Applications of Combinatorics and Graph Theory to the Biological and Social Sciences pp 103-137 | Cite as

# Uniqueness in Finite Measurement

## Abstract

This article surveys recent investigations of real sequences *(d* _{1},..., *d* _{ n }) which arise in the theory of measurement from considerations of uniqueness for numerical representations of qualitative relations on finite sets. The sequences we discuss arise from measurement problems which include measurement of subjective probability, extensive measurement, difference measurement, and additive conjoint measurement. The measurement problems lead to sequences with fascinating combinatorial and number-theoretic properties.

The unifying mathematical framework under which we analyze uniqueness of measurement in these diverse areas involves the analysis of the sequences (*d* _{1},...,*d* _{n})as the solutions of finite systems of linear equations. Different applications are translated into different restrictions on the types of linear equations that are admissible for each area.

Two primary concerns of measurement theory are involved in the work being surveyed: (1) axioms for the qualitative relation that are necessary and sufficient, or at least sufficient, for unique represent ability; (2) the structure of sets of unique solutions. The latter concern leads to combinatorial and number-theoretic problems involving characterizations of unique solutions, counts of numbers of unique solutions, and extreme-value questions. Definitive results and presently open problems are described for the areas covered by the basic theory.

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### Primary References

- [FMR]Fishburn, P.C., H.M. Marcus-Roberts & F.S. Roberts,
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