Sampling of pairwise comparisons in decision-making

Various decision-making techniques rely on pairwise comparisons (PCs) between the involved elements. Traditionally, PCs are provided by experts or relevant actors, and compiled into pairwise comparison matrices (PCMs). In highly complex problems, the number of elements to be compared may be very large. One of the issues limiting PC applicability to large-scale decision problems is the so-called curse of dimensionality, that is, many PCs need to be elicited from an actor, or built from a body of information. In general, when applied to a set of n elements to be compared, the number of PCs that have to be made is n(n−1)/2. When the information in the comparison matrix is complete, the priorities can be obtained. This is the case of decision-making with complete information. However, if there are missing entries due to uncertainty or lack of information, decision-making must be performed from the available incomplete information. The authors have addressed the issue of incomplete information in [5, 6], and have characterized the consistent completion of a PCM using graph theory in [6]. In this contribution, we claim that less than that number of comparisons may be suitable to develop sound decision-making. There is a trivial solution providing a lower bound for the sample size: just produce n − 1 PCs, for example comparing one element with the others. It can be shown that this is equivalent to give directly the priority vector. Here we reduce the number of pairwise comparisons in a decision-making problem by selecting just a sample of n PCs that are able to provide balanced and unbiased (incomplete) information that still produces consistent and robust decisions. Both the size of the sample and its distribution within the PCM are of interest. We address this research within a linearization theory developed by the authors [2] based on optimizing the consistency of reciprocal matrices.