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!)


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.


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