On strict ranking by pairwise comparisons

1. Problem Statement

The paper addresses a fundamental challenge in the Pairwise Comparison (PC) method: obtaining a strict ranking (where no two items share the same rank) from a matrix of pairwise comparisons, particularly when the input data is inconsistent.

• Context: In Multi-Criteria Decision Making (MCDM), alternatives are compared pairwise to derive weights. However, human judgment often leads to inconsistencies (violations of transitivity, e.g., $A > B$, $B > C$, but $A < C$).
• The Gap: Traditional methods often rely on "consistency procedures" (e.g., orthogonal projection) to force a matrix into a consistent state before deriving weights. The author argues that these standard procedures can destroy the inherent ordering of the data, potentially collapsing distinct items into equal ranks or altering the relative order of items.
• Goal: To establish a mathematical framework where a strict ranking can be derived directly from an inconsistent matrix without necessarily forcing global consistency first, and to propose a minimization problem that guarantees both consistency and strict ranking simultaneously.

2. Methodology and Mathematical Framework

The paper employs tools from Lie theory, topology, and optimization to analyze the structure of PC matrices.

A. The R-Condition and Strict Ranking

The author defines a Strict Ranking not merely as a set of weights, but as a specific topological property of the PC matrix.

• Definition: A matrix satisfies the R-condition if it allows for a permutation $\sigma$ of the items such that $w_{\sigma(1)} < w_{\sigma(2)} < \dots < w_{\sigma(n)}$.
• Mathematical Formulation: This is equivalent to the condition that for all $i, j$, $i = j \iff a_{i,j} = 1$. If $i \neq j$, then $a_{i,j} \neq 1$.
• Admissible Loci: The set of all PC matrices ($PC_n$) is partitioned into connected components. The author identifies admissible loci ($RPC_n$) where a strict ranking exists. These loci are characterized by the signs of the logarithms of the matrix entries (the "characteristic ranking matrix").
• Key Insight: A strict ranking can exist even if the matrix is inconsistent. Consistency is not a prerequisite for establishing a strict order; it is a prerequisite for the existence of a unique set of weights that perfectly reconstructs the matrix.

B. Topological Analysis of Ranking Loci

• The paper proves a bijection between the set of permutations $S_n$ (possible rankings) and the admissible loci in $RPC_n$.
• Theorem 3.6: For $n \geq 3$, there exist non-admissible loci in $RPC_n$. This means there are matrices where the pairwise comparisons are contradictory in a way that prevents any strict ranking (e.g., cyclic dominance that cannot be linearized).
• Instability of Consistency Procedures: The author demonstrates (Theorem 3.8) that standard consistency procedures (like orthogonal projection) are unstable regarding the R-condition. A matrix satisfying the R-condition can be transformed by a consistency procedure into a matrix that:
  1. No longer satisfies the R-condition (items become equal).
  2. Satisfies the R-condition but belongs to a different admissible locus (the ranking order changes).

C. The Proposed Solution: The Functional $\Phi$

To resolve the conflict between consistency and strict ranking, the author proposes a new minimization problem defined by a functional $\Phi: APC_n \to \mathbb{R}^+$.

• Domain: The functional is defined on $APC_n$ (the set of admissible matrices), explicitly excluding matrices where items are already tied ($RCPC_n$).
• Definition:
  $$\Phi(A) = \left( \prod_{i<j} \frac{ii(A)}{(\log a_{i,j})^2 + (ii(A))^{n(n+1)/2}} \right) \times \sum_{i<j} ((\log a_{i,j})^4 + 1)$$
  Where $ii(A)$ is a standard inconsistency indicator.
• Properties:
  1. $\Phi(A) = 0$ if and only if $A$ is consistent AND satisfies the R-condition (strict ranking).
  2. $\Phi(A)$ is undefined (or diverges) if $A$ contains equal items ($a_{i,j}=1$ for $i \neq j$), effectively penalizing non-strict rankings.
  3. The functional is designed to be minimized using gradient methods.

3. Key Results

1. Decoupling Consistency and Ranking: The paper proves that strict ranking is a topological property of the matrix's connected component (locus) and does not require the matrix to be consistent.
2. Failure of Standard Procedures: It is mathematically demonstrated that converting an inconsistent matrix to a consistent one via orthogonal projection can alter the strict ranking or eliminate it entirely.
3. Existence of Non-Admissible Loci: Not all inconsistent matrices can be ranked strictly; some contain inherent contradictions (cyclic structures) that prevent a linear order.
4. The Minimization Theorem: The functional $\Phi$ is constructed such that its global minimum (value 0) corresponds uniquely to a consistent matrix that preserves the strict ordering of the original data.
5. Divergence at Boundaries: The functional $\Phi$ diverges to $+\infty$ as the matrix approaches the boundary of the admissible set (where items become equal), ensuring that optimization algorithms naturally avoid collapsing ranks.

4. Significance and Contributions

• New Paradigm: The paper challenges the conventional wisdom that consistency must be achieved before ranking. It suggests that ranking is a primary structure that can be extracted from inconsistent data, provided the data lies in an "admissible locus."
• Robustness: By introducing the functional $\Phi$, the author provides a theoretical tool to find a "best fit" consistent matrix that respects the original strict ordering, avoiding the pitfalls of standard projection methods.
• Interdisciplinary Application: The work bridges Decision Science (ranking problems), Lie Theory (logarithmic maps, group structures), and Physics (references to Loop Quantum Gravity and holonomies).
• Psychological Insight: The conclusion notes that human judgment often relies on fuzzy linguistic values rather than precise numbers. The author suggests that while the current mathematical model works for technical applications (like wireless networks), human-centric problems might require more complex structures (e.g., non-Abelian Lie groups) to model the "fuzziness" of preference better than simple real numbers.

5. Conclusion

Jean-Pierre Magnot proposes a rigorous mathematical approach to strict ranking that prioritizes the preservation of order over immediate consistency. By defining the R-condition and the functional $\Phi$, the paper offers a solution to the problem of deriving a strict, consistent ranking from inconsistent pairwise comparisons, proving that standard consistency methods are often insufficient or detrimental to the ranking integrity. The paper leaves the practical implementation of the gradient descent on $\Phi$ and the extension to non-Abelian structures for future research.