Optimal sequences for pairwise comparisons: the graph of graphs approach

TL;DR

This paper identifies optimal patterns for filling incomplete pairwise comparison matrices to best approximate complete preferences, revealing that star and bipartite-like graphs are optimal, with implications for decision-making processes.

Contribution

It introduces the concept of optimal filling sequences and graph structures for incomplete pairwise comparisons, supported by extensive numerical simulations for up to six alternatives.

Findings

Star graph is optimal among spanning trees.

Optimal graphs are close to bipartite structures.

Regular graphs with balanced degrees are also optimal.

Abstract

In preference modelling, it is essential to determine the number of questions and their arrangements to ask from the decision maker. We focus on incomplete pairwise comparison matrices, and provide the optimal filling in patterns, which result in the closest (LLSM) weight vectors on average to the complete case for at most six alternatives and for all possible number of comparisons, when the underlying representing graph is connected. These results are obtained by extensive numerical simulations with large sample sizes. Many optimal filling structures resulted in optimal filling in sequences -- one optimal case can be reached by adding a comparison to a previous one -- which are presented on graph \color{black} of graphs. The star graph is revealed to be optimal among spanning trees, while the optimal graphs are always close to bipartite ones. Regular graphs also correspond to optimal cases, furthermore regularity is important for all optimal graphs, as the degrees of different vertices are always as close to each other as possible. Besides applying optimal filling structures in given decision making problems, practitioners can utilize the optimal filling sequences in the cases, when the decision maker can abandon the problem at any period of the process (e.g., in online questionnaires).