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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sorry to be clear here by wealth do you mean $ or log($) or something else?
if you mean $ then the first part is false
if you mean log($) then the second is false
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$'s - the amount of $'s you have at the end of the period [1,t] as you take the limit t->inf.
I do not think this is false, actually a long-time known result? It is a concrete claim so could you explain why you believe it is false?
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Replying to @_charlienoyes @danrobinson and 2 others
what do you mean by "optimal"?
I think (?) the following strategy will beat your proposed strategy more than 50% of the time:
a) bet 1.1x Kelly until you're in the 75th percentile outcome that Kelly would be at or the 25th percentile
b) afterwards, do Kelly
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I think you need to throw in a bunch more assumptions to make it true, e.g. that you are forced to bet the same % of your wealth at teach point and are trying to decide what constant % that should be
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That is simply not the case, as long as the outcomes are i.i.d. (which is why my claim is the expected log, not the conditionally expected).
Suggest a gamble dynamic and I will let you do literally anything that you want, and whatever you do will be dominated in the limit.
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I'm also happy to as well. How about we do either:
even money 60% bet
or
trading between cash and one risky asset which follows a GBM of mu=1.0, sigma=1.25
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let's go with even money 60% bet
note, though, the following:
twitter.com/SBF_Alameda/st
Alternately I *think* there's a simple strategy that doesn't require as many simulations, will post soon
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wait ok I think you probably misspoke or made a typo somewhere
your claim is false in way too many ways
I suspect that instead of saying EV($_1/$_2) you meant to say something like EV(log($_1/$_2))
If *that* is what you're saying then I agree with your statement 1 but disagree with statement 2