Note 7 - Ranking Elections

In-Class Voting

Here are the results of in-class voting:¹

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?

Recall preference relations from the previous lecture.

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

Borda Count Ranking:

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.

Unanimity: If every voter prefers candiate \(a\) to \(b\), then \(R\) should rank \(a\) higher than \(b\).
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.

“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).

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!)

Instant Runoff

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

Instant Runoff

Now, c wins according to plurality

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

Plurality: b wins. Runoff: a wins.