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Fig. 13.2 A computer model can be seen as the implementation of a function that transforms
probability distributions over the set of possible inputs into probability distributions over the set of
possible outputs

over the set of possible inputs into probability distributions over the set of possible
outputs (Fig. 13.2). The rest of the chapter is devoted to characterising this function.

13.5 Tools to Understand the Behaviour of Formal Models

Once it is settled that a computer model can be seen as a particular implementation
of a (potentially stochastic) function in a certain programming language, let us refer
to such a function as the ‘formal model’ that the computer model implements.
As mentioned before, this formal model can be expressed in many different
formalisms—in particular, it can always be expressed as a set of well-defined math-
ematical equations (Leombruni and Richiardi 2005)—and our objective consists
in understanding its behaviour. To do that, we count with two very useful tools:
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mathematical analysis and computer simulation.
The advantages and limitations of these two tools to formally study social
systems have been discussed at length in the literature (see, e.g. Axtell 2000; Axtell
and Epstein 1994; Edmonds 2005; Gilbert 1999; Gilbert and Troitzsch 1999; Gotts
et al. 2003; Holland and Miller 1991;Ostrom 1988). Here we only highlight the
most prominent differences between these two techniques (see Fig. 13.3).
In broad terms, when using mathematical analysis, one examines the rules that
define the formal model directly and tries to draw general conclusions about these
rules. These conclusions are obtained by using logical deduction; hence they follow
with logical necessity from the premises of the formal model (and the axioms of

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We use the term ‘mathematical analysis’ in its broadest sense, i.e. we do not refer to any particular
branch of mathematics, but to the general use of (any type of) mathematical technique to analyse a
system.