1. CRITERIA FOR A FAIR ELECTION
1st criteria for a fair election:
The Majority Criterion (page 6) - If a choice receives a majority of the first place votes in an election, then that choice should be the winner of the election.
2nd criteria for a fair election
The Condorcet Criterion (page 8) - If there is a choice that in a head-to-head comparison is preferred by the voters over every other choice, then that choice should be the winner of the election.

2. 3rd criteria for a fair election:
The Monotonicity Criterion (page 15). If choice X is a winner of an election and, in a reelection, the only changes in the ballots are changes that only favor X, then X should remain a winner of the election.
4th criteria for a fair election:
The Independence-of-Irrelevant-Alternatives Criterion (page 18) If choice X is a winner of an election and one (or more) of the other choices is removed and the ballots recounted, then X should still be a winner of the election.

3. Arrow’s Impossibility Theorem (page 3) - A method for determining election results that is democratic and always fair is a mathematical impossibility.
Methods used to find the winner of an election:
Plurality Method
Borda Count Method
Plurality-with-Elimination Method
Method of Pairwise Comparison

4. Example 1.2: The Math Club Election (Page 6)

5. I. THE PLURALITY METHOD
plurality method (page 6) - the candidate (or candidates) with the most first place votes wins.
A plurality does not imply a majority but a majority does imply a plurality.

6. What’s wrong with the plurality method?
Example The Band Election (page 7)
If we compare the Hula Bowl to any other bowl on a head-to-head basis, the Hula Bowl is always the preferred choice.

7. Which methods satisfy which criterion?

9. II. THE BORDA COUNT METHOD
The Borda count method (page 10) - Each place on a ballot is assigned points. In an election with N candidates we give 1 point for last place, 2 points for next to last place, …, and N points for first place. The points are tallied for each candidate separately and the candidate with the most points is the winner.
4 points 56 40 32 16 4
3 points 42 30 24 12 3
2 points 28 20 16 8 2 14 10 8 4 1
1 point

10. Math Appreciation Society
A gets =79 points
B gets =106 points
C gets =104 points
D gets =81 points

12. TABLE 1-5 What’s wrong with the Borda count method?
Example 1.6: The School Board Principal Election (page 11)
TABLE 1-5

13. The School Board Principal Election
Example The School Board Principal Election
TABLE 1-5
The School Board Principal Election
A gets =29 points
B gets =32 points
C gets =30 points
D gets 6+4+9=19 points
The Borda count method violates the majority criterion.

14. Which methods satisfy which criterion?

15. III. THE PLURALITY-WITH-ELIMINATION METHOD
Round 1. (page 12) Count the first-place votes for each candidate. If a candidate has a majority of first-place votes, that candidate is declared the winner. Otherwise eliminate the candidate (or candidates if there is a tie) with the fewest first place votes.

16. 37 voters so a majority requires at least 19 votes
TABLE 1-6
37 voters so a majority requires at least 19 votes

17. III. THE PLURALITY-WITH-ELIMINATION METHOD
Round 1. (page 12) Count the first-place votes for each candidate. If a candidate has a majority of first-place votes, that candidate is declared the winner. Otherwise eliminate the candidate (or candidates if there is a tie) with the fewest first place votes.
Round 2. Cross out the name(s) of the candidate eliminated from the preference schedule and recount the first-place votes. If a candidate has a majority of first-place votes, declare that candidate the winner. Otherwise, eliminate the candidate with the fewest first place votes.

18. 37 voters so a majority requires at least 19 votes
TABLE 1-6
37 voters so a majority requires at least 19 votes

19. III. THE PLURALITY-WITH-ELIMINATION METHOD
Round 1. Count the first-place votes for each candidate. If a candidate has a majority of first-place votes, that candidate is declared the winner. Otherwise eliminate the candidate (or candidates if there is a tie) with the fewest first place votes.
Round 2. Cross out the name(s) of the candidate eliminated from the preference schedule and recount the first-place votes. If a candidate has a majority of first-place votes, declare that candidate the winner. Otherwise, eliminate the candidate with the fewest first place votes.
Round 3,4, etc. Repeat the process, each time eliminating one or more candidates, until there finally is a candidate with a majority of first-place votes, which is then declared a winner.

20. 37 voters so a majority requires at least 19 votes
TABLE 1-6
37 voters so a majority requires at least 19 votes

21. 11 voters so a majority requires at least 6 votes
Example 1.9: (page 15)
TABLE 1-9
11 voters so a majority requires at least 6 votes

22. Winner MAS Voting Method
Alisha Plurality
Boris Borda Count
Dave Plurality-with-elimination
Example:

23. 26 voters so a majority requires at least 14 votes

24. 29 voters so a majority requires at least 15 votes
Example The summer Olympic games (page 15)
TABLE 1-9
Round 1
Candidate A B C
Number of first place votes 11 8 10
29 voters so a majority requires at least 15 votes
Round 2
Candidate A B C
Number of first place votes 11 18

25. 29 voters so a majority requires at least 15 votes
TABLE 1-10
29 voters so a majority requires at least 15 votes
Round 1
Candidate A B C
Number of first place votes 7 8 14
Round 2
Candidate A B C
Number of first place votes 15 14

26. What’s wrong with the plurality-with-elimination method?
3th criteria for a fair election:
The Monotonicity Criterion. If choice X is a winner of an election and, in a reelection, the only changes in the ballots are changes that only favor X, then X should remain a winner of the election.
The plurality-with-elimination method violates the monotonicity criterion.

27. Which methods satisfy which criterion?

28. Homework
Read Pages 6 – 23
Page 30: 18, 19, 24, 28, 29, 30, 33, 59, 60, 71