALLY TOTAL ______


Sample Solution

VECTOR AD AGENCY

You are a member of a creative team at the Vector Ad Agency, Athe agency with magnitude and direction@. SumWare Company is offering your firm a chance to market their newest theme park, CalcLand. SumWare wants to purchase ad-time during the most popular prime time situation comedy. It is important to Vector Ad Agency to snag the SumWare account from a rival company, Shakespeare Ltd. The SumWare account would generate over $2.1 million for your agency. Your boss has assigned the account to your creative group and to another group in the firm. Your boss will use the written and oral presentations to decide which group gets the $25,000 bonus.

Your group=s plan of action is:

a) To conduct a preferential survey of the top three situation comedies

HOME IMPROVEMENT, FRIENDS , and MURPHY BROWN .

b) To use Plurality, Borda Count, Run-Off, Sequential Run-Off, and Condorcet to determine winners.

c) To make a final choice using the results of your decision-making process.

Your creative team will

* submit a written report which contains

* a description of the problem

* a list of assumptions

* a data sheet with the results of the survey

* results of the decision-making processes

* a recommendation for the client

* an explanation of your decision

* give an oral presentation which covers everything in the written report


Our creative team, AdForce, was given an opportunity to work on the CalcLand proposal for SumWare. SumWare wants to advertise CalcLand, their newest theme park, during the most popular prime time situation comedy. Our action plan was to conduct a preferential survey of AFriends@, AHome Improvement@, and AMurphy Brown@ and use five different decision-making methods to determine the most popular show. We used the results of the survey as the basis of our recommendation.

Assumptions:

1) Our survey results are representative of young adults in the area.

2) The people surveyed were honest in their responses.

3) The three choices in the survey are the top three prime time comedy shows.


STUDENT TALLY SHEET

HOME IMPROVEMENT

HOME IMPROVEMENT

FRIENDS

MURPHY BROWN

MURPHY BROWN

FRIENDS

TALLY TOTAL 14

TALLY TOTAL 7

FRIENDS

FRIENDS

MURPHY BROWN

HOME IMPROVEMENT

HOME IMPROVEMENT

MURPHY BROWN

TALLY TOTAL 12

TALLY TOTAL 10

MURPHY BROWN

MURPHY BROWN

FRIENDS

HOME IMPROVEMENT

HOME IMPROVEMENT

FRIENDS

TALLY TOTAL 4

TALLY TOTAL 7


Results with Decision-Making Methods

Home Improvement

Home Improvement

Friends

Friends

Murphy Brown

Murphy Brown

Friends

Murphy Brown

Murphy Brown

Home Improvement

Friends

Home Improvement

Murphy Brown

Friends

Home Improvement

Murphy Brown

Home Improvement

Friends

14

7

12

10

4

7

Plurality Method: The winner is AFriends@.

Home Improvement = 14+7 = 21

Friends = 12+10 = 22

Murphy Brown = 4+7 = 11

Borda Counts Method: The winner is AFriends@.

Home Improvement = 14(3)+7(3)+12(1)+10(2)+4(1)+7(2) = 113

Friends = 14(2)+7(1)+12(3)+10(3)+4(2)+7(1) = 116

Murphy Brown = 14(1)+7(2)+12(2)+10(1)+4(3)+7(3) = 95

Run-Off Method: The winner is AHome Improvement@.

Home Improvement = 21

Friends = 22

Murphy Brown=s 4 goes to Friends.

Murphy Brown=s 7 goes to Home Improvement.

Thus, Home Improvement = 21+7 = 28

Friends = 22+4 = 26

Sequential Run-Off Method: The winner is AHome Improvement@.

Results are identical to those of the Run-Off method.

Condorcet Method: The winner is AHome Improvement@.

Home Improvement

Friends

Murphy Brown

Home Improvement

X

W

W

Friends

L

X

W

Murphy Brown

L

L

X

We recommend that SumWare advertise during AFriends@.

We made this decision because:

* AFriends@ won the Plurality Method. This means more people had it as their first choice.


* AFriends@ also won the Borda Count Method. This means, if points were assignment to the rankings, AFriends@ earned more points.

* Young adults will make time for their favorite show, so they would be more likely to see the commercial during AFriends@.

* AFriends@ revolves around a twenty-something crowd, while AHome Improvement@ (the winner of Run-Off, Run-Off Sequential and Condorcet methods) revolves around a family.

* Since AFriends@ and AHome Improvement@ are not shown on the same evening, the viewers do not have

to choose between them.

Funded in part by the National Science Foundation and Indiana University 1995