Kemeny-Young: Optimal rank aggregation minimising Kendall τ disagreement

Overview

Rank aggregation (Kemeny consensus, NP-hard optimisation). Output typically rank_position (lower value = preferred).

Strengths

• •Method-specific: Rank aggregation (Kemeny consensus, NP-hard optimisation)
• •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 KEMENY-YOUNG-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'KEMENY-YOUNG bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Input is a rank matrix (1=best, m=worst per voter)
• •Hatalı: 'KEMENY-YOUNG bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Each voter ranks all alternatives
• •Hatalı: KEMENY-YOUNG'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 K expert rankings R_k. Formül: \mathcal{R} = \{R_{1}, \ldots, R_{K}\} Anchor: Kemeny 1959, p.585
2. 2.Adım 2 (F2): Step 2: Pairwise disagreement count d(σ,R_k) for each candidate consensus σ. Formül: d(\sigma, R_{k}) = \#\{(i,l): \sigma_{i}<\sigma_{l}\ \text{but}\ R_{k,i}>R_{k,l}\} Anchor: Kemeny 1959, p.586
3. 3.Adım 3 (F3): Step 3: Find σ* minimising total Kemeny distance Σ_k d(σ,R_k). Formül: \sigma^{*} = \arg\min_{\sigma\in S_{m}} \sum_{k=1}^{K} d(\sigma, R_{k}) Anchor: Kemeny 1959, p.587 Eq.(2)
4. 4.Adım 4 (F4): Step 4: Kemeny consensus σ* is the final ranking. Formül: \text{ranking} = \sigma^{*} Anchor: Young-Levenglick 1978, p.288 Theorem 1

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