Condorcet Method: Pairwise majority winner from ranked ballots

Overview

Pairwise majority rule: winner beats every other alternative in pairwise contest. Output typically rank_position (lower value = preferred).

Strengths

• •Method-specific: Pairwise majority rule: winner beats every other alternative in pairwise contest
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

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

Assumptions to verify

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

When not to use

• •Cardinal preferences important → use a MAUT method

Edge cases

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

Common pitfalls

• •Hatalı: 'CONDORCET bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Input is a rank matrix (1=best, m=worst per voter)
• •Hatalı: 'CONDORCET bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Each voter ranks all alternatives
• •Hatalı: CONDORCET'yi 'Cardinal preferences important → use a MAUT method' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Collect rankings: ranking matrix R where R[i,k] is the rank of alternative i under ranking source k. Formül: R = [r_{ik}]_{m\times n},\ r_{ik}\in\{1,2,\ldots,m\} Anchor: Condorcet 1785, Discours préliminaire
2. 2.Adım 2 (F2): Step 2: For each ordered pair (i,j), count the number of rankings that prefer i over j. p_ij = #{k : r_ik < r_jk}. Formül: p_{ij} = \#\{k : r_{ik} < r_{jk}\},\quad \forall i\neq j Anchor: Condorcet 1785, Eq.(maj.)
3. 3.Adım 3 (F3): Step 3: Build the pairwise dominance digraph: edge i→j iff p_ij > p_ji (strict majority of rankings prefers i to j). Formül: i \succ_{\text{maj}} j \iff p_{ij} > p_{ji} Anchor: Condorcet 1785, Théorème majoritaire
4. 4.Adım 4 (F4): Step 4: Identify the Condorcet winner: alternative that beats every other alternative pairwise. If none exists (Condorcet cycle), return partial preorder. Formül: i^{*}\ \text{is Condorcet winner} \iff p_{i^{*}j} > p_{ji^{*}}\ \forall j\neq i^{*} Anchor: Condorcet 1785, Théorème du vainqueur

How to cite