# 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:
- Broda:
- Instant Runoff:
- 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 forces :
- A/B/C - 45
- B/C/A - 30
- C/B/A - 25

- Scenario forces :
- A/B/C - 45
- B/C/A - 30
- B/C/A - 25

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

## 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 should not change the outcome of the election.

**Monotonicity**: If a voter moves a particular candidate, say , higher up within rankings, should not drop in social ranking.**Condorcet Winner Criterion**/**Condorcet Loser Criterion**:**IIA of Preference Strengths**: Preference strength for a particular ranking, vs , is number of spots is above . If two spots have the same preference ranking for all voters, they should be the same in social ranking.**Cancellation of Ranking Profiles**: Suppose . If we have the preference profiles , , , 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 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 is a strategy-proof voting rule on , it must be a dictatorship.