It's not log(log(wealth)) either! You can repeat the process as much as you want; I will always disregard the infinitesimal outcome no matter how high. I am not trying to maximize an expectation value at all!
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What are you trying to maximize?
I claim that if you are trying to end up with almost surely higher f(wealth), you are maybe also maximizing E[log(f(wealth))]?
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I'm not sure there's a way to frame it as maximization of some value. Here's basically the axiom of the decision theory: if strategy A results in higher wealth than strategy B with probability 1 (en.wikipedia.org/wiki/Almost_su), then I prefer strategy A to strategy B.
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It turns out that under certain assumptions (most unrealistically: infinite time), having the above heuristic means that at each step, I need to maximize the rate of growth of my wealth.
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But that log appears in the OUTPUT of the theory, not as an input assumption (the way it does in Bernoulli's theory of log utility of wealth).
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Yes - even if you aren't *deliberately* trying to maximize the expectation of anything, it may be possible to construct a random variable X whose EV you are equivalently but *inadvertently* maximizing
(I think may be doing this fairly explicitly now)
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And it may be reasonable to call this random variable X your 'utility'
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I think "utility" has misleading connotations because historically it has been used to refer to subjective preferences and circumstances (like what I would be able to spend that wealth on)
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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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to be explicit: are you saying that you are always trying to maximize ev(log(wealth)), or only in the case of infinitely repeating log-identical bets where you can only choose a single % of your wealth to bet each time?
I think the model's assumptions are somewhat weaker than that. The bets don't have to be identical. And I can pick any independent strategy for each one.
But yes, the result is only guaranteed to be true in the model. I wouldn't maximize EV(log(wealth)) in my last ever gamble.
If the outcomes are i.i.d., maximizing ev(log(wealth))) yields an optimal fixed fraction strategy.
If they are not i.i.d., with no restrictions on the distribution of the process, maximizing the conditionally expected log given current information is asymptotically optimal.
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So, to be clear, the fixed "single % of your wealth" strategy follows from the "log-identical bet" assumption (assuming you mean i.i.d. outcomes), not the other way around.
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