## In-Class Voting

Here are the results of in-class voting:1

• S/W/B - 22
• S/B/W - 8
• B/S/W - 10
• B/W/S - 3
• W/B/S - 2
• W/S/B - 16

Problem: Idea of a “median” voter is good if the population is not polarized. If it is, a “middle-of-the-road” candidate satisfies no one!

Main Question: How do we aggregate the individuals to see who the aggregate best candidate should be?

### Preference Relations

Recall preference relations from the previous lecture.

Question: Does there always exist a voting rule that will give rise to any ranking rule?

### Our Preferences

Borda Count Ranking:

• Sanders: 147
• Warren: 120
• Buttigieg: 97

Borda Count outcome gives $$S \triangleright W \triangleright B$$
Pairwise outcomes also gives $$S \triangleright W \triangleright B$$

Exercise: Come up with a voting rule that comes up with a different ranking rule that gives a different outcome than Borda Count.

## Properties of a Fair Ranking Rule

1. Unanimity: If every voter prefers candiate $$a$$ to $$b$$, then $$R$$ should rank $$a$$ higher than $$b$$.
2. Independence of Irrelevant Alternatives: Candidate $$c$$ should not affect the ordering on $$a$$ and $$b$$.

The idea is that manipulating the outcome of the election should not be possible.

## Plurality Voting, Runoff

• “First Past the Post”
• Works if there are 2 candidates; the one with the most top ranks wins.
• If more than 3 candidates, can lead to results like 1992 US elections: elected candidate was the least favorite of the majority of voters.
• Encourages insincere voting.
• Runoff: plurality with elimination. This is expensive!
• Could use instant runoff: preference ballots used, and candidate with fewest first place is eliminated and the votes redistributed to second choice (used in Australia, Ireland).

## Condorcet Rule

• Winner of pairwise elections should win. That is, whichever candidate would win in (most) head-to-head contests, with greatest difference.
• Not easily manipulated, but vulnerable to paradox of a non-transitive result. (Can have no winner!)

### Example

Instant Runoff

• a/b/c: 45
• b/c/a: 30
• c/b/a: 25

b wins 55 votes when it’s b vs a
Plurality goes to a.

### Example 2

Instant Runoff

• c/a/b: 10
• a/b/c: 35
• b/c/a: 30
• c/b/a: 25

Now, c wins according to plurality

### Example 3

n = 15 b/a/c: 7 a/c/b: 5 c/a/b: 3

Plurality: b wins. Runoff: a wins.

## Footnotes

1. S: Sanders, W: Warren, B: Buttigieg