Measurement-Theoretic Foundations of Questions
鈴木 聡 (Satoru SUZUKI)
駒澤大学総合教育研究部非常勤講師

The theory of a question and its answers is one of the most popular topics in the philosophy oflanguage (cf. Cross and Roelofsen [3]). In this talk, we would like to argue about the crossroads of thetheory of a question and its answers, decision theory, and information theory in terms of measurement
theory (cf. Krantz et al. [4]). According to van Rooij [7], the relevance of a question and its answers
can be determined in terms of how much it contributes to solving a decision problem. The expectedutility (decision value) EUR (Q) of a question Q with respect to a partition R is defined by the average
expected utility of its answers A:
EUR (Q) := ∑ P(A) ·UR (A).
A∈R

On the other hand, the information value IVR (A) of A with respect to R:
IVR (A) := H(R) − HA (R),where H(A) is the entropy of R and HA (R) is the entropy of R with respect to the probability function
conditionalized on A. The expected information value EIVR (Q) of Q with respect to R is defined as
the average information value of A:
EIVR (Q) := ∑ P(A) · IVR (A).
A∈R

There are three kinds of frequently-used proper utility functions (scoring rules):
1. logarithmic utility function,
2. quadratic utility function, and3. spherical utility function.
On the basis of Bernardo (1979)’s theorem, van Rooij concludes that if U is a proper (i.e., ∑A∈R P(A) ·UR (P, A) ≥ ∑A∈R P′ (A) ·UR (P′ , A) for any P and P′ ) and local (i.e., U is defined only by P(A)(P′ (A))
where A ∈ R but not by P(P′ )) and R has more than two cells, then U is a logarithmic utility function
and so EUR (Q) = EIVR (Q) (i.e., the decision value of a question = its information value). Boththe quadratic and spherical utility functions are not local. Among these three types of functions, thelogarithmic utility functions only are both proper and local. Can the logarithmic utility functions be1

more appropriate than the other utility functions? Bickel [1] criticizes the quadratic and sphericalutility functions, whereas Selten [6] criticizes the logarithmic utility functions. According to Carvalho[2], the choice of the most appropriate proper utility function is dependent on the desired properties,
which in turn are dependent on the underlying context. Then the following problem arises: What is anunderlying context to determine whether EUR (Q) = EIVR (Q) or not? In this talk, we try to cope withthis problem in terms of measurement theory. Measurement theory provides such important concepts
as scale types, representation and uniqueness theorems, and measurement types. Scale types have such
categories as
1. ratio scale (unique up to φ (x) = α x(α > 0)),2. interval scale (unique up to φ (x) = α x + β (α > 0)), and
3. ordinal scale (unique up to order).
When U := ψ (P), the forms of ψ can be considered to create an underlying context to connect P to U
and to determine whether EUR (Q) = EIVR (Q) or not. Luce [5]’s theorems may be originally intended
to determine the psychophysical laws that connect the physical scales to psychological scales in termsof measurement theory. But they have more general applicability. According to Luce’s theorems,
when U := ψ (P), if P is considered to be a ratio scale in a wide sense and U is also an ratio scale, thenψ (x) = α xβ . However, it is sufficient that U is as weak as an interval scale (cardinal utility function)
to calculate EUR . So since the derivative of an interval scale is a ratio scale, if P is considered to bea ratio scale in a wide sense and U is an interval scale, then either ψ (x) = β α+1 xβ +1 + γ if β ̸= −1 orα log x + γ if β = −1. We conclude that when P is considered to be a ratio scale in a wide sense andU is an interval scale and R has more than two cells, if ψ (x) = α log x + γ , then U is a logarithmic
function and so EUR (Q) = EIVR (Q), and if ψ (x) = β α+1 xβ +1 + γ , then U is a quadratic or sphericalutility function and so EUR (Q) ̸= EIVR (Q) In this way, the (in)equality of β with −1 in the context

ψ (x) = α xβ that connects a ratio scale P to another ratio scale U determines the (in)equality between
the decision value of a question and its information value. （使用言語：日本語）

参考文献[1] J. E. Bickel. Some comparisons among quadratic, spherical, and logarithmic scoring rules. Decision Analysis,
4:49–65, 2007.
[2] A. Carvalho. An overview of applications of proper scoring rules. Decision Analysis, Articles in Advance,
2016.
[3] C. Cross and F. Roelofsen. Questions, 2020. Stanford Encyclopedia of Philosophy.
[4] D. H. Krantz et al. Foundations of Measurement, volume 1. Academic Press, New York, 1971.
[5] R. D. Luce. On the possible psychophysical laws. The Psychological Review, 66:81–95, 1959.
[6] R. Selten. Axiomatic characterization of the quadratic scoring rule. Experimental Economics, 1:43–62, 1998.
[7] R. van Rooij. Utility, informativity and protocols. Journal of Philosophical Logic, 33:389–419, 2004.

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