Dodgson Method: Condorcet completion by minimum pairwise swaps

Overview

Pairwise-swap distance: alternative needing fewest swaps to become Condorcet winner. Output typically rank_position (lower value = preferred).

Strengths

• •Method-specific: Pairwise-swap distance: alternative needing fewest swaps to become Condorcet winner
• •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

• •See F.steps and D.parameters for DODGSON-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'DODGSON bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Input is a rank matrix (1=best, m=worst per voter)
• •Hatalı: 'DODGSON bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Each voter ranks all alternatives
• •Hatalı: DODGSON'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 R[i,k]; build pairwise preference count matrix p_ij = #{k: r_ik < r_jk}. Formül: R=[r_{ik}]_{m\times n},\ p_{ij}=\#\{k: r_{ik} < r_{jk}\} Anchor: Dodgson 1876, Sec.1
2. 2.Adım 2 (F2): Step 2: For each alternative i, find the minimum number of adjacent swaps (across all rankings) required so that i becomes the Condorcet winner. Sum of swaps over rankings = Dodgson score s_i. Formül: s_{i} = \min_{\pi\in S_{m}^{n}} \#\{\text{adjacent swaps in } \pi^{(1)},\ldots,\pi^{(n)}\}\ \text{s.t.}\ p_{ij}^{\pi} > p_{ji}^{\pi}\ \forall j\neq i Anchor: Dodgson 1876, Sec.2; Black 1958 Appendix A
3. 3.Adım 3 (F3): Step 3: Dodgson winner = arg min_i s_i (alternative needing fewest swaps). Rank others by ascending swap count. Formül: i^{*} = \arg\min_{i\in\{1,\ldots,m\}} s_{i};\quad \text{Rank}(i) \propto s_{i} Anchor: Dodgson 1876, Sec.3

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