Unique Nontransitive Additive Conjoint Measurement on Finite Sets.

Nontransitive additive conjoint measurement for a binary relation > on a set X sub 1 x X sub 2 x ... x X sub n of n-tuples (x sub 1,...,x sub n), (y sub 1,...,y sub n),... is concerned with the representation (x sub 1,...,x sub n) > (y sub 1,.. ., y sub n) sum from n to i=1 phi sub i (x sub i, y sub i) > 0, where phi sub i is a skew symmetric real valued function on X sub i x X sub i. The representation is said to be unique if, wherever (phi sub 1,..., phi sub n) and (phi' sub 1,... ,phi' sub n) satisfy it, there is a positive constant lambda such that phi' sub t = lambda phi sub t for all i. This paper investigates unique representations for nontransitive additive conjoint measurement when every X sub i is finite. It begins with background on measurement theory, followed by conditions for uniqueness that are based on equations sum phi sub i(x sub i, y sub i) = 0 that correspond to the symmetric complement of >. We then examine aspects of sets of unique solutions for n=2, for arbitrary n with |X sub i| = 2 for all i, and for n = 3. Various combinatorial and number-theoretic arguments are used to derive results that count and characterize sets of unique solutions.