1. Impossibility and Other Alternative Voting Methods
MAT 105 Spring 2008
Impossibility and Other Alternative Voting Methods

2. Which Method to Use?
We have seen many methods, all of them flawed in some way
Which method should we use?
Maybe we shouldn’t use any of them, and keep searching for a better way

3. Arrow’s Theorem
In 1951, Kenneth Arrow proved that this search will be in vain
Specifically, he proved that there is no voting system that satisfies all of the following conditions:
is not a dictatorship
voters rank their candidates in order
independence of irrelevant alternatives
Pareto condition

4. Getting Around Arrow’s Theorem
Instead of giving up hope, we might look for ways around Arrow’s Theorem
Since we know we can’t have a voting method with all of those conditions being true, we might look for one condition to drop
We certainly don’t want a dictatorship or a method that doesn’t satisfy the Pareto condition
If we drop IIA, we could use Borda Count

5. Dropping Preference Orders
If we drop the preference order condition, we have two options:
allow voters to express less information
allow voters to express more information
These choices lead to approval voting and range voting, respectively

6. Approval Voting
Voters cast a single vote for each candidate they approve of
The candidate who receives more votes than any other is the winner

7. Properties of Approval Voting
We can assume that voters are still using (mental) preference lists, but simply have a “cutoff” above which they approve of a candidate and below which they do not
For example, suppose we have two voters both with preference A>B>C>D
One of these voters might approve of A, B, and C, and the other might approve only of A

8. Approval Conditions
Voters
Preference Order 1
A > B > C
B > A > C
C > A > B
Approval voting does not satisfy the Condorcet winner criterion
Consider the profile shown here, where red indicates that the voter approves of the candidate
A is the Condorcet winner, but B wins the approval vote

9. Range Voting Voters rate each candidate on a scale Examples
scale from 0 to 10
1 star to 5 stars
-2 to 2 thumbs
The candidate with the most points wins

10. Properties of Range Voting
Again we assume that voters are using an internal ranking to cast votes
If a voter likes A more than B, they will give A at least as many points as they give B
However, there are lots of possibilities for how a voter with preference A>B>C>D could fill out a ballot

11. Range Conditions Range voting satisfies the Pareto condition
If every voter prefers A over B, then every ballot will give A at least as many points as B, so A’s total will be at least B’s total
The only way this could make B win is if every single voter gave A the same score as B, and A and B tied for the win

12. Range Conditions
In fact, range voting satisfies many important conditions (though notably not Condorcet winner)
Range voting appears to be the closest to “fair” that we may be able to get