Cancellation conditions for finite two-dimensional additive measurement
The author partially settles a delicate issue in two-component additive conjoint measurement theory. It is known that a weak order on a finite set \(X_1\times X_2\) admits an additive real-valued order-preserving representation if and only if it satisfies a denumerable set of cancellation conditions \(C(k),\;k=2,3,\dots\). The question of concern here is, given cardinalities \(m,n\) of \(X_1,X_2\), what is the number \(f(m,n)\) of cancellation conditions that need to be checked? Apart for some known special cases of small \((m,n)\), the existing result for general \((m,n)\) is that \(f(m,n)\leq m+n-1\). The present paper shows that for all odd \(m\) and \(n\) greater than \(11\) one also has \(f(m,n)\geq m+n-10\). Thus for large odd \(m,n\), one needs to check approximately \(m+n\) cancellation conditions.
scientific article; zbMATH DE number 1749370
scientific article