• riccardo

Decision making through intuition_1/3_Power Law distribution

Updated: Dec 22, 2018



 

We often try to calculate exact numbers without realizing that they seldom constitute actionable information once we consider their possible variance. Most of the times numbers manifest themselves and what is left to us are questions like “will it keep rising or go back down?”, “is it going to last longer or end soon?”; ranges are often the correct actionable thing to look for.


In life we encounter new personal and business challenges every day but, their nature is likely to be of only few types. In particular, there are three main different situations we often face and we want to discuss them in this 3-post series. Knowing those few different types of situation can allow us to make immediate right intuitive decisions. Let’s go through the first one.


The video that we will link at the end of the post is from the helmet-camera of arguably the best enduro rider in the world, Graham Jarvis. While riding through a single-track passage on his dirt-bike he gets stopped by another rider stuck in the middle of the path by a combination of mud and rocks. This is a common situation in enduro racing where it is possible to have very narrow passages with many riders going through it at the same time. After few seconds waiting for the other rider to proceed, Graham starts noticing an alternative path on his right and probably wondering whether to take the slightly longer path or keep waiting for the other rider to proceed.


That is a recurring situation in our lives: "should we stay with the current losing situation ( investment, customer, partner etc… ) because it will soon turn a winning one or should I accept a loss and change?". Note, it may be the opposite: "should I keep exploiting the current winning condition or get out before it turns a losing one?". In Graham’s case, after waiting few additional seconds, he decides to turn right through the longer path and bypass the other rider slowing him down.


Graham probably made the right decision and that is because he had the right intuition about the possible Power-Law nature of that kind of events. The Power-Law distribution characterizes the first of the three types of situations we will discuss in this series and it describes phenomena like wealth distribution: we may have an average person earning about $75k per year and few individuals with multi-million, or billion, earnings. Those phenomena are characterized by a snow-balling effect where, once you surpass the mean, you are likely to go much further. A person earning $1MM today is likely to go to $2MM the next year because he may have more opportunities and more means to catch them. On the contrary, a person earning $75k today is likely to stay around that number next year. That characteristic also implies a fat-tail distribution where tail events are not that rare. It is possible to see the tail effect from the picture below where the Power-Law distribution is compared to the right part of the Gaussian distribution; the latter clearly drops-off faster than the former, so its probabilities. As we will see, the different tail has big effect on the right decision to make in those situations.



We think Graham acted on the intuition that if an enduro rider gets stuck, he either gets out of it within about 10 seconds or he is likely to take much longer ( inexperienced rider requiring outside help, technical problem etc … ). So, after few seconds Graham decides to go for the few extra seconds it would take him to go around the obstacle being confident it would be just few predictable seconds.


Actually, a more rigorous reasoning about the possible Power Law nature of the situation may go like this: with the length in time of the event characterized by that distribution, it is possible to predict only that if the event has lasted n seconds, it will last other n seconds -- at each moment in time. Therefore, if the rider is able to quantify the few extra seconds that the alternative route would take -- e.g. 10 seconds more -- he may have a good intuition in taking that route after waiting about 10 seconds the other rider. That's because we can expect the stuck rider will be there other 10 seconds, therefore we are indifferent between waiting or bypassing him; conversely, immediately after that moment, we prefer the bypassing route because waiting will take a little more that 10 seconds since we’ve already waited a little more than that.


When facing similar situations, a possible wrong decision is the one based on the following consideration: “well, I have waited so long … it is worth waiting a little more”. Not really, in those situations we should wait only for the average time we think it would take to solve the issue, not much more. After that limit, we should then think about changing because it could take much longer and not just a little more. Note, in the same way, if it is a positive outcome, we should consider staying longer because it may be possible to gain more.


In some way, events that are power-law distributed remind us that life is not always fair.


The key takeaway is: if the possible outcomes are power-law distributed, after something has gone a little longer than we expected (not just in time), we should not expect it will stay close to the mean, it could be one of those big outliers and continue further. In those cases, staying in a negative event or leaving a positive one could be costly.


To conclude, we want to discuss how to understand whether a possible scenario is Power-Law distributed? Mathematical considerations could be made ( e.g. check whether the log-log plot results in a linear graph… ) but, since we want to rely on intuition here, we can ask ourselves: is it possible to have events where the number associated with this phenomenon may rise to different order of magnitude above the mean? An example and anticipation to the next posts: personal wealth goes from tens of thousands of dollars to billions, therefore, Power-Law characteristics. On the contrary, human life-expectancy does not present an average of 75 years and then outliers of 500 years old … it is more likely to be 100 years max, therefore stay close to the mean.


We will discuss this and more in the second post of the series.


 

Graham’s video; the part discussed here starts at 2:05:

https://www.youtube.com/watch?v=J0-0OnNVY-0


We encourage the reader to look at books with similar and more discussion; a possible one:

"Algorithms to Live By: The Computer Science of Human Decisions",

by Brian Christian and Tom Griffiths

 

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