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.


  • 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)