On Fishburn’s questions about finite two-dimensional additive measurement

Abstract

Cancellation conditions play a central role in the representational theory of measurement for a weak order ≾ on a finite two-dimensional Cartesian product set X=X1×X2. The order has an additive real-valued representation if and only if it satisfies a sequence of cancellation conditions C(2),C(3),.... Given fixed cardinalities m and n for X1 and X2, there is a largest K, denoted by f(m,n), such that some ≾ on X satisfies C(2) to C(K−1) but violates C(K). In a pivotal paper, Fishburn, mentioning some results reported by Krantz, Luce, Suppes and Tversky, shows that f(3,3)=3, f(3,4)=f(4,4)=4. He gives lower and upper bounds for f(m,n), including 4≤f(3,5)≤7, and asks for the exact values of f(m,n) for some small (m,n) such as (3,5), (4,5) and (5,5). In this article, we obtain f(3,5)=4 and make explicit a minimal chain of cancellation violating sequences adequate for the detection of all not additively representable weak orders for (m,n)=(3,3), (3,4) and (3,5).