1. 8.2 Voting Possibilities and Fairness Criteria
Unit 4- Discrete Mathematics

2. The Fairness Criteria of Voting
What do we mean by fair?
Over the years, those who study voting theory have proposed numerous criteria which most people would expect a 'fair' preferential election method to satisfy. There are four Fairness Criteria:
The Majority Criterion
The Condorcet Criterion
The Irrelevant Alternatives Criterion
The Monotonicity Criterion
Kenneth Arrow
Mathematical economist who proved in 1952 that there is NO consistent method of making a fair choice among three or more candidates with preferential voting.
Borda Winner = Boston

3. The Majority Criterion
The voting methods we’ve been practicing are all used in real world situations but none are perfect.
Majority Criterion- if a candidate has majority of the first place votes, then that candidate should be the winner
Lets show that the BORDA METHOD fails the majority criterion.
Look at the following voter
profile with 20 members below
voting on Boston, Chicago, Philly,
and Orlando.
Borda Winner = Boston 8 4
1st C B
2nd P O
3rd
4th

4. The Majority Criterion
Borda Winner = Boston
This method does not select the candidate with the “majority” of the votes, who should be Chicago.
The Plurality, Pairwise, and Hare
Methods do satisfy the Majority criterion. 8 4
1st C B
2nd P O
3rd
4th

5. The Majority Criterion
Plurality- always satisfies
Pairwise- always satisfies
Borda-does not satisfy
Hare- always satisfies

6. The Condorcet Criterion
The candidate who can win a pairwise comparison with EVERY other candidate is called the Condorcet candidate.
Condorcet Criterion- should be the winner of an election if they win in pairwise comparison. (This is sometimes called HEAD TO HEAD)
Lets show that the PLURALITY method fails the Condorcet criterion.

7. The Condorcet Criterion
Let’s try to prove the Condorcet Criterion again with an example.
Remember, the Condorcet Winner is the winner of the pairwise comparison voting method.

8. The Condorcet Criterion
Condorcet Criterion (aka Head to Head)
Plurality- may not satisfy
Pairwise- always satisfies
Borda- may not satisfy
Hare- may not satisfy

9. The Monotonicity Criterion
Monotonicity Criterion- If a candidate wins an election, and, in a reelection, the only changes are changes that favor the candidate, then the same candidate should win the reelection.
For example, if a candidate wins a straw vote (pre-election poll), then gains additional support in the actual election without losing any original support, that candidate should win the actual election.
This criterion can actually be violated by the Hare Method, which can produce a different winner from
the straw vote winner.

10. The Monotonicity Criterion
Monotonicity Criterion- If a candidate wins an election, and, in a reelection, the only changes are changes that favor the candidate, then the same candidate should win the reelection.
Ex: Using the Hare Method, C is the winner
In the actual election, the 3 votes in the last column were changed to C, A, B, adding to the first column.
Hare makes B the winner, even though C actually gained more votes through the switch.

11. The Monotonicity Criterion
Plurality- always satisfies
Pairwise- always satisfies
Borda- always satisfies
Hare- does not satisfy

12. The Irrelevant Alternatives Criterion
Irrelevant Alternatives Criterion- If a candidate wins an election and, before the results are announced, one or more candidates
withdraw from the running, the same candidate should be the winner on the recount
All four voting methods may violate this criterion.
Ex:
Borda – C wins
If A drops out, then B wins by Borda.

13. The Irrelevant Alternatives Criterion
Plurality- may not satisfy
Pairwise- may not satisfy
Borda- may not satisfy
Hare- may not satisfy

14. Summary! Criteria → Majority Condorcet Monotonicity Irrelevant Alt
Plurality
Satisfies
VIOLATES
Pairwise
Borda
Hare

15. Do Now Please!
Practice Problems Worksheet
Please complete #9!