In my humble opinion, mathematical modelling is the most creative side of applied mathematics. To me, mathematical modelling is about looking into the real world; translating it into mathematics, solving that mathematics and then applying that solution back into the real world. However, a secret of the trade is that every mathematical model will have its assumptions and limitations.
It is a widely held belief that if you remove a model’s assumption you remove some of it limitations. Some say the more simple the model the better it is. In both cases I believe that these statements are not always true. A simple model with let’s say one thousand assumptions may not be fit for purpose. Why? If a model is true iff all of its assumptions are true and/or robust then in then its results may proved to be unstable. Conversely a model that is so complex, nobody can relate to its results nor connect it to it input may proved to be not fit for purpose.
In my 25+ years as a professional mathematical modeller, I have seen (and some times written) the good, the bad and the ugly models! The Bad is a model too simple to be of use. The ugly way too complicated to be understood and the results to be analysed. A Good mathematical model will have the right amount of transparent and challengeable assumptions, which coupled with an good mathematical approach provides the framework for potential insight into real world problems.
For example, a commercial organisation may use a forecast model to predict market share through time. However a good mathematical model will question the hidden assumption that its competitors will sit back and do nothing while one company take its customers. Added to the further dynamics of a good customer/ bad customer. By definition a “good” customer makes a company profitable while a “bad” customer adds extra costs not profit.
Does a company really want to be number one in the market but the majority of its customer are bad? By extending the gambler’s ruin problem we have a framework of modelling this real world problem. Let’s look at the following figures which shows the predicted market share of six commercial companies and the percentage of good customer. Each company has approximately 50% good customers through time and the ranking of the six companies remain the same.
Now let’s say the lowest performing commercial company in terms of market share wishes to become number one. It may use a simple model that states that its market share will grow independent of what ever its competitors do. Hence, it this was true the respected figures may look like this.
Under these circumstances the future looks bright for the company represented by the Green line. However, there is an underlying assumption that this model prediction is based on; the company’s market share will grow independent of what ever its competitors do. Really? One of the beauties of a good mathematical model is the ability to test the assumptions that underpins it. One strategy the competitors could apply is that they will fight hard to keep its own good customer’s but will persuade its bad customers to leave. What are the implications of this?
Our lowest performing company may become number one but at a cost. It could lose most of its good customers to its competitors making them more profitable.
In conclusion, a model with weak assumptions is a weak model. Nevertheless, a transparent and challengeable assumptions couple with a good mathematical approach, helps with insight into real world problems and will create the right conditions for the model to evolve.