ugh sorry try this on for size:
compare Kelly to "all in every time".
a.s. Kelly > all in
but in EV all in > Kelly
so as t --> inf, EV(all in/ kelly) --> infinity even though probability(kelly > all in) --> 1
Conversation
I'm not convinced EV(all in / kelly) -> infinity
I think it can be consistent that
(a) EV[all in] > EV[kelly]
(b) kelly / all in -> infinity almost surely
(c) EV[all in / kelly] -> 0
2
1
limsup[n->inf] (1/n log (any-other/ kelly)) <= 0 a.s.
by KKT conditions we have
E[any-other/kelly] <= 1
did not forget a log
and then yea i think E[all-in/kelly] = 0 in particular
1
2
1
yep I always agreed with the EV statement
(I also just found projecteuclid.org/download/pdf_1, which has the same statement and a different looking proof, starting page 72)
2
1
the proofs above do appear to assume the same dice rolls btw
1
1
which is clearly the correct assumption for the purposes of comparing the strategies
separately, since we got into this debate via Uniswap, Bitcoin is not a different price tomorrow for Sam than for me (regardless of what that price ends up being)
1
1
. do you take issue with that assumption? It can also be argued as objectively - not preferentially - correct, and then I’m not sure there are any other outstanding points of contention?
1
twitter.com/SBF_Alameda/st
while the assumption does turn out to be true in the uniswap case it's accidentally so and I claim that the fact that this proof relies on it means that the thing it's maximizing for can't be a reasonable thing to maximize for
1
In particular the property is non-transitive!
So I claim that E(wealth(A)/wealth(B)) is not something you can care about
2
i.e. it would lead you to say things like:
E(wealth(all in on ETH)/wealth(kelly on ETH)) <= 1
E(wealth(all in on BTC)/wealth(kelly on ETH)) > 1
and in fact:
E(wealth(all in on ETH)/(all in on BTC)) > 1
E(wealth(all in on BTC)/(all in on ETH)) > 1