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I wonder when can complex simulations be formulated as possibly complex but still analytical equations. I'm particularly interested in agent based models, that are frequently used in evolutionary biology.

These models are composed of individuals on a regular grid of cells, and each individual have chances to migrate to neighbouring cells or stay where they are, chances to survive, chances to mate and produce offspring etc. all with probabilities coming from some known basic distributions.

These models look really nice and seem to be very intuitive and flexible (for non-technical people at least) so that you can resemble reality as close as you like. But I get the feeling like we are fooling ourselves for two reasons:

First, these simulations really take a long time, even in optimal conditions with well implemented algorithms with suitable programming languages on powerful computers.

Second, we try to deduce a general system behaviour based on replicates of the simulation and also using several alternative values for some simulation parameters. Selection of alternative parameter values and number of replicates for each parameter set is often based on intuition. Quantitative changes in parameter values may lead to qualitative changes in system behaviour and we may not be able to see whole picture with a few parameter values, and also a couple thousand replicates may not be enough for a reliable inference.

So, back to my question, are there any rules of thumb regarding when a simulation is effectively irreducible to closed-form equations? And as a side questions, are there methods to assess reliability of agent based models?

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The role of an agent based model is primarly one of theory not as an empirical description of reality; it is a "story" of why certain results could happen not a "description" of things we have seen. The purpose of a theory is to give a possible explanation of why certain things happen. Ideally your model is extremely simple and still produce the feature you are looking to explain. A good example is Schelling's segregation model.

The fact that small changes in parameters can lead to large changes in outcomes can be part of the story, but you need to be substantively interested in both the parameter and the outcome, and after being initially surprised, the reader should gain from your model an idea of why such instability happens. Moreover, after you have written your theory you (or some other group) still needs to go into the field and test whether your theory corresponds to reality.

If your simulation is too complex, such that it no longer tells a story that is understandable for humans, then that is a problem. You either need to simplify your simulation, typically by making sure you really focuss on one outcome and one mechanism and trim everything out of your model that is not related to that outcome or mechanism. Alternatively, you could look for a analytical solution. Even if you are not succesful in finding such a analytical solution, the process of looking for one may be enough to give you inspiration on how to simplify or interpret the model.

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