Unique nontransitive additive conjoint measurement on finite sets

Abstract

Nontransitive additive conjoint measurement for a binary relation>on a setX ₁ ×X ₂ ×...×X _n ofn-tuples(x ₁,...,x _n ), (y ₁,...,y _n ),... is concerned with the representation

$$(x_1 ,...,x_n ) > (y_1 ,...,y_n ) \Leftrightarrow \sum\limits_{i = 1}^n {\phi _i (x_i ,y_i ) > 0,} $$

whereφ _φ is a skew symmetric real valued function onX _i ×X _i . The representation is said to be unique if, whenever (φ ₁,...φ _n ) and (φ ^′₁ ,...φ ^′ _n ) satisfy it, there is a positive constant λ such thatφ ^′ _i =λφ _i for alli. This paper investigates unique representation for nontransitive additive conjoint measurement when everyX _i is finite. It begins with background on measurement theory, followed by conditions for uniqueness that are based on equations ∑φ _i (x _i ,y _i )=0 that correspond to the symmetric complement of >. We then examine aspects of sets of unique solutions forn=2, for arbitraryn with |X _i |=2 for alli, and forn=3. Various combinatorial and number-theoretic arguments are used to derive results that count and characterize sets of unique solutions.