[Paper Review] Theoretical properties of the eigenvector method

Sándor Bozóki, László Csató|arXiv (Cornell University)|Mar 25, 2026

Multi-Criteria Decision Making0 citations

TL;DR

This chapter analyzes five theoretical weaknesses of the right eigenvector method for deriving weights from pairwise comparisons, illustrated by examples, and discusses related open questions.

ABSTRACT

A classical proposal to derive weights from a pairwise comparison matrix is the right eigenvector. The literature has identified some potential weaknesses of this method in previous decades. This chapter discusses five of these issues. First, right-left asymmetry emerges because of the difference between the right and inverse left eigenvectors. Second, group incoherence for choice means that, in group decision-making problems, the ranking given by the aggregated individual weight vectors is not guaranteed to coincide with the ranking derived from the aggregated pairwise comparison matrix. Third, the ranking based on the right eigenvector may depend on the intensity of the preferences, represented by taking a positive power of all comparisons. Fourth, both the ranking position and the normalised weight of an object might change counter-intuitively after modifying a particular comparison. Fifth, the right eigenvector is not necessarily Pareto efficient: a dominating weight vector that approximates each pairwise comparison at least as well, with an improvement in at least one position, could exist. All violations of the theoretical properties are highlighted by illustrative examples. We also present several open questions in order to inspire future research.

Motivation & Objective

• Motivate why the eigenvector method is popular but imperfect for deriving weights from pairwise comparison matrices.
• Identify and describe five core theoretical weaknesses of the right eigenvector method.
• Provide illustrative examples to demonstrate the weaknesses and their practical implications.
• Discuss related concepts (inconsistency, group coherence, scale invariance, monotonicity, Pareto efficiency) and open questions for future work.

Proposed method

• Present the eigenvector method and the left eigenvector relationship, including the reciprocal left eigenvector in the consistent case.
• Review and summarize five identified weaknesses: right-left asymmetry, group incoherence for choice, sensitivity to intensity of preferences, non-monotonicity, and Pareto inefficiency.
• Use explicit matrix examples to illustrate violations and compare right eigenvector to inverse left eigenvector.
• Discuss inconsistency measures (e.g., CI, CR) and how they interact with the properties.
• Reference Monte Carlo simulations and prior studies to support the discussion of differences between weighting methods.

Experimental results

Research questions

• RQ1 What theoretical properties does the right eigenvector method satisfy or violate in comparing alternatives?
• RQ2 Under what conditions do right eigenvector and inverse left eigenvector yield different rankings or weights?
• RQ3 How do issues like group coherence, scale invariance, monotonicity, and Pareto efficiency relate to right-left asymmetry?
• RQ4 To what extent do perturbations of pairwise comparisons affect rankings and weights in the eigenvector method?
• RQ5 What open questions remain for understanding the practical impact of these theoretical weaknesses?

Key findings

• Right-left asymmetry can occur because the right eigenvector and the inverse left eigenvector can yield different weights and rankings in inconsistent matrices.
• Group incoherence for choice can arise in group decision-making when aggregating individual matrices versus aggregating the matrices themselves.
• The eigenvector method is not scale-invariant; scaling the entries of the matrix can change the resulting ranking, unlike the row geometric mean method.
• Rank monotonicity and weight monotonicity can be violated by the eigenvector method, with violations more likely as inconsistency grows.
• The right eigenvector can be Pareto inefficient; there exist dominating weight vectors that better approximate pairwise comparisons in at least one position.
• The row geometric mean method remains Pareto efficient and satisfies several monotonicity properties, highlighting its robustness relative to the eigenvector approach.