Combining Mathematical and Simulation Approaches to Understand the Dynamics of Computer Models
Why Read This Chapter?
To learn how to better understand the dynamics of computer models using both simulation and mathematical analysis. Our starting point is a computer model which is already implemented and ready to be run; the objective is to gain a thorough understanding of its dynamics. Combining computer simulation with mathematical analysis can help to provide a picture of the model dynamics that could not be drawn by using only one of the two techniques.
This chapter shows how computer simulation and mathematical analysis can be used together to understand the dynamics of computer models. For this purpose, we show that it is useful to see the computer model as a particular implementation of a formal model in a certain programming language. This formal model is the abstract entity which is defined by the input-output relation that the computer model executes, and can be seen as a function that transforms probability distributions over the set of possible inputs into probability distributions over the set of possible outputs.
It is shown here that both computer simulation and mathematical analysis are extremely useful tools to analyse this formal model, and they are certainly complementary in the sense that they can provide fundamentally different insights on the same model. Even more importantly, this chapter shows that there are plenty of synergies to be exploited by using the two techniques together.
The mathematical approach to analyse formal models consists in examining the rules that define the model directly. Its aim is to deduce the logical implications of these rules for any particular instance to which they can be applied. Our analysis of mathematical techniques to study formal models is focused on the theory of Markov Chains, which is particularly useful to characterise the dynamics of computer models.
In contrast with mathematical analysis, the computer simulation approach does not look at the rules that define the formal model directly, but instead tries to infer general properties of these rules by examining the outputs they produce when applied to particular instances of the input space. Thus, conclusions obtained with this approach may not be general. On a more positive note, computer simulation enables us to explore formal models beyond mathematical tractability, and we can achieve any arbitrary level of accuracy in our computational approximations by running the model sufficiently many times.
Bearing in mind the relative strengths and limitations of both approaches, this chapter explains three different ways in which mathematical analysis and computer simulation can be usefully combined to produce a better understanding of the dynamics of computer models. In doing so, it becomes clear that mathematical analysis and computer simulation should not be regarded as alternative – or even opposed – approaches to the formal study of social systems, but as complementary. Not only can they provide fundamentally different insights on the same model, but they can also produce hints for solutions for each other. In short, there are plenty of synergies to be exploited by using the two techniques together, so the full potential of each technique cannot be reached unless they are used in conjunction.
KeywordsComputer Simulation Markov Chain Computer Model Formal Model Mathematical Analysis
The authors have benefited from the financial support of the Spanish Ministry of Education and Science (projects DPI2004-06590, DPI2005-05676 and TIN2008-06464-C03-02), the Spanish Ministry for Science and Innovation (CSD2010-00034), and the JCyL (projects VA006B09, BU034A08 and GREX251-2009). We are also very grateful to Nick Gotts, Bruce Edmonds, Gary Polhill, and Cesáreo Hernández for many extremely useful discussions.
- Arthur WB, Holland JH, LeBaron B, Palmer R, Tayler P (1997) Asset pricing under endogenous expectations in an artificial stock market. In: Arthur WB, Durlauf S, Lane D (eds) The economy as an evolving complex system II. Addison-Wesley Longman, Reading, pp 15–44Google Scholar
- Axelrod RM (1997a) Advancing the art of simulation in the social sciences. In: Conte R, Hegselmann R, Terna P (eds) Simulating social phenomena (Lecture notes in economics and mathematical systems, 456). Springer, Berlin, pp 21–40Google Scholar
- Axtell RL (2000) Why agents? On the varied motivations for agent computing in the social sciences. In: Macal VM, Sallach D (eds) Proceedings of the workshop on agent simulation: applications, models, and tools, Argonne National Laboratory, Argonne, pp 3–24Google Scholar
- Axtell RL, Epstein JM (1994) Agent based modeling: understanding our creations. The Bulletin of the Santa Fe Institute, Winter 1994, pp 28–32Google Scholar
- Balzer W, Brendel KR, Hofmann S (2001) Bad arguments in the comparison of game theory and simulation in social studies. J Artif Soc Soc Simul 4(2), http://jasss.soc.surrey.ac.uk/4/2/1.html
- Edmonds B (2001) The use of models: making MABS actually work. In: Moss S, Davidsson P (eds) Multi-agent-based simulation (Lecture notes in artificial intelligence, 1979). Springer, Berlin, pp 15–32Google Scholar
- Edmonds B (2005) Simulation and complexity: how they can relate. In: Feldmann V, Mühlfeld K (eds) Virtual worlds of precision: computer-based simulations in the sciences and social sciences. Lit-Verlag, Münster, pp 5–32Google Scholar
