Unique nontransitive measurement on finite sets

Two themes in the theory of measurement that have been studied extensively in the past few years are numerical representations of nontransitive binary comparison structures and uniqueness in finite measurement systems. This paper brings the two together by exploring the solutions to a nontransitive, additive model that are unique up to multiplication by a positive constant. The model relates to various contexts including decision under risk, evaluation of objectives, comparative probability, and voting theory. The family of unique solutions for the model is shown to be extremely rich and varied.