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Uniqueness in Finite Measurement

  • Peter C. Fishburn
  • Fred S. Roberts
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 17)

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Primary References

  1. [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
  2. [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
  3. [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
  4. [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
  5. [FR2]
    Fishburn, P.C. & F.S. Roberts, Unique finite conjoint measurement, Mathematical Social Sciences 16 (1988), 107–143.MathSciNetMATHCrossRefGoogle Scholar
  6. [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
  7. [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

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Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • Peter C. Fishburn
    • 1
  • Fred S. Roberts
    • 2
  1. 1.AT&T Bell LaboratoriesMurray HillUSA
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickUSA

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