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.

## Keywords

Subjective Probability Independent Equation Regular Sequence Extensive Measurement Fibonacci Sequence## Preview

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## Primary References

- [FMR]Fishburn, P.C., H.M. Marcus-Roberts & F.S. Roberts,
*Unique finite difference measurement*, SIAM Journal on Discrete Mathematics**1**(1988), 334–354.MathSciNetMATHCrossRefGoogle Scholar - [FO]Fishburn, P.C. & A.M. Odlyzko,
*Unique subjective probability on finite sets*, Preprint, AT&T Bell Laboratories, Murray Hill, NJ, 1986 (Journal of the Ramanujan Mathematical Society, to appear).Google Scholar - [FOR]Fishburn, P.C., A.M. Odlyzko & F.S. Roberts,
*Two-sided generalized Fibonacci sequences*, Preprint, AT&T Bell Laboratories, Murray Hill, NJ, 1987 (Fibonacci Quarterly, to appear).Google Scholar - [FR1]Fishburn, P.C., & F.S. Roberts,
*Axioms for unique subjective probability on finite sets*, Preprint, AT&T Bell Laboratories, Murray Hill, NJ 1987 (Journal of Mathematical Psychology, 33 (1989), in press).Google Scholar - [FR2]Fishburn, P.C. & F.S. Roberts,
*Unique finite conjoint measurement*, Mathematical Social Sciences**16**(1988), 107–143.MathSciNetMATHCrossRefGoogle Scholar - [FRM]Fishburn, P.C. F.S. Roberts & H.M. Marcus-Roberts,
*Van Lier sequences*, Preprint, AT&T Bell Laboratories, Murray Hill, NJ, 1987 (Discrete Applied Mathematics, to appear).Google Scholar - [VL]Van Lier, L.,
*A simple sufficient condition for the representability of a finite qualitative probability by a probability measure*, Discussion paper 8708, Centre d’Economie Mathematique et d’Econometrie, Université Libre de Bruxelles, Brussels, 1987 (Journal of Mathematical Psychology, to appear).Google Scholar

## References

- Brown, J.L., Jr. (1964),
*Zeckendorf’s theorem and some applications, The Fibonacci Quarterly*2, 163–168.MATHGoogle Scholar - De Finetti, B. (1931),
*Sul significato soggettivo della probabilità, Fundamenta Mathematicae*17, 298–329.Google Scholar - Fishburn, P.C. (1982),
*Nontransitive measurable utility, Journal of Mathematical Psychology*26, 31–67.MathSciNetMATHCrossRefGoogle Scholar - Fishburn, P.C. (1986),
*The axioms of subjective probability, Statistical Science*1, 335–345.MathSciNetCrossRefGoogle Scholar - Fishburn, P.C. (1988), Nonlinear
*Preference and Utility Theory*, Baltimore, MD: Johns Hopkins University Press.MATHGoogle Scholar - Jensen, N.E. (1967),
*An introduction to Bernoullian utility theory. I. Utility functions, Swedish Journal of Economics*69, 163–183.CrossRefGoogle Scholar - Kraft, C.H., J.W. Pratt & A. Seidenberg (1959),
*Intuitive probability on finite sets, Annals of Mathematical Statistics*30, 408–419.MathSciNetMATHCrossRefGoogle Scholar - Krantz, D.H., R.D. Luce, P. Suppes & A. Tversky (1971),
*Foundations of Measurement*, Vol. 1, New York: Academic Press.MATHGoogle Scholar - Luce, R.D. (1967),
*Sufficient conditions for the existence of a finitely additive probability measure, Annals of Mathematical Statistics*38, 780–786.MathSciNetMATHCrossRefGoogle Scholar - Roberts, F.S. (1979),
*Measurement Theory with Applications to Decisionmaking, Utility and the Social Sciences*, Reading, MA: Addison-Wesley.MATHGoogle Scholar - Roberts, F.S. (1985),
*Issues in the theory of uniqueness in measurement, Graphs and Orders*(I. Rival, ed.), 415–444. Amsterdam: Reidel.Google Scholar - Roberts, F.S. & R.D. Luce (1968),
*Axiomatic thermodynamics and extensive measurement, Synthese*18, 311–326.MATHCrossRefGoogle Scholar - Roberts, F.S. & Z. Rosenbaum (1988),
*Tight and loose value automorphisms, Discrete Applied Mathematics*22, to appear.Google Scholar - Scott, D. (1964),
*Measurement models and linear inequalities, Journal of Mathematical Psychology*1, 233–247.MATHCrossRefGoogle Scholar