In other words, log(wealth) is the ONLY utility function that is consistent with the prefers-an-almost-surely-dominant-strategy heuristic (in the model, in the long run)
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I'm all for log wealth utility, but that doesn't justify its use in this context. More likely you're looking at log(W + w) ~ log(W) + w / W.
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See this is why I don’t want to use the word utility—it sounds like I’m talking about marginal benefit of additional wealth
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If you're looking at a marginal benefit, you definitely don't want to be maximising the log of that benefit.
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Of course not. But again, I am not talking about maximization of subjective enjoyment. I am talking about managing a portfolio in order to maximize wealth
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The model doesn't assume that the portfolio is all of your wealth, but it does assume that it never interacts with the rest of your wealth. More realistic in some cases (like a Roth IRA) than others
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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 actually a math thing now
if you have linear utility you just do the EV max thing
if you have log utility, you would:
a) search out relatively uncorrelated bets for each pot
b) bet waaaay more than Kelly on each one individually
this is what *Kelly* on the overall portfolio would imply!
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Yeah...this is a rather odd statement to make from Dan
Any concavity of the utility function would require an understanding of the correlation structure of the bets for the optimization, else for linear indeed shoving everything into the max EV one across bets is optimal