This is why I don't like the Kelly criterion despite loving log utility, it pushes people towards this fallacy.
Conversation
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.
4
1
3
“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...
2
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)
2
1
even if you can't literally xfer $ between pots you can always have 1 pot look at the performance of the other and change its strategy based on that
1
This kind of thing is why I said this, but you told me it didn't matter
Quote Tweet
Replying to @hasufl @SBF_Alameda and 3 others
Awesome—sounds like we’ve hit on a real disagreement (although I think we have to delineate the question more formally because there are probably some underlying different assumptions around what kind of active management is allowed)
1
er ok sorry about that! you might be right.
I think the assumptions necessary for yours to hold are:
1) no xfers
2) can't see how the other is doing
3) goes on literally infinitely
4) assets grow unboundedly upwards in the median case
1
1
1
without any of these yours fails horribly though and they're each ridiculously bad assumptions to make
1
I don’t think 2 is required. Knowing about the other pots doesn’t change the strategy for each individual pot. (If each pot obeys the Kelly criterion—betting a constant proportion of its bankroll—then the total portfolio will also obey the Kelly criterion, right?)
2
1
Quote Tweet
Replying to @danrobinson @elliot_olds and 2 others
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
1
yeah agreed, I think ^ wins if each pot is able to updated based on how the other pots are doing
I think that's only true if the different pots have access to different opportunities, right?
Basically my intuition is that every pot will make the exact same Kelly bets, scaled to their current portfolio size, and that adds up to portfolio Kelly