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Mathematical models and methods 557
(as opposed to ‘inductive’), that it makes use of set theory, etc. On the other
hand, if we take Thurstone’s multi-factorial analysis, it would have to be called
a multi-dimensional (as opposed to unidimensional) model, unverifiable, etc.,
so that the two systems of classification do not coincide.
What seemed the simplest and clearest answer to the problem was to class
models by their functions in research.
It is therefore proposed to begin by distinguishing two major groups of
models : descriptive models and non-descriptive models. The descriptive models
will be deemed to include measurement, classification and dimensional analysis
models, etc., which obviously do not have an explanatory or theoretical func-
tion: they simply classify or put in order large masses of data.
Among the many non-descriptive models, a distinction is made between
theoretical and inductive models. Inductive models are those seeking to explain
an observed reality: the task may be, for instance, to explain why - to take
Bortkiewicz’s famous example - the proportion of cases in the Prussian army
corps in which horses kicked out proved fatal shows a highly characteristic
distribution: the Poisson type. On the other hand, a theoretical model - or at
least what we are here terming a theoretical model - may be for analysing a
notion (as when Arrow shows that the conception of the general interest implies
contradictory exigencies) or for analysing the consequences of a collection of
axioms expressing a given reality (as in game theory, where there is no question
of forecasting the course of actual games, but rather of determining the prop-
erties of a game defined by a particular set of axioms).
Naturally these distinctions need to be clarified by examples, and in any case
can never be deemed absolute. Further, it wil be necessary to introduce addi-
tional subdivisions within the major groups of models we have defined. Thus it
is obvious that in the class of inductive models, it is relevant to distinguish the
models used in experimental situations from those used in the observational
sciences.
A. Theoretical models
Two types of theoretical models can be distinguished: general models and
specific models. The former deal with subjects defined in very wide terms: thus
we get action theory, diffusion theory, learning theory or imitation theory. The
second group is theoretical inasmuch as these models do not set out directly
to explain observed data, and specific inasmuch as they analyse the consequences
of hypotheses which express specific situations. A few examples wil clarify
these distinctions.
An example of a general theory is the theory of games, originated by von
Neumann and Morgenstern. It may be looked on as a range of models represent-
ing variations on the following simple situation: let us imagine two players A
and B, each having the choice between two decisions: a and b in the case of the
former, c and din the case of the second. Let us assume that at the instant when
each of the players chooses between his two alternative lines of action he is