# Note 8 - More on Ranking Elections

## Table of Contents

## 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 Voting
^{1}: 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:
- Plurality
- Broda Count
- Instant Runoff (a
*mixed method*)

- Pairwise Ranking Methods:
- Condorcet (repeated rounds of binary contests) – Number of pairwise wins
- 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

**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.

**Monotonicity**: If a voter moves a particular candidate, say \(A\), higher up within rankings, \(A\) should not drop in social ranking.**Condorcet Winner Criterion**/**Condorcet Loser Criterion**:**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.**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.