Reciprocal matrices with random coefficients

Abstract

A method is developed to estimate the average opinion (or core) of a group of people. The method elicits judgments from a smaller group of individuals than the total population. What we obtain is a scattering of values around a core value that is being estimated. Some of those values will be closer to the core and others will lie away from it. The method presented here allows us, given the density of concentration of the judgments, to use to a greater extent those values closer to the core. The method generates a surface which is more like a probability distribution that can be used to estimate the core without treating the data as if it were direct estimates of it. The shape of the distribution that we have shown to be the relevant one is that corresponding to a Dirichlet distribution. Here we show that the only distribution of judgments which yields this type of result is the gamma distribution. Under the assumption of total consistency, if the judgments are gamma distributed, the principal right-eigenvector of the resulting reciprocal matrix of pairwise comparisons is Dirichlet distributed. If the assumption of consistency is relaxed, then the hypothesis that the principal right-eigenvector follows a Dirichlet distribution is accepted if inconsistency is 10% or less.