[Paper Review] Preference fusion when the number of alternatives exceeds two: indirect scoring procedures

Pavel Chebotarev, Elena Shamis|arXiv (Cornell University)|Feb 8, 2006

Multi-Criteria Decision Making34 references21 citations

TL;DR

This paper analyzes indirect scoring procedures for aggregating incomplete preferences among more than two alternatives, focusing on whether they satisfy the self-consistent monotonicity (SCM) axiom. It introduces the class of win-loss combining scoring procedures and proves they universally violate SCM, while establishing a sufficient condition for SCM satisfaction. The generalized row sum procedure is shown to satisfy SCM, offering a robust method for preference aggregation under intransitive and incomplete data.

ABSTRACT

We consider the problem of aggregation of incomplete preferences represented by arbitrary binary relations or incomplete paired comparison matrices. For a number of indirect scoring procedures we examine whether or not they satisfy the axiom of self-consistent monotonicity. The class of {\em win-loss combining scoring procedures} is introduced which contains a majority of known scoring procedures. Two main results are established. According to the first one, every win-loss combining scoring procedure breaks self-consistent monotonicity. The second result provides a sufficient condition of satisfying self-consistent monotonicity.

Motivation & Objective

• To evaluate whether indirect scoring procedures satisfy the self-consistent monotonicity (SCM) axiom in preference aggregation with more than two alternatives.
• To examine the limitations of widely used win-loss combining scoring procedures in maintaining consistency under incomplete preference data.
• To identify a sufficient condition for scoring procedures to satisfy SCM, ensuring logical consistency in ranking outcomes.
• To compare known scoring procedures—especially the generalized row sum procedure—under the SCM criterion.
• To lay the groundwork for axiomatic derivation of robust preference aggregation methods in incomplete decision contexts.

Proposed method

• Models individual preferences as incomplete paired comparison matrices with entries 1 (i > j), 0 (j > i), 1/2 (equivalence), or undefined (no opinion).
• Defines a scoring procedure as a neutral and anonymous function mapping preference profiles to real-valued scores for each alternative.
• Introduces the self-consistent monotonicity (SCM) axiom, requiring that if alternative i dominates j in a consistent way across comparisons, then i's score should not be less than j's.
• Proposes a sufficient condition for SCM satisfaction based on a function f defined on multisets of real triples (ap_ij, si, sj), with monotonicity properties in ap_ij, si, and sj.
• Applies the sufficient condition to verify that the generalized row sum procedure satisfies SCM, while proving that all win-loss combining procedures fail it.
• Uses graph-theoretic and Markov chain interpretations to justify the generalized row sum procedure’s consistency and stability.

Experimental results

Research questions

• RQ1 Do common indirect scoring procedures preserve self-consistent monotonicity when preferences are incomplete?
• RQ2 Why do win-loss combining scoring procedures universally fail to satisfy the self-consistent monotonicity axiom?
• RQ3 What mathematical condition ensures that a scoring procedure respects self-consistent monotonicity in incomplete preference aggregation?
• RQ4 How do the generalized row sum and other nonlinear scoring methods compare in satisfying SCM?
• RQ5 Can additional axioms like macrovertex independence or splitting balance further refine the selection of consistent scoring procedures?

Key findings

• All win-loss combining scoring procedures—such as the row sum and generalized row sum—fail to satisfy the self-consistent monotonicity (SCM) axiom, as proven in Theorem 8.
• The generalized row sum procedure satisfies SCM, as shown in Proposition 11, due to its implicit structure satisfying the sufficient condition in Theorem 12.
• A sufficient condition for SCM satisfaction is established: a function f on multisets of (ap_ij, si, sj) triples must be increasing in ap_ij and sj, decreasing in si, and satisfy strict monotonicity when replacing 1 or 0 entries.
• The generalized row sum procedure is derived from a system of linear equations involving a parameter ε and a matrix of relative performance, ensuring stability and consistency.
• The generalized row sum procedure is the only one among the four analyzed that satisfies SCM and is defined on all preference profiles, while others are restricted to indivisible profiles.
• The paper demonstrates that SCM is not overly restrictive, as it allows for some preconceived rankings (e.g., favoring a known strong player), but can be further constrained by axioms like splitting balance.