Abstract
As more and more human-related variables such as preference, subjective decision making, sense of taste or smell, and so on, are involved in the discussion of measurement and evaluation, the theoretical exploration into the foundation for measurement from a broad perspective is becoming important.
One remarkable character of human-related variables can be seen in their binary relationship. Almost always intransitivity appears. For example, from among a set of objects {A, B, C, …}, take a pair of them and ask the evaluator: “Which do you prefer?” (the pairwise comparison). The evaluator may prefer A to B and B to C among many such comparisons. There is often the case, however, when A and C are paired later casually, the evaluator chooses C rather than A! In such a case, the data are represented as an asymmetric directed graph known as “tournament” which has cycles.
Here, we present an analysis of the strongly connected tournament, sometimes too complicated and difficult to analyse. Paying attention to the transitive subsets appearing along a Hamiltonian cycle we discuss how to construct the Hamiltonian cycle along which more transitive subsets apper. Our purpose is, through such graph-theoretic formalization, to provide a theoretical framework to consider the relationship between a local transitivity, and the grobal intransitivity in the course of evaluation.
We use the operation of “addition of a point”, i.e. insertion of a point to a subcycle, and point out two distinct types of insertions: the degenerate insertion and the generic insertion. As a result, some aspect of strongly connected tournament can be featured by the degree of degeneracy.
