Preference Analysis Using Random Spanning Trees: A Stochastic Sampling Approach to Inconsistent Pairwise Comparisons

Abstract

Eliciting preferences from human judgements is inherently imprecise, yet most decision analysis methods force a single priority vector from pairwise comparisons, discarding the information embedded in inconsistencies. We instead leverage inconsistency to characterise preference uncertainty by examining all priority vectors consistent with the decision maker's judgements. Spanning tree analysis enumerates combinations of evaluation and weighting vectors from pairwise comparison subsets, each yielding a distinct preference vector and collectively defining a distribution over possible preference orderings. Since exponential growth renders complete enumeration prohibitive, we propose a stochastic random walk sampling approach with sample sizes formally established via statistical sampling theory. This enables two key metrics: Pairwise Winning Indices (PWIs), the probability one alternative is preferred to another, and Rank Acceptability Indices (RAIs), the probability an alternative attains a given rank. A notable advantage is applicability to incomplete pairwise comparisons, common in large-scale problems. We validate the methodology against complete enumeration on a didactic example, then demonstrate scalability on a telecommunications backbone infrastructure selection case study involving billions of spanning tree combinations. The approach yields probabilistic insights into preference robustness and ranking uncertainty, supporting informed decisions without the burden of exact enumeration.