A few points to consider. By a mathematical approach, I do not mean esoteric equations that I expect you just to accept. Rather, to me the essence of a mathematical approach, to anything, is represented by the two figurines at the top of this page. Math is about distilling the main idea. It's the precision strike, timed just right, selecting words and actions carefully, delivering maximum impact and winning the fight.
The combative figures are also, more literally, appropriate as representations of the competition between companies to gain market share. I have recently introduced a model, built upon a well-established foundation for modeling the choices that consumers make, that takes into account the decisions companies make, as far as whom to market to, in a way that allows companies to optimize their marketing allocation based upon expected return.
The essential idea of the model is exceedingly simple, and to anyone in the real-world, will seem like a no-brainer. Likely for this reason, the idea has not appeared in more academic approaches to this problem. The idea is simply that when considering a marketing allocation, one should consider the effect that the allocation has on the choices of those in the network, rather than simply the choices being currently made on the network.
There may be some skepticism about a model, as opposed to say data, or intuition. From my perspective, a model is simply an organized framework for predicting, collecting, storing, and analyzing data. If you are reading this, you likely understand the importance of software, or computer programs, in performing tasks necessary to the operation of your business. In order to write code, one needs a set of directions to follow that dictate what should be done to which pieces of data, where particular pieces of data should be stored, when they should be recalled from memory, when intermediate computations can be discarded, and so on.
Your intuition dictates what pieces of data your organization collects. The models upon which my method rest are data-driven in the sense of predicting the observed data but making no additional assumptions. The question, then, is whether a model that is optimal (i.e., data-driven) for data that was collected based on intuition is able to predict the outcomes or events that your organization desires to. If so, great, you can rest assured that you are doing things well. Of course, even in this case, there will likely be tweaks that can be made here or there. On the other hand, if models predicated upon collected data are not able to predict events of interest, this means that different pieces of data need to be collected, which may require a modification to one's intuition.
In general, system architecture and analysis is my specialty, as discussed on my website.
Ann Arbor, Michigan, United States