Imagine you divided your $1b into 1 billion pots of $1 each, managed independently. (A really bad idea!)
The right strategy for each one would be to maximize its own log growth.
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This is why I don't like the Kelly criterion despite loving log utility, it pushes people towards this fallacy.
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I am asserting that if I manage each independent portfolio in the way that maximizes log growth, and you pick some other strategy, then asymptotically I will have exponentially more wealth than you after infinity amount of time with probability 1.
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“In the way that maximizes log growth” of each portfolio individually? Then SBF can pick a different strategy which maximizes log growth of the portfolio of portfolios and he will be wealthier than you.
Imagine you have a billion coins that land heads with p=.51...
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The strategy of maximizing asymptotic log growth in wealth of each pot IS the strategy that maximizes asymptotic log wealth growth of the total set of pots. (GIVEN that we cannot rebalance between pots)
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This seems not true?
Take two pots with wealth x,y
Give them independent coins with p=0.75 of heads
Let m = 2p-1
Pot-Kelly says bet mx, my in each pot
Portfolio-Kelly says bet min(x, (x+y)*m/(1+m^2)) in the first pot
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(There's a derivation of two-simultaneous-bet Kelly here eecs.harvard.edu/cs286r/courses, in example 6.1)
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er I think maybe it's different if for some reason your bets in each pot can't look at how well the other pot is doing in real time, which for some reason I think might be something is assuming?
Agree that's a weird assumption
Even with that assumption it's not at all obvious to me that pot-kelly and portfolio-kelly are the same
e.g. maybe there's a strategy with higher growth rate where I bet more heavily on pots depending on how 'unlucky' they have been historically
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Here's how I'd beat Dan in the one billion coins with p=.51 example even if I couldn't look at results:
On the first flip (or first ten flips) I'd bet each coin's entire pot on the result. Then I'd do Kelly from there on out.
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