CLASS OF PROBABILISTIC CONJOINT MEASUREMENT MODELS - SOME DIAGNOSTIC PROPERTIES

Let Pax,by be the probability of picking the two-dimensional object ax in the set {ax, by}. This paper investigates necessary and/or sufficient conditions for a number of representations for these probabilities: (f, g, F, K, k, G, u, and v are functions) 1. 1. Pax,by ≤ 1 2 iff f(a) + g(x) ≤ f(b) + g(y), 2. 2. Pax,by = F[f(a) + g(x), f(b) + g(y)], 3. 3. Pax,by = K{k[f(a) + g(x)] - k[f(b) + g(y)]}, 4. 4. Pax,by = K[u(a) v(x) - u(b) v(y)], 5. 5. Pax,by = G[(f(a) + g(x)) (f(b) + g(y))]. The results concerning 1, 2, and 3 are essentially trivial consequences of known facts. The results concerning 4 and 5 are new, and amount to representation theorems for these equations. The cases in which k is a convex or concave function in 3 (of which 4 and 5 are special cases) are also analyzed. © 1979.