The incomplete Analytic Hierarchy Process and Bradley-Terry model: (in)consistency and information retrieval

TL;DR

This paper compares pairwise comparison models, especially the incomplete Analytic Hierarchy Process and Bradley-Terry model, showing they yield identical priority vectors under consistency and exploring optimal comparison subsets.

Contribution

It proves the equivalence of priority vectors from AHP and Bradley-Terry models for consistent comparisons and identifies optimal comparison subsets for small item sets.

Findings

AHP and Bradley-Terry models produce the same priorities for consistent data.

Optimal comparison subsets are consistent across models for small item sets.

Surprising coincidence suggests a more general underlying principle.

Abstract

Several methods of preference modeling, ranking, voting and multi-criteria decision making include pairwise comparisons. It is usually simpler to compare two objects at a time, furthermore, some relations (e.g., the outcome of sports matches) are naturally known for pairs. This paper investigates and compares pairwise comparison models and the stochastic Bradley-Terry model. It is proved that they provide the same priority vectors for consistent (complete or incomplete) comparisons. For incomplete comparisons, all filling in levels are considered. Recent results identified the optimal subsets and sequences of multiplicative/additive/reciprocal pairwise comparisons for small sizes of items (up to n = 6). Simulations of this paper show that the same subsets and sequences are optimal in case of the Bradley-Terry and the Thurstone models as well. This, somehow surprising, coincidence suggests the existence of a more general result. Further models of information and preference theory are subject to future investigation in order to identify optimal subsets of input data.