If each pot has the same quality of opportunity (coins with same bias, but different coins), my intuition is you want to bet more than Kelly because the risk of a string of bad luck putting you into a deep hole is lower.
The more coins you have the more certain outcomes are.
Conversation
The issue with that logic is that you can’t rebalance between pots. So when your pot gets in a hole, it is stuck in it
2
My intuition agrees with - note that you can bet *more* than Kelly and still have positive growth rate (but less than Kelly)
So maybe we can just bet Kelly fraction + epsilon in each pot, and while the *individual* pots have lower growth rate, their *sum* might not
2
i.e. E[log(X)], E[log(Y)] < E[log(Z)] does not imply that E[log((X+Y)/2)] < E[log(Z)]
Let me see if I can turn this into a proof either way
1
3
OK so I think:
1) is wrong given his assumptions
2) if you *also* assume that there aren't two pots which have coins with *exactly* the same growth rate, is right given his assumptions
3) his assumptions are stupid
4) given good assumptions, he's wrong
2
3
If there are two pots with the same growth rate, and you can’t rebalance between them, you think you can beat the growth rate?
3
And the objectionable assumption that I’m making is the infinite time horizon, right?
And I guess the assumption “I prefer A to B if A outperforms B with probability 1” but that seems really hard to object to (apart from objecting to the infinite time horizon)
1
nope, there are a lot of bad assumptions!
1) infinite time horizon
2) the prob dist of the coins: your model relies on the claim that with p --> 1, ETH > $999999999999999 eventually
3) that you can't rebalance
4) that you can't choose a better way to trade
...cont'd
1
1
5) that you assign prob 0 to blowing out to ~$0 and staying there with an AMM
6) that you lose only epsilon to each arb
7) (that you are log-maxing but while I think that's wrong it's not *stupid*)
8) and, yes, the thing you said it's hard to object to.
you still haven't responded to
1
Fair, I didn’t mean every assumption in the post about rebalancing portfolio growth, just in this separate hypo about pots. But I see why you think it’s an unrealistic scenario.
1
1
Show replies