## 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 (plurality1), 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)