I mostly disagree with this: log isn't linear.
So if you separately have $1b on the side and are considering what to do with $1k, then the growth rates are going to be roughly 0.00005% instead of 50%, and the nonlinear terms are going to be much weaker
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Totally, it’s irrelevant. But given that you are managing this $1k as its own pot, might as well manage it well. If you robotically maximize EV for it, then it will get St. Petersburged and end up at 0 with very high probability
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Imagine you divided your $1b into 1 billion pots of $1 each, managed independently. (A really bad idea!)
The right strategy for each one would be to maximize its own log growth.
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This is why I don't like the Kelly criterion despite loving log utility, it pushes people towards this fallacy.
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I am asserting that if I manage each independent portfolio in the way that maximizes log growth, and you pick some other strategy, then asymptotically I will have exponentially more wealth than you after infinity amount of time with probability 1.
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“In the way that maximizes log growth” of each portfolio individually? Then SBF can pick a different strategy which maximizes log growth of the portfolio of portfolios and he will be wealthier than you.
Imagine you have a billion coins that land heads with p=.51...
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The strategy of maximizing asymptotic log growth in wealth of each pot IS the strategy that maximizes asymptotic log wealth growth of the total set of pots. (GIVEN that we cannot rebalance between pots)
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even if you can't literally xfer $ between pots you can always have 1 pot look at the performance of the other and change its strategy based on that
This kind of thing is why I said this, but you told me it didn't matter
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Replying to @hasufl @SBF_Alameda and 3 others
Awesome—sounds like we’ve hit on a real disagreement (although I think we have to delineate the question more formally because there are probably some underlying different assumptions around what kind of active management is allowed)
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er ok sorry about that! you might be right.
I think the assumptions necessary for yours to hold are:
1) no xfers
2) can't see how the other is doing
3) goes on literally infinitely
4) assets grow unboundedly upwards in the median case
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