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Voting Methods

Voting Methods. Introduction. What we will learn. T. Serino. The Mathematics of Voting Preference Ballots and Preference Schedules The Plurality Method The Borda Count Method The Plurality with Elimination Method The Method of Pairwise Comparisons. The Candidates. T. Serino.

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Voting Methods

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  1. Voting Methods Introduction

  2. What we will learn T. Serino The Mathematics of Voting • Preference Ballots and Preference Schedules • The Plurality Method • The Borda Count Method • The Plurality with Elimination Method • The Method of Pairwise Comparisons

  3. The Candidates T. Serino Introducing, the candidates!

  4. The voters T. Serino Beatrice is ok, but Carl gets my vote. I’m picking Beatrice! I like Amy. Just NOT Carl. Ewww boys! Go Bea! Carl is the best! I like Carl. Amy’s my girl. Anyone but Carl! Carl is NOT qualified.

  5. Preference Ballots T. Serino Although most people in a democratic society believe in the “one man, one vote” system, sometimes it is convenient to know more than a voter’s top pick. Preference ballots are used to give us more information. A preference ballot puts candidates in order of preference rather than declaring a voter’s top preference only.

  6. Preference Ballots T. Serino This voter’s ballot may look like this.

  7. Preference Ballots T. Serino A shorter form of this ballot may look like this. The only problem with preference ballots is that there can be many different (unique) ballots for only a few candidates.

  8. Preference Ballots T. Serino With 3 candidates, there are 6 possible unique ballots. Multiply! With more than 3 candidates, it may get difficult to list every possible unique ballot. It will be easier to use the following method to count the possible unique ballots. Start with 3 choices for first pick. which leaves 2 choices for second pick which leaves 1 choice left for last. This makes 6 possible unique ballots. X X =

  9. Counting Principle T. Serino Counting Principle: With n candidates there are n! possible unique ballots. Factorial Factorial: A symbol in mathematics denoted with and exclamation point (!) n! = (n) x (n-1) x (n-2) x (n-3) x ... x (1) So, for 3 candidates: 3! = 3 x 2 x 1 = 6

  10. Organizing the Ballots T. Serino With so many possible unique ballots, it becomes important to organize (or stack) them. Ballots that are exactly alike get stacked on top of each other and we use the stacked ballots to make a preference table (or preference schedule).

  11. Organizing the Ballots T. Serino The Mathematics Anxiety Club (MAC) The 37 members of the MAC are electing a new president. Their preference ballots are as follows: Preference Ballots

  12. Organizing the Ballots T. Serino If we were to organize the 37 MAC preference ballots by stacking like ballots, our stacks would look like this.

  13. Organizing the Ballots T. Serino We use our sorted (stacked) ballots to make a preference table (or preference schedule).

  14. Assumptions T. Serino A couple of assumptions we will make while studying voting theory. If a voter prefers C to B and prefers B to D, then the voter will also prefer C to D. If a candidate is eliminated, then a voter’s ballot will change as shown.

  15. Example T. Serino Remember our voters? Try to guess what each voter’s preference ballot would look like based on what they are saying and thinking.

  16. Example T. Serino Were there any problems? If you found that two of the voters’ 2nd and 3rd preferences could not be determined, you were correct.

  17. Solution T. Serino A B C C ? ? A B C B A C C ? ? B A C C B A

  18. Example T. Serino Use the preference table to answer the questions. 10 4 8 6 2 1 1stB C A D A A 2nd D B D B D C 3rd C D B C B B 4thE E E E C D 5thA A C A E E 1. How many people voted? 2. How many unique ballots were cast? With 5 candidates, how many possible unique ballots could have been cast in the election? 31 6 120

  19. M athematical D ecision M aking

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