1. Discrete Mathematics
Election Theory Unit

2. Introduction Voting on the surface seems really easy:
We hold an election.
We count the ballots.
We determine the outcome in a consistent and fair manner.
In this unit you will learn:
Methods for collecting and recording votes
Methods for counting votes and ranking candidates
“Fairness Criterion” – what it means to be fair

3. Small Group Activity
In your groups write these five soft drinks on a piece of paper:
Coke
Dr. Pepper
Mountain Dew
Pepsi
Sprite

4. Group Activity continued
Within your group rank the soft drinks your group likes best…..write a 1 beside the top choice, 2 beside the next favorite….etc.
Choose someone to report out your group’s rankings.
Everyone should write down these results and we’ll answer some questions.

5. Group Activity Exercises
Do all the group rankings have the same drink ranked first? If not, which is ranked first most often?
Second?
Third?
Fourth?
Fifth?

6. Group Activity Exercise
Let’s talk about how each group made their decisions on the rankings.

7. INDEPENDENTLY Rank These
Rank order the following sports in your groups from your favorite to your least favorite.
Basketball, Softball/Baseball, Football, Soccer
What’s another word for your “favorite”?

8. Preference Schedules
In Election Theory, a “preference schedule” is sometimes used to represent the preferences of one or more individuals. The example below simply represents a ranking of B first, C second, D third, and A fourth.
_______B
_______C
_______D
________A

9. Preference Schedules
In your groups, YOU ALREADY created a preference schedule of your results for the five soft drinks…..
Within your groups, were there people who didn’t agree with your “group” ranking?
Those individuals would represent a different preference schedule.

10. Preference Schedules continued
Sometimes people (or groups) have similar preference schedules and in those cases, mathematicians will write the number of voters under the arrow as below in a group of 26 people…..
____A ____B ____C ____D Total number of voters =
= 26
____B _____C _____B _____B
____C ____D _____D ____C
____D ____A _____A ____A
Let’s take a look at your results….

11. Preferential Ballot
A preferential ballot is one that allows voters to rank the choices. Where might you see another example in the real world?
Ideas/Examples?
Most of our elections in the US are not preferential ballots. Do you think they are a good idea? Why? Why not?

12. You Practice
A survey was taken last or high school students asking them to rank order their three favorite fast food restaurants in the Concord area. The results were as follows: 35 students ranked Chik-fil-a first, Bojangles second and Cook-out third. 26 ranked Bojangles first followed by Zaxby’s and then Chick-fil-a. 17 ranked McDonalds first and then Wendy’s and Chick-fil-a.
Create the three Preference Schedules reflecting these survey results.

13. Practice Activity
There are three choices in a situation that uses preferential ballots. Call the choices A, B, and C. The figure below gives the six possible preferences a voter can express:
____A ____A ____B ____B ____ C ____C
____B ____C ____A ____C _____A ____B
____C ____B ____C _____A _____B _____A
If a fourth choice D is attached to the bottom of each of these schedules, there are 6 schedules with D at the bottom. Similarly, there could be six schedules with D third, six with D second, and six with D first, or a total of 4(6) = 24 schedules. Thus, the total number of schedules with four choices is four times to the total number of schedules with three choices.
There are 24 possible schedules with four choices. How many are there with five choices? Six choices?

14. Review
Remember this example….notice that choice A is selected first by eight voters? Does this make A the clear winner? Why? Why not?
____A ____B ____C ____D Total number of voters =
= 26
____B _____C _____B _____B
____C ____D _____D ____C
____D ____A _____A ____A

15. Plurality Method
If A is the winner purely on the basis that A got the most first place votes, then A is called the “Plurality Winner” and the election method being used is called the “Plurality Method”.
What percent of the vote did A receive?
If A receives over 50% of the first place vote, then A is considered to be the “Majority Winner?”

16. Ideas ____A ____B ____C ____D ____C ____D _____D ____C
Are there other ways of thinking about this vote and how to find the winner?
Total number of voters = = 26
____A ____B ____C ____D
____B _____C _____B _____B
____C ____D _____D ____C
____D ____A _____A ____A

17. Borda Method
Borda, a French mathematician and soldier was opposed to the plurality method and proposed what is now called the Borda Method or Borda Count.
Simply put, Borda assigned points to each place with the highest number of points being assigned to the top choice and reducing the points by one for each place.
Put that in mathematical terms…..
If you have “n” choices, then you assign “n” points to a first-place ranking, n-1 to second and so on…..

18. Let’s Refer Back
Remember this example….so consider this previous example and apply the Borda Method and see what happens….
____A ____B ____C ____D Total number of voters =
= 26
____B _____C _____B _____B
____C ____D _____D ____C
____D ____A _____A ____A

19. continued
A: B: C: D:

20. continued
Compare your results… A: 8(4) + 5(1) + 6(1) + 7(1) = 50 B: 8(3) + 5(4) + 6(3) + 7(3) = 83 C: 8(2) + 5(3) + 6(4) + 7(2) = 69 D: 8(1) + 5(2) + 6(2) + 7(4) = 58 So what do you notice? Thoughts? Fair or not fair?

21. Let’s Take an Example From Class
Step 1: Preference Schedule Step 2: Plurality Method Results Step 3: Impose the Borda Method Step 4: Borda Method Results

22. If these are the results of a rank order of soft drinks24 22 23 18 19 25 28 P D M S C

23. Literacy/research activity
Pick one of the following topics related to Election Theory and research the topic.
Preference Balloting
Preference Schedules
Borda Method
Sequential Runoff Method
Plurality versus Majority
Marquis de Condorcet
Kenneth Arrow
Arrow’s Conditions
Weighted Voting
Approval Voting
Then identify the five most important things you learn about your topic and create a Tri-Fold study guide or Poster that will help other students learn those five things.
Your product should be informative, descriptive, colorful, and contain examples/graphic organizers for the information you want to teach and remember.
Due Date: Tuesday, September 19.