Maybe one way to frame this: ignore the infinite time horizon point—imagine it’s a one-shot.
And ignore utility as a function of wealth—let’s just talk in terms of raw utility (the output of the utility function).
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If presented a gamble, I would not necessarily want to maximize EV(utility).
I am not indifferent between 99% chance of 0 utility and 1% chance of 100, and 100% chance of 1.
That preference is NOT a statement about my function of wealth->utility. Right?
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So my objection to the maximize-EV(wealth) heuristic is that over time it will tend inevitably to box me into an outcome where almost all of my utility is enjoyed by a version of me that lives in a vanishingly unlikely world—regardless of what my utility function on wealth is
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Your heuristic is that you prefer strategy A to strategy B if A has higher EV(utility). My heuristic is that I always prefer strategy B to strategy A if B leads to higher utility with probability (1 - epsilon)—regardless of how high my utility would be in the epsilon case.
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Now, whether Kelly actually outperforms your strategy with probability (1 - epsilon) does depend on some assumptions, including that we take the limit as t goes to infinity. But I think that is a separate disagreement
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Do you agree that it’s coherent (and indeed kinda reasonable) for me to prefer the gamble where my utility is almost surely higher, even if my average utility across all outcomes is lower because I miss out on a very rare outcome with astronomical utility?
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(I agree with )
There are 2 different threads here.
A) how risk seeking should you be?
B) should there exist some function F that you're trying to maximize the EV of?
on (A) I disagree with you but don't think it's obvious, and isn't the primary thing here.
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(In particular I think the correct decision *is* to bite the bullet on twitter.com/danrobinson/st, but think that's not at all necessary for my argument here)
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Replying to @danrobinson @SBF_Alameda and @elliot_olds
So my objection to the maximize-EV(wealth) heuristic is that over time it will tend inevitably to box me into an outcome where almost all of my utility is enjoyed by a version of me that lives in a vanishingly unlikely world—regardless of what my utility function on wealth is
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Moving on to (B):
Say there are 3 possible outcomes; o_1 < o_2 < o_3
You have two options:
i) 100% o_2
ii) p_2 chance of o_1, (1-p_2) chance of o_3
for what p_2 are you indifferent between (i) and (ii)?
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This creates a linear relationship between o_1, o_2, and o_3.
Now take some other option, o_4.
You have two options:
i) 100% o_3
ii) p_3 chance of o_2, (1-p_3) chance of o_4
for what p_3 are you indifferent between (i) and (ii)?
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Keep going for all options.
If you call D(n) = D(o_n, o_{n+1}) the difference in your preference for o_n and o_{n+1}, this will then give you the size of all D(n) relative to each other; in other words it gives a constant k_n for each n such that D(n) = k_n * (D(1))
now define f(n) = d(1) + d(2) + ... + d(n-1).
So f(n) is the difference between o_n and o_1.
I claim that f(n) is the function you're trying to maximize the linear EV of.
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I also claim that if you disagree with this, then you're going to end up being forced to believe weird things like:
"I'd prefer some distribution of outcomes w_1 to another w_2, but I'd prefer 50% o_18, 50% w_2 to 50% o_18, to% w_1"
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