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?
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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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what do you mean by "optimal"?
I think (?) the following strategy will beat your proposed strategy more than 50% of the time:
a) bet 1.1x Kelly until you're in the 75th percentile outcome that Kelly would be at or the 25th percentile
b) afterwards, do Kelly
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Optimal in that it induces terminal wealth W* such that E(W/W*) <=1 for all other strategies W (Cover's "Elements of Info Theory" 16.33).
Then by Markov's inequality, P(W>= tW*) <= 1/t, t>=1: an opponent can't outperform W* by a factor t with p > 1/t.
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I think you and are using different notions of optimality
These statements look true to me
But "an opponent can't outperform W* by a factor t with p > 1/t" doesn't mean much, maybe when the opponent outperforms by a factor of t, it outperforms by *much* more than t
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er to be clear does E(W/W*) mean "the linear expected value of ($ made by SBF's strategy / $ made by Kelly)"?
Because if so you can get that to be > 1
Yes, I think's the correct interpretation
The Thorpe paper claims
"Given a strategy A which maximizes E[log X_n(A)] and any other “essentially different” strategy B (not necessarily a fixed fractional betting strategy), then
lim n→∞ X_n(A)/X_n(B) = ∞ almost surely"
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yeah if each strategy is a coin flip then E(w/w*) = infinity and also E(w*/w) is infinity :P
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