Every time some famous or achieved person talks, there will be disciples eagerly listening to words of wisdom and religiously put them into use.
Let us be skeptical. I have written about prosecutor's fallacy in my post Official Nonsense. In essence, while the probability of success conditioning on a good strategy/talent is high, the probability of good strategy/talent conditioning on success might not be high. In other words, we could not infer that much from success. What they did might not get you anywhere.
Let me do a calibration. Suppose there are four strategies: a,b, c, d. You can also think there are four talent groups. Each individual in the group will produce a random outcome following a normal distribution. The mean (average level) and standard deviation (dispersion) will be different for each groups.For group A, it is N(3,1), B---N(2,2), C---N(1,3), D---N(0,4). In other words, in expectation, group A is the best, not just in terms of average outcome, but also in terms of risks. Group A has the lowest risks. From a mean-variance analysis point of view, group A dominates the other groups. No question. However, when we do the simulation, and we wish to calculate, among those in 90, 95 and 99 percentiles, how much of them come from each group. Here are my results.
90 95 99
A 0.0025 0 0
B 0.0525 0 0
C 0.1050 0 0
D 0.8400 1.0000 1.0000
The results are striking! Most of the successful people come from group D! and all of the super-successful come from group D!
This simulation uses 1000 people in each group.
Let us be robust, what about only 100 people in each group? Here is the result:
90 95 99
A 0 0 0
B 0.0250 0 0
C 0.1000 0 0
D 0.8750 1.0000 1.0000
The moral of the study is NOT that people who are successful took the inferior path. This simulation only cautions against inferring too much from outcomes. The reason I used different means for each group is to demonstrate that, given enough risks, you could expect people to stand out even if they are not the type of people who would normally succeed. It is the triumph of risks in ex post observation against expectation.
Basically the take-away is : worship is unnecessary, and above all, do not worship yourself.
I have attached the Matlab code I used in case curious readers wanna play with different specifications. It is quite robust.
rng(1)
a=normrnd(3,1,100,1);
b=normrnd(2,2,100,1);
c=normrnd(1,3,100,1);
d=normrnd(0,16,100,1);
complete=[a;b;c;d];
achievement90=quantile(complete,0.9);
achievement95=quantile(complete,0.95);
achievement99=quantile(complete,0.99);
m=[a';b';c';d'];
summary=[mean((m>achievement90),2)/0.4,mean((m>achievement95),2)/0.2,mean((m>achievement99)/0.04,2)]
a=normrnd(3,1,1000,1);
b=normrnd(2,2,1000,1);
c=normrnd(1,3,1000,1);
d=normrnd(0,16,1000,1);
complete=[a;b;c;d];
achievement90=quantile(complete,0.9);
achievement95=quantile(complete,0.95);
achievement99=quantile(complete,0.99);
m=[a';b';c';d'];
summary=[mean((m>achievement90),2)/0.4,mean((m>achievement95),2)/0.2,mean((m>achievement99)/0.04,2)]
In general, those with low expectation will be more willing to take risks, since in some sense they have little to lose - they are expected to be less successful anyways. However, the problem is, do all people's outcomes follow normal distribution, especially group D? Certainly they have the potential to achieve higher things, but is the probability of success really the same as the probability of failure for them? I personally think that the probability of falling around and below their expectation would be much higher. While I agree with your point (that probability of good method/expectaion conditioning on success is not necessarily high), I think the model fails to reflect the point that the probability of success conditioning on low expectation should be considerably lower.
ReplyDeleteLet us consider some other distribution other than normal distribution. Let us assume that in group D, Pr(X0.5, that is people from group D are more likely to perform below average. What this assumption will necessarily lead to is that group D has a fat tail to the right---people who do perform above average do much better. This result is due to the fact, we have to make the mean the same. Given this, the result will be only more extreme. The key to the model is that it is not just lucky people who perform above average that enters the fortune list, it is those extreme outliers in the tail that matters. If you are skeptical, use the log-normal case. the correction term of variance in the exponents is about this.
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