Notes on the existence of a solution in the pairwise comparisons method using the heuristic rating estimation approach

Pairwise comparisons (PC) is a well-known method for modeling the subjective preferences of a decision maker. The method is very often used in the models of voting systems, social choice theory, decision techniques (such as AHP - Analytic Hierarchy Process) or multi-agent AI systems. In this approach, a set of paired comparisons is transformed into one overall ranking of alternatives. Very often, only the results of individual comparisons are given, whilst the weights (indicators of significance) of the alternatives need to be computed. According to Heuristic Rating Estimation (HRE), the new approach discussed in the article, besides the results of comparisons, the weights of some alternatives can also be a priori known. Although HRE uses a similar method to the popular AHP technique to compute the weights of individual alternatives, the solution obtained is not always positive and real. This article tries to answer the question of when such a correct solution exists. Hence, the sufficient condition for the existence of a positive and real solution in the HRE approach is formulated and proven. The influence of inconsistency in the paired comparisons set for the existence of a solution is also discussed.