Marc Le Menestrel
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Papers on the Foundations of Measurement

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by Marc Le Menestrel (18/06/2008)

In these papers, Bertrand Lemaire and myself develop an approach to measurement where an empirical binary relation is represented by both a scale and a factor. The scale is interpreted as measuring the objects over which the relation is defined while the factor is interpreted as measuring the way these objects are measured.

In our first joint-paper “Biased Extensive Measurement: the Homogeneous Case” (published in the Journal of Mathematical Psychology in 2004), Bertrand Lemaire (UMR 8628 du CNRS, Université de Paris-Sud, Mathématiques) and I consider the case of one-dimensional objects and show that we can measure objects with a function unique up to multiplicative transformation (i.e. a ratio-scale) even when the observed relation lacks transitivity of indifference. This lack of discrimination, which prevents the maximization process, is characterized by a unique and constant multiplicative factor which remains distinct from the ratio-scale.

In “Biased Extensive Measurement: the General Case” (published in the Journal of Mathematical Psychology), we generalize these results to sets of multi-dimensional objects. We propose axioms that allow for the representation of interval orders and semiorders with a ratio-scale and a unique multiplicative factor, which is not necessarily constant. These results extend the classical results of extensive measurement [1]

In these papers, the main property required for a relation to be represented by a ratio-scale and a unique factor which “distorts” this ratio-scale is termed “homotheticity” and corresponds to the intuition of scale invariance. A more detailed mathematical analysis in terms of the theory of algebraic representations is presented in “Homothetic Interval Orders” (published in Discrete Mathematics).

In “Ratio-scale Measurement with Intransitivity or Incompleteness: the Homogeneous Case” (published in Theory and Decision in 2006), we return to the restrictive setting of one-dimensional objects and we extend further our first results by relaxing more properties of the standard model. We do not assume completeness nor transitivity and show how we can model a rational individual who strictly prefers an object with a lower utility because of a unique factor that influences his preference towards that object relative to the other.

In “The biased balance: observation, formalism and interpretation of a dissymmetric measuring device” (submitted to the Journal of Mathematical Psychology), I am showing how this approach contributes to the foundations of measurement. Using the example of the biased balance, I apply my approach to the relation between observation, formalism and interpretation, three basic steps of scientific methodology.

Footnotes

[1] For the theory of extensive measurement, see Krantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. (1971), Foundations of measurement, Vol. I. San Diego: Academic Press. Seminal works on semiorders are due to Luce, R.D. (1956), “Semiorders and a theory of utility discrimination”, Econometrica and on interval orders to Fishburn, P.C., (1973), “Interval representations for interval orders and semiorders”, Journal of Mathematical Psychology. and have an economic interpretation with the idea of “thick indifference curves”.


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