## Review – In-Class Voting

Recall the results of the previous section’s in-class voting:

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

We get:

• Plurality: $$S \triangleright W \triangleright B$$
• Broda: $$S \triangleright W \triangleright B$$
• Instant Runoff: $$S \triangleright W \triangleright B$$
• Approval Voting1: Sanders

It seems that Sanders (at least within our class) is the Condorcet Winner. In other words, he wins pairwise against any competitor. There is no paradox.

## Another Example

100 voters with Plurality Voting:

• Scenario $$\pi_1$$ forces $$A \triangleright B \triangleright C$$:
• A/B/C - 45
• B/C/A - 30
• C/B/A - 25
• Scenario $$\pi_2$$ forces $$B \triangleright A \triangleright C$$:
• A/B/C - 45
• B/C/A - 30
• B/C/A - 25

Note that Independence of Irrelevant Alternatives was violated! Changing the positions of $$B$$ and $$C$$ for the third class of voters changed the outcome of the positions of $$B$$ and $$A$$.

## Ranking Methods

• Positional Methods:
1. Plurality
2. Broda Count
3. Instant Runoff (a mixed method)
• Pairwise Ranking Methods:
1. Condorcet (repeated rounds of binary contests) – Number of pairwise wins
2. Copeland Index (kind of a modified Condorcet) – Number of pairwise wins - number of pairwise losses

Exercise: Think of situation where a plurality winner is the Condorcet loser.

## Other Desirable Properties

1. Symmetry/Anonymity: Who the voter is should not affect the result.
• Mathematically, permuting the preference profile $$\pi$$ should not change the outcome of the election.
2. Monotonicity: If a voter moves a particular candidate, say $$A$$, higher up within rankings, $$A$$ should not drop in social ranking.
3. Condorcet Winner Criterion/Condorcet Loser Criterion:
4. IIA of Preference Strengths: Preference strength for a particular ranking, $$A$$ vs $$B$$, is number of spots $$A$$ is above $$B$$. If two spots have the same preference ranking for all voters, they should be the same in social ranking.
5. Cancellation of Ranking Profiles: Suppose $$A = {A, B, C}$$. If we have the preference profiles $$A/B/C$$, $$B/C/A$$, $$C/A/B$$, then removing all three should not change the outcome.

Stretegically Vulnerable: A strategy is strategically vulnerable if it violates IIA and leaves itself open to manipulation.

Arrows Impossibility Theorem: Any transitive ranking rule on $$\vert A \vert > 3$$ that satisfies IIA and unanimity must be a dictatorship. (Only one voter dictates social ranking.)

Gibbard Satterthwaite Theorem: A strategy-proof voting rule, where no voter can gain from misrepresenting their true preference, is impossible without

If $$F$$ is a strategy-proof voting rule on $$A, \vert A \vert \geq 3$$, it must be a dictatorship.

## Footnotes

1. Does not use the outcomes of the previous. We conduct another vote in class. Details can be found here