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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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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
In addition to winning > 50% of the time I think my proposed strategy will also end up with E(w/w) >= 1
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wait sorry actually I think given how you phrased it just going all in every time ends up with E(W/W*) > 1
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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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"which is why my claim is the expected log, not the conditionally expected"
what do you mean by this? there isn't any log in the metric you proposed:
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