- Edwards M, Huet S, Goreaud F, Deffuant G (2003) Comparing an individual-based model of behaviour diffusion with its mean field aggregate approximation. J Artif Soc Soc Simul 6(4), http://jasss.soc.surrey.ac.uk/6/4/9.html
- Ehrentreich N (2002) The Santa Fe Artificial Stock Market re-examined: suggested corrections (Betriebswirtschaftliche Diskussionsbeiträge Nr. 45/02). Wirtschaftswissenschaftliche Fakultät, Martin-Luther-Universität, Halle/Saale, http://econwpa.wustl.edu:80/eps/comp/papers/0209/0209001.pdf
- Epstein JM (2006) Remarks on the foundations of agent-based generative social science. In: Judd KL, Tesfatsion L (eds) Handbook of computational economics, vol. 2: agent-based computational economics. North-Holland, Amsterdam, pp 1585–1604Google Scholar
- Epstein JM, Axtell RL (1996) Growing artificial societies: social science from the bottom up. Brookings Institution Press/MIT Press, CambridgeGoogle Scholar
- Evans A, Heppenstall A, Birkin M (2013) Understanding simulation results. Chapter 9 in this volumeGoogle Scholar
- Flache A, Hegselmann R (2001) Do irregular grids make a difference? Relaxing the spatial regularity assumption in cellular models of social dynamics. J Artif Soc Soc Simul 4(4), http://jasss.soc.surrey.ac.uk/4/4/6.html
- Galán JM, Izquierdo LR (2005) Appearances can be deceiving: lessons learned re-implementing Axelrod’s ‘Evolutionary approach to norms’. J Artif Soc Soc Simul 8(3), http://jasss.soc.surrey.ac.uk/8/3/2.html
- Galán JM et al (2009) Errors and artefacts in agent-based modelling. J Artif Soc Soc Simul 12(1), http://jasss.soc.surrey.ac.uk/12/1/1.html
- Gilbert N (1999) Simulation: a new way of doing social science. Am Behav Sci 42(10):1485–1487Google Scholar
- Gilbert N (2007) Agent-based models, vol 153, Quantitative applications in the social sciences. Sage, LondonGoogle Scholar
- Gilbert N, Troitzsch KG (1999) Simulation for the social scientist. Open University Press, BuckinghamGoogle Scholar
- Gotts NM, Polhill JG, Adam WJ (2003b) Simulation and analysis in agent-based modelling of land use change. In: Online proceedings of the first conference of the European social simulation association, Groningen, The Netherlands, 18–21 Sept 2003, http://www.uni-koblenz.de/~essa/ESSA2003/proceedings.htm
- Gotts NM, Polhill JG, Law ANR, Izquierdo LR (2003d) Dynamics of imitation in a land use simulation. In: Dautenhahn K, Nehaniv C (eds) Proceedings of the second international symposium on imitation in animals and artefacts, University of Wales, Aberystwyth, 7–11 April 2003, pp 39–46Google Scholar
- Grinstead CM, Snell JL (1997) Chapter 11: Markov chains. In: Grinstead CM, Snell JL (eds) Introduction to probability (Second revised edition). American Mathematical Society, Providence, pp 405–470, http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html
- Hegselmann R, Flache A (2000) Rational and adaptive playing. Anal Krit 22(1):75–97Google Scholar
- Holland JH, Miller JH (1991) Artificial adaptive agents in economic theory. Am Econ Rev 81(2): 365–370Google Scholar
- Huet S, Edwards M, Deffuant G (2007) Taking into account the variations of neighbourhood sizes in the mean-field approximation of the threshold model on a random network. J Artif Soc Soc Simul 10(1), http://jasss.soc.surrey.ac.uk/10/1/10.html
- Izquierdo SS, Izquierdo LR (2006) On the structural robustness of evolutionary models of cooperation. In: Corchado E, Yin H, Botti VJ, Fyfe C (eds) Intelligent data engineering and automated learning – IDEAL 2006 (Lecture notes in computer science, 4224). Springer, Berlin/Heidelberg, pp 172–182Google Scholar
- Izquierdo LR, Polhill JG (2006) Is your model susceptible to floating point errors? J Artif Soc Soc Simul 9(4), http://jasss.soc.surrey.ac.uk/9/4/4.html
- Izquierdo LR, Galán JM, Santos JI, Olmo R (2008a) Modelado de sistemas complejos mediante simulación basada en agentes y mediante dinámica de sistemas. Empiria 16:85–112Google Scholar
- Izquierdo SS, Izquierdo LR, Gotts NM (2008b) Reinforcement learning dynamics in social dilemmas. J Artif Soc Soc Simul 11(2), http://jasss.soc.surrey.ac.uk/11/2/1.html
- Izquierdo LR, Izquierdo SS, Galán JM, Santos JI (2009) Techniques to understand computer simulations: Markov chain analysis. J Artif Soc Soc Simul 12(1), http://jasss.soc.surrey.ac.uk/12/1/6.html
- Mabrouk N, Deffuant G, Lobry C (2007) Confronting macro, meso and micro scale modelling of bacteria dynamics. In: M2M 2007: Third international model-to-model workshop, Marseille, France, 15–16 Mar 2007, http://m2m2007.macaulay.ac.uk/M2M2007-Mabrouk.pdf
- Richiardi M, Leombruni R, Saam NJ, Sonnessa M (2006) A common protocol for agent-based social simulation. J Artif Soc Soc Simul 9(1), http://jasss.soc.surrey.ac.uk/9/1/15.html
- Suber P (2002) Formal systems and machines: an isomorphism (Electronic hand-out for the course “Logical Systems”, Earlham College, Richmond, IN) http://www.earlham.edu/~peters/courses/logsys/machines.htm
- Takadama K, Suematsu YL, Sugimoto N, Nawa NE, Shimohara K (2003) Cross-element validation in multiagent-based simulation: switching learning mechanisms in agents. J Artif Soc Soc Simul 6(4), http://jasss.soc.surrey.ac.uk/6/4/6.html
- Vilà X (2008) A model-to-model analysis of Bertrand competition. J Artif Soc Soc Simul 11(2), http://jasss.soc.surrey.ac.uk/11/2/11.html
- Wikipedia (2007) Structured program theorem, http://en.wikipedia.org/w/index.php?title=Structured_program_theorem&oldid=112885072
- Wilensky U (1999) NetLogo, http://ccl.northwestern.edu/netlogo