A further simplification of the measurement function may be achieved by requiring a special kind of non-interaction of the components which has become known as additive independence. This requires that the equation for decomposable structures is reduced to:

(R, P) >= (R', P' ) <=> [[Phi]]1 (R ) + [[Phi]]2 (P ) >= [[Phi]]1 (R' ) + [[Phi]]2 (P' )

where F is simply the addition function. An example of a non-decomposable structure is given by:

(R, P) >= (R', P') <=> [[Phi]]1 (R ) + [[Phi]]2 (P ) + [[Phi]]1 (R ) [[Phi]]2 (P ) >= [[Phi]]1 (R' ) + [[Phi]]2 (P' ) +

+ [[Phi]]1 (R' )[[Phi]]2 (P' )

* It can be shown that (starting at the other end) given an additively independent representation the properties defined in 1 and 3, and the Archimedean property are necessary. The structural conditions 4 and 5 are sufficient.

Here the term [[Phi]]1 [[Phi]]2 is referred to as the interaction term, its absence accounts for the non-interaction in the previous condition.

We are now in a position to state the main representation theorem.

Theorem

Suppose <R x P, >= > is an additive conjoint structure, then there exist functions, [[Phi]]1 from R, and [[Phi]]2 from P into the real numbers such that, for all R, R' [[propersubset]] R and P, P' [[propersubset]] P:

(R, P) >= (R', P' ) <=> [[Phi]]1 (R ) + [[Phi]]2 (P ) >= [[Phi]]1 (R' ) + [[Phi]]2 (P' )

If [[Phi]]i['] are two other functions with the same property, then there exist constants [[Theta]] > 0, [[gamma]]1, and [[gamma]]2 such that

[[Phi]]1['] = [[Theta]][[Phi]]1 + [[gamma]]1 [[Phi]]2['] = [[Theta]][[Phi]]2 + [[gamma]]2

The proof of this theorem may be found in Krantz et al.[15].

Let us stop and take stock of this situation. So far we have discussed the properties of an additive conjoint structure and justified its use for the measurement of effectiveness based on precision and recall. We have also shown that an additively independent representation (unique up to a linear transformation) exists for this kind of relational structure. The explicit form of [[Phi]]i has been left unspecified. To determine the form of [[Phi]]i we need to introduce some extrinsic considerations. although the representation F = [[Phi]]1 + [[Phi]]2 , this is not the most convenient form for expressing the further conditions we require of F, nor for its interpretation. So, in spite of the fact that we are seeking an additivelyindependent representation we consider conditions on a generalF.