this is never going to get anywhere if people keep confusing the difference between a.s. and 1, and between "tends to zero" and "is zero", and stop assuming that 0/0=0.
Can we please just talk about numbers less than the number of atoms in the universe?
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Also Kelly isn't defined by "almost surely" either, that's an emergent property of it in some very specific infinite scenarios.
Kelly is defined by optimizing for log(wealth)
en.wikipedia.org/wiki/Kelly_cri
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This part really is a language debate, but saying that it is defined by optimizing for log(wealth) IMO makes it sound like it’s driven by preferences about marginal utility of wealth, which it isn’t.
Would prefer “maximizing log return” or “maximizing annualized rate of return.”
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"maximizing log return" is obviously equivalent to maximizing log wealth, it's just subtracting log(start_wealth) from all outcomes. Really not sure why you think that's an important thing to add!
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Perhaps the more important is the latter (annualized return), or equivalently “average log return over time.”
All equivalent. But with these latter terms it is much more obvious why you might want to optimize for them if you care about compound interest
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twitter.com/SBF_Alameda/st
to be clear I think you're not understanding correctly what Kelly is here.
You're saying we should define Kelly as a thing which is *not* Kelly but is instead the other different thing that you believe.
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Replying to @SBF_FTX @danrobinson and 4 others
But "maximizing annualized rate of return" is *not* kelly in general! It's only Kelly in some very specific situations.
e.g. if you only have a single coin flip ever and it's in 1 year then max annualized rate of return = max return = max linear EV != kelly
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The formula for annualized rate of return is (W_t/W_0)^(1/t), right?
After not too many gambles the expectation of this converges pretty quickly.
But you’re right, the formal definition for that does require t->infinity, so withdrawn
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How about “maximizing expected geometric growth rate”?
Or “maximizing average log return over time.”
All equivalent, but this gets at the WHY. I like Kelly because I want to maximize compounding growth, not because I have preferences about the log of my wealth.
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