Note 7  Ranking Elections
Table of Contents
InClass Voting
Here are the results of inclass 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 “middleoftheroad” 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
 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.
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) headtohead contests, with greatest difference.
 Not easily manipulated, but vulnerable to paradox of a nontransitive 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

S: Sanders, W: Warren, B: Buttigieg ⤴