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lambdaclass.com
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Here’s a much more intuitive explanation that doesn’t require any fancy math. Since there’s a 50% chance each time, for any sequence of 2 tosses you can expect to get 1 of each. As it happens, regardless of the order (because multiplication is commutative), you end up with 90% of what you had before at the end of 2 tosses: 0.6 * 1.5 = 0.9. if you lose 10% every 2 tosses, you eventually end up with roughly zilch.

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In this case the wealth converges to a scalar, but that might not be the case and there you need a little bit more complicated math. What we are trying to show is that it is important to calculate the time average instead of the ensemble average. In many social sciences the ensemble average is used instead of the time average. Obviously this has enormous consequences.

If you liked the post I recommend you read this post by Taleb: https://medium.com/incerto/the-logic-of-risk-taking-107bf41029d3

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0.6 * 1.5 = 0.9

I didn’t understand from the article (because I didn’t really understand the article) where the values 0.6 and 1.5 come from. Could you (or @unbalancedparentheses) please explain that?

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You either win 50% (i.e. multiply your current wealth by 1.5) or lose 40% (i.e. multiply your current wealth by 0.6).

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Thank you, that makes sense :)

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You should maximize geometric mean of outcomes, not arithmetic mean. This is called Kelly betting.

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Exactly, Kelly was one of the few that found that. Ole Peters generalized the Kelly criterion with ergodicity economics: https://ergodicityeconomics.com/

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In order to decide if we would accept to play this game for the rest of our lives we have to check how Wn behaves when n→∞.

This is a common mistake, but I’ve never found it presented in this hilarious way before. If I ask you to argue for this method, what is the life expectancy you based it on?

I’m not saying it makes a difference to the conclusion, but if the point of the article is to not make fallacious assumptions, I would at least like an argument for why we assume a person lives an infinite amount of time.

A more sensible argument would maybe show that the wealth is inversely correlated with time, or something. Does anyone know how to produce this?

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That’s actually a good point. In some cases, it can make a whole lot of difference: for example, sum of a finite/infinite number of values of convergent series.

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You just proved that median does not equal to mean, and you stated that you would bet on median instead of mean.

In other words, in parallel universes, there are larger number of you losing money than the number of you earning money. Although if you pool all of your money from all the universes, and redistribute it evenly among all of you, you gain.

I would definitely bet, and I will bet on a fixed dollar amount each time.

By the way, you also proved that E[X] ≠ exp(E[log(x)])