Decision-making and Fractals v0.5

As I mentioned before in the most broad terms, the process of decision-making could be considered slightly differently At the moment the journal Integrative Psychological and Behavioral Science already published my academic article "Evaluation of decision-making chains and their fractal dimensions".

In fact, a decision-making as a sequence of actions (decisions) to reach certain goal naturally fuses with an idea of translating from one conditional point to the other; all this is happening inside some abstract space. Points or nodes mark up the moments of actual decision-making, while the movement better be compared with a directional vector of a certain length.  

That said, all these philosophical-theoretical considerations along with a contact with a real world, directly point out to the fact that a decision assessment is made on the basis of several criteria; the best way is to consider them as coordinate axis inside our abstract decision space. Hence, supposing them to be independent (it is reasonable theoretically and practically), such space possesses a dimension, called in math a topological dimension.

In that case a decision-making really looks as a chain — a broken line trajectory. In the article I discuss these details with lot of attention, but a real “rout” of such chain is not uniquely defined due to the presence of different random factors; not so precise also, since a real-world problems frequently use quality-driven criteria, even subjective and emotionally affected.

Then “zigzags” of real trajectory could drastically differ from a supposed one. Moreover, a chain is armed on the basis of the “final” decision at each point. Otherwise we try to estimate the consequences, building decision trees and studying different options.

To a very last degree this process has a fractal nature. Both for human-driven and machine-oriented (AI) — decision trees and forests especially. It happens that “designing” options and their exact assessment results in too many questions and a tiny number of answers. At the same time a “side” characteristic of a decision-making process — fractal dimension, became usable in theory and practice.

In this post I shall mention only two aspects, but truly important ones:

1. Fractal dimension of the decision tree always is less than a topological dimension; even it could be one for two criterion case and binary tree (i.e. Yes/No solutions). In other words — the famous practical wisdom that actually we never had a choice...
   
2. Considering longer decision chains, they have a fractal dimension =2 for any topological dimension ≥2. Translating it from mathematical to commoner’s language, it does not matter how many criterions were used for current evaluation (at each separate step). To build strategic (long-lasting) solutions we must select only two truly important and definitely independent criteria, measuring their “far-sight” notes.

   

   An example of "wandering" trajectory on a plane

As a preliminary announce for further posts, I’d recommend to read about fractal dimensions and how to estimate them numerically for a raw data sets.

An example to consider