Schulze Method: Beat-path Condorcet-consistent rank aggregation

Overview

Rank aggregation (beat-path, polynomial time, Condorcet-consistent). Output typically rank_position (lower value = preferred).

Strengths

• •Method-specific: Rank aggregation (beat-path, polynomial time, Condorcet-consistent)
• •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 SCHULZE-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'SCHULZE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Input is a rank matrix (1=best, m=worst per voter)
• •Hatalı: 'SCHULZE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Each voter ranks all alternatives
• •Hatalı: SCHULZE'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: Schulze 2011, p.273 Sec.2
2. 2.Adım 2 (F2): Step 2: Pairwise preference matrix d[a,b] = #{k: a ≻_k b}. Formül: d[a,b] = \sum_{k=1}^{K} \mathbb{1}[a \succ_{k} b] Anchor: Schulze 2011, p.273 Eq.(1)
3. 3.Adım 3 (F3): Step 3: Strongest path p[a,b] via widest-path Floyd-Warshall. Formül: p[a,b] = \max_{\text{paths}\ a\to b} \min_{(u,v)\in\text{path}} d[u,v] Anchor: Schulze 2011, p.274 Eq.(2)
4. 4.Adım 4 (F4): Step 4: Schulze winner: a beats b iff p[a,b] > p[b,a]; derive ranking. Formül: a \succ_{S} b \iff p[a,b] > p[b,a] Anchor: Schulze 2011, p.275 Theorem 1

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