1. Voting systems
Chi-Kwong Li

2. Outline... How do we define a “good” voting system?
Voting between two candidates
Voting among three or more candidates
Arrow’s Impossibility Theorem
Discussion

3. Social choice theory:
How can we measure individual interests
and preferences and combine them into one
collective decision?
Goal:
Finding an outcome that reflects “the will of
the people”

4. Preference List Ballot
A rank order of candidates: often pictured as a vertical list with the most preferred candidate on top and the least preferred on the bottom 1
Green 2
Orange 3
Yellow 4
Red 5
Purple

5. Choosing Between Two Candidates
Majority rule:
Each voter indicates a preference for one of the two candidates and the candidate with the most votes wins.
Assumption:
The number of voters is odd.

6. Advantages of Majority Rule
All voters are treated equally.
2. Both candidates are treated equally.
3. It is monotone: If a new election were held and a single voter changed her ballot from the loser of the previous election to the winner, but everyone else voted exactly as before, the outcome of the new election would be the same.

7. Can we find a better voting system?
May’s Theorem
Among all possible two-candidate voting systems that never result in a tie, majority rule is the only one that treats all voters equally, treats both candidates equally, and is monotone.
Proven by Kenneth May in 1952
Mark Fey extended the theorem to an infinite number of voters in 2004

8. What about elections with three or more candidates?
Several different possibilities for voting systems exist:
Condorcet’s Method
Plurality Voting
The Borda Count
Sequential pair-wise voting
The Hare System

9. 1. Condorcet’s Method First A B C Second Third
A candidate is a winner if he or she would defeat every other candidate in a one-on-one contest using majority rule.
First A B C
Second
Third
A defeats B (2 to 1)
A defeats C (2 to 1) Therefore, A wins!
B defeats C (2 to 1)

10. Condorcet’s Voting Paradox
With three or more candidates, there are elections in which Condorcet’s method yields no winners!
First A B C
Second
Third
A defeats B (2 to 1)
C defeats A (2 to 1) No winner!
B defeats C (2 to 1)

11. 2. Plurality Voting Only first place winners are considered
The candidate with the most votes wins
Fails to satisfy the Condorcet Winner Criterion, e.g US presidential election
Manipulability
Condorcet’s Winner Criterion
For every possible sequence of preference ballots, either there is no Condorcet winner, or the voting system produces exactly the same winner for this election as does Condorcet’s Method.

12. 3. The Borda Count
Assigns points to each voter’s rankings and then sums these points to arrive at a group’s final ranking.
Each first place vote is worth n-1 points, each second place vote is worth n-2 points, and so on down.
Method: count the number of occurrences of other candidate names that are below this candidates name.
Applications: senior class rank, sports hall of fame, track meets, etc.
First A B
Second C
Third

13. Problem with the Borda Count...
Does not satisfy the property known as “independent of irrelevant alternatives”.
Independence of Irrelevant Alternatives
It is impossible for a candidate B to move from non-winner status to winner status unless at least one voter reverses the order in which he or she had B and the winning candidate ranked.

14. Failure of the IIA
First A B
Second C
Third
Borda scores: A = 6, B = 7, C = 2  B is the winner!
First A C
Second B
Third
Borda scores: A = 6, B = 5, C = 4  A is the winner
But no one has changed his or her mind about whether B is preferred to A!

15. 4. Sequential Pairwise voting
Start with a (non-ordered) list of the candidates.
Pit the first candidate against the second in a one-on-one contest
The winner then moves on to the third candidate in the list, one-on-one.
Continue this process through the entire list until only one remains at the end.
Example: choosing a favorite color:

16. Problem with Sequential Pairwise Voting...
It fails to satisfy the “Pareto Condition.”
Pareto Condition
If everyone prefers one candidate A to another candidate B, then B should not be the winner!

17. 5. The Hare System
“[The Hare System] is among the greatest improvements yet made in the theory and practice of government.” ~ John Stuart Mill
Arrive at a winner by repeatedly deleting candidates that are “least preferred”, in the sense of being at the top of the fewest ballots.
If a single candidate remains after all others have been eliminated, it alone is the winner (otherwise, it is a tie).

18. Applying the Hare System
Rank 5 4 3 1
First A C B
Second
Third
Eliminate B, C. So, A is the winner. However, if the last column changes to [A,B,C]^T, then…..
Rank 5 4 3 1
First A C
Second
Then, C wins!

19. Problem with the Hare System
It fails to satisfy the property of monotonicity.
Monotonicity
If a candidate is a winner, and a new election is held in which the only ballot change made is for some voter to move the former winning candidate higher on his or her ballot, then the original winner should remain the winner!

20. A summary of voting systems for three or more candidates
Problem
Condorcet’s Method
Not always a winner
Plurality Voting
Fails to satisfy the Condorcet Winner Criterion, manipulability
Borda Count
Fails to satisfy the Independence of Irrelevant Alternatives (IIA) property
Sequential Pairwise Voting
Fails to satisfy the Pareto Condition
Hare System
Fails to satisfy monoticity

21. Can we do better?
Is it possible to find a voting system for three or more candidates as “ideal” as majority rule for two candidates?
Arrow’s Impossibility Theorem
With three or more candidates and any number of voters, there does not exist - and there will never exist - a voting system that always produces a winner, satisfies the Pareto condition and independence of irrelevant alternatives, and is not a dictatorship.

22. Another possibility? Approval Voting
We have seen that any search for an idealvoting system of the kind we have discussed is doomed to failure.
One alternative possibility is approval voting: Instead of sing a preference list ballot, each voter is allowed to give one vote to as many of the candidates as he or she finds acceptable.
No limit is set on the number of candidates for whom an individual can vote.
The winner under approval voting is the candidate who receives the larges number of approval votes.
Approval voting is used to elect new members to the National Academy of Science and the Baseball Hall of Fame.

23. Discussion Other applications?
Which method do you think is most easily manipulated?
Which might be a good method for electing the US president, as an alternative to the Electoral College, if any?