Condorcet Method - Pairwise majority winner from ranked ballots

Overview

The Condorcet criterion is the gold standard for pairwise voting: a Condorcet winner - when it exists - is universally considered the legitimate group choice. The drawback is the Condorcet paradox: cyclic majorities (A>B>C>A) can prevent any winner. Use Schulze, Kemeny, or Copeland to break ties when no Condorcet winner exists.

Output

rank position, higher is better

Data

Crisp, complete rank

Size

2+ alternatives, 3-12 criteria works best

Used for

Rank aggregation, social choice, preference fusion

How it works

1. 1

   Collect rankings: ranking matrix R where R[i,k] is the rank of alternative i under ranking source k.

   $$R=[r_{ik}{]}_{m\times n},r_{ik}\in \{1,2,\dots ,m\}$$

   Condorcet 1785, Discours préliminaire

2. 2

   For each ordered pair (i,j), count the number of rankings that prefer i over j. p_ij = #{k : r_ik < r_jk}.

   $$p_{ij}=\#\{k:r_{ik}<r_{jk}\},\forall i\ne j$$

   Condorcet 1785, Eq.(maj.)

3. 3

   Build the pairwise dominance digraph: edge i→j iff p_ij > p_ji (strict majority of rankings prefers i to j).

   $$i{\succ }_{\text{maj}}j⟺p_{ij}>p_{ji}$$

   Condorcet 1785, Théorème majoritaire

4. 4

   Identify the Condorcet winner: alternative that beats every other alternative pairwise. If none exists (Condorcet cycle), return partial preorder.

   $$i^{\ast }\text{is Condorcet winner}⟺p_{i^{\ast }j}>p_{j{i}^{\ast }}\forall j\ne i^{\ast }$$

   Condorcet 1785, Théorème du vainqueur

Fits when / Look elsewhere when

Assumptions to verify

• Input is a rank matrix (1=best, m=worst per voter)
• Each voter ranks all alternatives

Edge cases and pitfalls

• •If none exists (Condorcet cycle), return partial preorder.

• Condorcet paradox: 3 voters with rankings (A,B,C), (B,C,A), (C,A,B) produce a cycle - no winner. Check is_cycle_flag in output.

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