Discrete Mathematics Election Theory Unit

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

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

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.

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?

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

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”?

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

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.

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 = 8+5+6+7 = 26 ____B _____C _____B _____B ____C ____D _____D ____C ____D ____A _____A ____A 8 5 6 7 Let’s take a look at your results….

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?

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.

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?

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 = 8+5+6+7 = 26 ____B _____C _____B _____B ____C ____D _____D ____C ____D ____A _____A ____A 8 5 6 7

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?”

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 = 8+5+6+7 = 26 ____A ____B ____C ____D ____B _____C _____B _____B ____C ____D _____D ____C ____D ____A _____A ____A 8 5 6 7

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…..

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 = 8+5+6+7 = 26 ____B _____C _____B _____B ____C ____D _____D ____C ____D ____A _____A ____A 8 5 6 7

continued A: B: C: D:

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?

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

If these are the results of a rank order of soft drinks 24 22 23 18 19 25 28 P D M S C

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.