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It is important to note that the fundamental distinction between mathematical
analysis and computer simulation as presented here is not about whether one
uses pen and paper or computers to analyse formal models. We can follow either
approach with or without computers, and it is increasingly popular to do mathemat-
ical analysis with computers. Recent advancements in symbolic computation have
opened up a new world of possibilities to conduct mathematical analyses (using, e.g.
Mathematica©). In other words, nowadays it is perfectly possible to use computers
to directly examine the rules that define a formal model (see Fig. 13.3).
Finally, as so often in life, things are not black or white but involve some shade
of grey. Similarly, most models are not tractable or intractable in mathematical
terms; most often they are partially tractable. It is in these cases where an adequate
combination of mathematical analysis and computer simulation is particularly
useful. We illustrate this fact in Sect. 13.8, but first let us look at each technique
separately. The following two sections provide some guidelines on how computer
simulation (Sect. 13.6) and mathematical analysis (Sect. 13.7) can be usefully
employed to analyse formal models.

13.6 Computer Simulation: Approximating the Exact
Probability Distribution by Running the Model

The previous sections have argued that any statistic obtained from a (stochastically
or deterministically) parameterised model follows a specific probability distribution.
The statistic could be anything as long as it is unambiguously defined; in particular,
it could refer to one or several time-steps and to one or various subcomponents
of the model. Ideally, one would like to calculate the exact probability distribution
for the statistic using mathematical analysis, but this will not always be possible.
In contrast, using computer simulation we will always be able to approximate this
probability distribution to any arbitrary level of accuracy; this section provides basic
guidelines on how to do that.
The output probability distribution—which is fully and unequivocally deter-
mined by the input distribution—can be approximated to any degree of accuracy
by running enough simulation runs. Note that any specific simulation run will be
conducted with a particular certain value for every parameter (e.g. a particular initial
location for every agent) and will produce one and only one particular certain output
(see Fig. 13.3). Thus, in order to infer the probability distribution over the set of
outputs that a particular probability distribution over the set of inputs leads to, there
will be a need to run the model many times (with different random seeds); this is
the so-called Monte Carlo method.
The method is straightforward: obtain as many random samples as possible (i.e.
run as many independent simulations as possible), since this will get us closer and
closer to the exact distribution (by the law of large numbers). Having conducted a