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
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.
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Fishburn, P.C. Unique nontransitive additive conjoint measurement on finite sets. Ann Oper Res 23, 213–234 (1990). https://doi.org/10.1007/BF02204847
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DOI: https://doi.org/10.1007/BF02204847