On linear classifications under varying choice probabilities

The paper deals with certain problems connected with the assumption that choice probabilities p_s(x, y) depend on the subject s. A set of postulates is given, which implies the existence of sequences of “classification standards”, i.e., sequences {z_j} such that whenever we have 0 < p_s0(x, z_i) < 1 for some s₀ and i, then p_s(z_i+k, x) = p_s(x, z_i?k) = 1 for all s, and k ≥ 1. Elements of any such sequence {z_j} can serve as boundaries between successive categories of classification based on the following rule: Assign x to jth category if you feel it is “to the right” of z_j and “to the left” of z_j+1. Under the condition stated above this rule is unambiguous, and the resulting classification has the property that every element is assigned to one of the two neighboring categories, regardless who performs the classification.Next, the postulates are enriched so as to imply the existence of “tightest” among such sequences {z_j}, hence leading to a classification with largest number of categories.