Some notes on non-reciprocal matrices in the multiplicative pairwise comparisons framework

Abstract

In most pairwise comparisons methods such as the Analytic Hierarchy Process (AHP) it is assumed that pairwise comparisons are reciprocal, since this is a necessary condition for consistent judgments. However, several empirical studies have shown that the condition of reciprocity is not satisfied when dealing with real human preferences, which might be significantly non-reciprocal due to inherent cognitive biases. This empirical evidence indicates that the study of non-reciprocal pairwise comparisons matrices should not be neglected when dealing with real decision-making processes. However, the literature on this topic is scarce and fragmented. The aim of our study is to fill this gap by discussing advantages and disadvantages of using non-reciprocal judgements multiplicative pairwise comparisons (MPCs), reviewing existing literature and introducing a new measure of non-reciprocity with some natural and desirable properties. In addition, we perform Monte Carlo simulations on randomly generated non-reciprocal MPC matrices and provide percentile tables allowing decision makers to decide whether a level of non-reciprocity of a given MPC matrix is acceptable or not.

Keywords:

• Pairwise comparisons
• multiple-criteria decision making
• reciprocity
• consistency

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 It should be noted that the generation of truly “random” values is rather impossible in general and MS Excel uses what is called a pseudorandom number generator, the Mersenne Twister algorithm

Additional information

Funding

Grant Agency of the Czech Republic (no. 21-03085S).