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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This is making the assumption that the pots do not have access to the same coins, right? Why would there be an investment opportunity available to one pot that isn't available to the others? That's an extra assumption (and a fairly arbitrary one).
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I think it's a natural assumption!
you have impact on coins. You can't go arbitrarily big.
At some point you have to split off some of your capital.
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Having impact on coins has nothing to do with the pots having heterogenous investment opportunities, right?
You could still have each subpot just make the exact same bets the macro portfolio would make, but scaled down.
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If bets have an effect on the market, then sure, the subportfolio manager would find it useful to know how large the total portfolio is (to know how much capital is following the same strategy).
But it will still be maximizing its own log wealth IN THAT CONTEXT.
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Uh actually, since the log-wealth-maximizing strategy is identical for every pot, all the pots will have identical performance, so technically the question of transparency is irrelevant. The pots will just exactly mirror what the full portfolio would do.
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So they know that the total portfolio is 1 billion * the pot's current value
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"ah yes let's put 0.0015 copies of our master bathroom into each of pots 1, 2, 3, 4, 5, ..."
sure you can do this! but then you're not allowed to have a pot which can be on uniswap in the first place because there is no toilet paper on uniswap
Yes absolutely, "divide $1b cash into $1 portfolios and manage them independently" was not an especially realistic version of how people divide their portfolios
But it's the example that made you say I was unambiguously wrong. Do you take back that statement?
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Yup! I was wrong about you being unambiguously wrong.
Though I still strongly think you're wrong ;)
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Replying to @SBF_FTX @danrobinson and 3 others
hm ok so I think my conclusion here is that we were, in fact, not disagreeing on the math -- we were just disagreeing on the assumptions
though as I understand it your assumptions now include "thinks such that my personal preference function would be equivalent to log-wealth"
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