Developing Mathematical Models using Archetypal Behaviour
Jon Warwick
London South Bank University
The development of mathematical models to assist in the exploration and understanding of problems that are not, at first sight, obviously mathematical has long been of interest to mathematicians. High School mathematics students, for example, will often be told about the work of Leonard Euler in developing the topology of networks. His starting point, in 1735, was the famous problem of the Konigsberg Bridges, in which the question is asked whether it is possible to walk around the city of Konigsberg crossing each of its seven bridges exactly once. The development of a mathematical solution to this rather practical of problems leads to ideas of vertices, nodes, graphs, edge traversal and Euler trails around graphs. However, the application of mathematical models to areas beyond the realm of the natural sciences had not really been extensive and systematic until the middle of the last century.
During the Second World War, the British Government set up a small group of quantitative scientists with the intention of applying mathematical and statistical methods to military operations, so that best use of very scarce resources could be made through enhanced planning of military operations. This was the birth of Operational Research (OR), and during the years following the war, the ideas that evolved from the OR unit were applied, naturally, to business and management since organisations were also trying to optimise performance using scarce resources. So, OR spread into the management domain, giving rise to the many fields of application of mathematical techniques in business and management which flourish today (we now have not only OR but also management science, business decision support, quantitative decision making, statistical modelling and so on).
The important point to note here is that OR has undoubtedly been successful in helping to organise and optimise the performance of business, and a glance at any OR textbook will give a feel for the vast array of methods and modelling approaches that now exist. The OR field is rich in mathematical models, and a secondary consequence of this has been that many of these approaches have been adopted for use in areas which are not the conventional home territory of business and management. This cross-fertilisation of ideas is a perfectly natural phenomenon, and there are sound reasons as to why it happens. If one is faced with the problem of trying to develop a model to explore a particular system under study, then there are a number of approaches that may be taken (see Morris, 1967). One is to develop new models specific to the field of study, but this is likely to be difficult (mathematical modelling is a skilled and artful process which many students or inexperienced modellers find difficult) and also expensive in terms of time for model development, validation, data capture, experimentation, etc. A second approach, however, is to use the power of analogy and seek out existing models (perhaps from other disciplines) that have been developed for systems that display a similar structure and then customise these models if necessary. The advantage here is that these models are usually well understood in terms of their structure and solution and will have a known set of assumptions and limitations. For example, in the area of library science there are many analogies of the book borrowing process that have been explored, including the use of a predator-prey model taken from biological population modelling. In this analogy we would model a student population (predator population) borrowing (devouring) books (prey population) and later returning the books to the library (prey population reproducing). The availability of the books (prey) governs the number of students using the library (the size of the predator population), so that clear links between the two systems can be made (Warwick 1992).
Another form that analogy might take is to look at the behaviour of the system (or parts of the system) under study over time and see whether the patterns of behaviour observed might suggest a particular existing model structure that we could use to reproduce it. In this article, we will look at two examples in which system dynamics models based on the engineering principles of feedback and control have been used to investigate social interactions where particular types of behaviour have been modelled.
To begin with, we need to say a little bit about feedback loops and the types of behaviour they produce. Engineers generally recognise that feedback occurs in two forms – positive feedback and negative feedback. Positive feedback is a reinforcing process in which a variable might increase or decrease without limit over time. This could be good or bad depending on the variable concerned (producing virtuous or vicious circles) in that, for example, increasing quality is good but increasing costs are not. Negative feedback is a balancing process so that the feedback loop seeks to reach a stable, unchanging position. Thus, observing the behaviours shown in figure 1 gives an indication of the types of feedback structure in operation and hence a clue to possible model structure.
Figure 1: Feedback Types
Of course, these feedback processes would not be acting in isolation, and a number of authors have identified combinations of feedback loops that produce certain types of behaviour. In his highly regarded book on the learning organisation, Peter Senge (Senge, 1990) lists a number of system ‘archetypes’ that produce certain types of behaviour. For example, there is the ‘Limits to Growth’ archetype (Figure 2), in which a process feeds on itself producing accelerating growth and expansion, but this eventually comes to a halt and sometimes may reverse into accelerating decline and collapse. This structure is brought about by the combination of a positive feedback process producing growth and a slower acting negative feedback process eventually halting the growth (perhaps through limited resources). Decay occurs if the positive feedback process is caused to reverse and becomes an ever declining rather than ever increasing loop.
Figure 2: Limits to Growth Archetype (Senge, 1990)
A second archetype is the ‘Tragedy of the Commons’, in which individuals or departments have use of a commonly available but limited resource. Initially they are rewarded for using it, but returns start to diminish and so they work harder and harder for no additional return, and eventually the resource is used up. The structure is shown in Figure 3.
