# Note 6 - Social Choice

## Table of Contents

## Main Question

How do we aggregate the preferences of individuals in a society? Two options: majority rule. If more than two options, might have inconsistent results, assuming voters are rational.

## Study of Voting Mechanisms

Borda was dissatisfied with the voting mechanism in place (plurality^{1}), so he proposed a system, Borda Count, that Condorcet demonstrated could result in the election of a candidate who was undesirable to the majority of the voters.

**Borda Count**: Suppose there are \(N\) different choices. The 1st choice gets \(N\) points, the 2nd gets \(N-1\) points, and the \(i\)th person gets \(N-i\) points.

### Example

- 40 chose A, C, B
- 35 chose B, A, C
- 25 chose C, B, A

Tallying this up, we get:

- A: 215
- B: 195
- C: 190

It seems that society prefers A to B to C. Pairwise, however, B is preferred to A, A is preferred to C, and C is preferred to B. It appears that *transitivity doesn’t hold*!

This is called the **Condorcet Paradox**.

## Preference relations

- Let \(A = \{1, 2, 3, ... , m\}, 2 < m\), be a finite set of candidates.
- For each vote \(i\), we can specify a
**preference**for \(i\): a relationship that, for any pair \(a,b \in A\), specifies which candidate is preferred by voter \(i\), denoted by \(\succ_i\) - \(a \succ_i b\) indicates that \(a\) is preferred to \(b\), with no ties allwoed.
- Assume that the preference relation is
**transitive**and**complete**(specified for every pair of candidates) - Suppose a society consists of \(n\) individuals, each with a
*transitive*preference over the set of candidates \(A\). - A
**preference profile**\(\pi\) is an \(n\)-tuple of preferences \((\succ_1, \succ_2, ...)\). - A
**voting rule**(\(f\)) assigns a unique winner frm \(A\), given a preference profile \(m\). - A ranking rule (\(R\)), given \(\pi\), assigns a
**social ranking**to the members of \(A\). This is a complete and transitive preference relation for the society, denoted \(\triangleright\).

We can see that ranking rule -> voting rule. We can see voting rule -> ranking rule. Does every ranking rule have a voting rule that would give it?

## Properties of Ranking Rules

- Fair ranking rule
- Transitivity (more, list is incomplete)