Quality & Quantity (2006) 40:897–910 © Springer 2006
Optimal Social Design: Estimating a Social
Welfare Function from Questionnaire Data
Institute of Educational Technology, Open University, Walton Hall, Milton Keynes, MK7
6AA, UK. E-mail: email@example.com
Abstract. A social design x evokes a response y from a set of individuals. The value of the
design is expressed in terms of a social welfare function which is derived from Arrow’s for-
mulation of social choice. Making certain simplifying assumptions the social welfare func-
tion can be expressed in terms of individuals’ ideal designs. A method for estimating the
social welfare function from quite limited empirical evidence is developed. The method is
applied to an educational case study. There was considerable variation in individuals’ ideal
designs. The components of the social welfare were estimated: the welfare ideal, the popula-
tion sensitivity, the population variation, the deviation from the ideal and the welfare ceiling.
Methodological problems are discussed.
Key words: Optimal control, quality, questionnaire data, social choice, social welfare
The mathematical modelling of a situation involves an interplay between
‘ordinary’ understanding and mathematical understanding, and between
theory and empirical evidence. In particular this applies to the mathemat-
ical modelling of control systems (Dutton et al., 1997, p. 27). For exam-
ple optimal control requires a knowledge of the reward function, both in
terms of its general form and in terms of the values of its parameters. This
knowledge can come about partly by making simplifying assumptions and
partly by obtaining empirical evidence. There may be difﬁculties in obtain-
ing evidence about a wide variety of system states and so there is a need
to consider what one can do on the basis of quite limited information.
Consider Huang’s (2001) discussion of the Taguchi (1986) quality selection
model with input x, output y and a quadratic loss function L. Chen and
Chou (2004) have noted that the quadratic assumption may not always be
realistic and proceed to consider an asymmetric loss function by adjoin-
ing a ‘left quadratic’ with a ‘right quadratic’. However no empirical evi-
dence is presented by either Huang or Chen and Chou as to whether or
not these functional forms are found in practice. Nor is there any empirical
data which would enable the estimation of the parameters of the models.