Figure 3: Tragedy of the Commons (Senge, 1990)
It is also possible to combine the archetypes into more complex meta-archetypes which exhibit complex, but still predictable, behaviour. Senge likens the simple feedback loops to words which can be combined into sentences (archetypes) which can themselves be combined into paragraphs (meta-archetypes), each structure conveying more complex meaning than the previous. We will restrict our discussion to archetypes only. Examples of the two archetypes described above are not difficult to find within the literature, and we now consider two areas from which examples may be drawn to illustrate the use of these engineering principles within social settings.
The need to make best use of scarce resources has been particularly acute within the Higher Education sector, and there have been many examples of the use of modelling in the planning of effective resource management (for example Kennedy and Clare, 1999). Two (of the many!) fundamental problems in Higher Education planning are how to try and forecast student demand for particular subject areas and how to deal with the allocation of essentially fixed and limited resources. Each of these problems can exhibit archetypal behaviour of the types mentioned above. In the case of forecasting demand for courses, we can consider the case of modelling the pattern of applications to Information Technology (IT) Higher Education courses within the UK. Despite the generally upward trend in overall course applications over the last 15 years, applications to IT courses have, in recent years, seen a very significant reduction, having been through a period of rapid expansion during the late 1990’s. More specifically, if we consider just applications for courses in computer science and software engineering (UCAS, 2005), then during the period 1994 to 2000 these rose by over 68% (from 15,465 to 26,043), and then from 2000 to 2004 fell by nearly 44% (back to 14,632). This, of course, is behaviour that we can recognise as of the ‘Limits to Growth’ type in that the rapid expansion of IT courses fuelled by interest in the internet and all things ‘dot-com’ produced strong reinforcing feedback behaviour. The expansion of internet activity leading up to the year 2000 opened up new fields of IT application within business which generated new Higher Education courses, heightened public interest, and produced plenty of IT graduates to fill job vacancies and push more organisations down the internet path. This expansion process could not, however, continue for long and eventually the balancing loop would assert itself, as the number of IT graduates started to exceed the job opportunities and interest subsided. Of course, the dot-com crash of 2000-2002 halted the growth in demand for IT graduates and, in fact, as demand reduced, the positive feedback loop began to work in reverse, accelerating the decline in student numbers as IT graduates could not find jobs and public perceptions of the IT industry changed. Thus the ‘Limits to Growth’ archetype is a feedback structure that should feature in a model for subject demand and therefore gives an indication of how the model development may start.
The ‘Tragedy of the Commons’ archetype has also been much in evidence in Higher Education Planning (Galbraith, 1998a, 1998b). Much of university planning revolves around the allocation of limited funds to faculties or departments by the university, or to universities by the Government. Unfortunately, this allocation process is often on the basis of performance. Thus, faculties, departments or universities are essentially in competition with each other for the limited funds. The behavioural archetype can be described by considering an example from Galbraith (1998b). He describes a situation quite common in universities in which faculties are encouraged to undertake research by rewarding them with an increased fraction of operating grants based on the number of papers published, research awards obtained, students graduating etc. The tragedy unfolds when all faculties increase their research output so that each activity has a reducing marginal return in terms of incentive achieved (the funds to be allocated are limited). Faculties must increase their activity just to maintain the same level of income, and a faculty working at maximum efficiency has nowhere to go but down! This type of behaviour, in which individuals and groups work harder and harder for less return per effort, can be observed at all levels: nationally between universities, between faculties within universities and between departments within faculties.
Illustration 2: The Growth and Decay of Society
System dynamics has been quite successful in modelling systems which include qualitative variables and human interaction (see for example Sitompul, Tasrif and Taufik, 1997). In the paper “Simulation of a System Collapse: the Case of Easter Island” (Mahon, 1997) we have an example of modelling in which the ‘Limits to Growth’ archetype is played out. The Easter Island population was one which experienced a period of exponential growth after a long period of consolidation, followed by overuse of available but limited resources, and finally collapse. The model produced is actually in the form of a Learning Environment within which the learner can change parameters and learn about the dynamics of the Easter Island society through a series of experiments. The simulation model itself has four main components: a definition of sustainability based on the core concepts of natural capital, produced assets, human capital and social capital; a model of exponential population growth and its relation to sustainability (these two features are the core of the ‘Limits to Growth’ concept); a basic simulation model of the feedback structures; and an allocation model that embodies the standard economic assumptions regarding scarce resources and diminishing returns.
Thus, the model in total looks at the interaction between population dynamics, social capital, natural resources, food production, production of statues, and resource allocation. Although the modelling is simple, it is in the nature of dynamic models such as this that quite complex behaviour can be generated.
In this article we have looked at the way in which the modelling of two essentially social activities (education and the development of society) can be informed by studying dynamic behaviour and relating this to archetypal modelling structures. This process of analogy allows modellers to start building models using existing frameworks rather than begin modelling from scratch, and it means that well understood notions from engineering control theory can be used in a wide variety of settings and contexts.
In his original work developing graph theory, Euler was able to show that the bridges of Konisberg could not be traversed once and once only. He had the ability to develop the necessary mathematical theory to model his problem – these days we have an extensive body of models and archetypes to draw on which makes life rather easier!
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