top bet problem
- Thread starter beyondbj
- Start date
London Colin
Well-Known Member
Strictly speaking, I think it must depend on the count (as well as the number of decks and the rules), all of which will affect the variance.beyondbj said:
.76 implies a variance of 1/.76 = 1.32.
Hi Colin,London Colin said:
It makes perfect sense, you can ignore pushes if you rescale the probabilities correspondingly. I.e. p(Win) -> p(Win) / (1 - p(Push)), and p(Loss) -> p(Loss) / (1 - p(Push)).
Then you indeed have a Kelly bet for the simplest 3-outcome bet (Win/push/loss).
The reason is, Kelly betting maximizes log-utility bankroll estimate (that is best estimated bankroll growth).
For a 2-outcome bet it is
maximize p*log(1 + f*b) + q * log(1 - f) for f
with the Kelly result
f = (b p - q)/b (with p+q=1)
If there is a push, and p is the probability to win, q to lose (and p+q<1), the Kelly criterion will be (see WolframAlpha result)
f = (b p - q)/(b p + b q)
which is elimininating the push by conditional (no push) probabilities p -> p/(p+q) and q -> q/(p+q).
Hence, the generalized Kelly criterion for a 3-outcome (again pushes) bet is simple, where p1,p2,p3 are the probabilities for result 1,2,3. and b1,b2,b3 are the payouts (p1+p2+p3=1)
maximize 1*log(1 + f*b1) + p2*log(1 + f*b2) +p3*log(1 + f*b3) for f
The Kelly-equivalent result will be
f = (p1 b1 + p2 b2 + p3 b3 - p2 b1 - 1) / (p1 b2 b3 + b1 p2 b3 + b1 b2 p3 - 1)
For an N-outcome bet, we could then assume it is
f = (p1 b1 + p2 b2 + ... + pN bN - 1) / (b1*b2*...*bN * (p1/b1 + p2/b2 + ... + pN/bN) - 1)
(without proof)
blackjack avenger
Well-Known Member
kelly is wrong?
Kelly considers growth and not bank preservation. The safer the bank the more certain the growth. Also, let's not forget Kelly is theory.
Kelly considers growth and not bank preservation. The safer the bank the more certain the growth. Also, let's not forget Kelly is theory.
The question is, what does bank preservation really mean. Maximum expected bank preservation is not to play. If you could give a bank preservation utility function, optimum bet is not much of a problem.blackjack avenger said:
Of course Kelly bet is far from practical, first of all by minimum bet.
Second, bankroll growth is a nice, but only if you don't ever spend that money for anything. Eventually you might want/need your bankroll for something different.
Maybe one could formulate a more practical utility function as: maximum expected bankroll on any random moment during course of play (say in the next 20 years).
jnrwilliam
Active Member
for instance, if u have $100k, one n the only 1 Br.,beyondbj said:
u bet at EV 2%, 1 spot, how much?
3%, 4%,... 10%.... how much?
Rather than betting your advantage, multiply the advantage by .76.jnrwilliam said:
With a $100k BR, and playing one spot, your optimal bet size will be $760 for a 1% advantage, $1520 for 2%; etc.
This is assuming that beyondbj is correct about the .76 figure (and I have no reason to believe that he's not).
zengrifter
Banned
That's the number Thorp used in BTD. zgSucker said:
Thank you, Zen; I will remember this number. (I think that the .6 number that I incorrectly quoted is the figure for playing 2 hands). I guess that means that for jnrwilliam's example; if he wanted to play 2 spots, the optimal bet size for a 1% advantage would be .76 x .6, or $456 PER SPOT. Does this sound about right?
zengrifter
Banned
My most aggressive sizing is basically to bet full advantage up to 1%, but then not bet higher regardless of how high the EV goes.
I rationalize that I therefore average the .70+ full-K. zg
I rationalize that I therefore average the .70+ full-K. zg
blackjack avenger
Well-Known Member
Semantics?
Kelly theory
1% advantage bet 1% of bank.
Kelly theory applied to bj
1% advantage bet 1% of bank times .76 (approximate, to account for variance of spl & doubles)
I believe on this site about everyone is applying kelly theory to bj whenever we write about kelly bets. Including multi hand bets and betting fractions of kelly.
Kelly theory
1% advantage bet 1% of bank.
Kelly theory applied to bj
1% advantage bet 1% of bank times .76 (approximate, to account for variance of spl & doubles)
I believe on this site about everyone is applying kelly theory to bj whenever we write about kelly bets. Including multi hand bets and betting fractions of kelly.
jnrwilliam
Active Member
Risk of ruin of full kelly = ?beyondbj said:
personal qn, ur bankroll can be replaced/ refilled fr other income?
apology if u feel bothered.
jnrwilliam
Active Member
u really did it in reality ?beyondbj said:
if u don't resize, 39 rounds prepared[6k/ round].
u r as brave as a lion but the lion must be well ready to be resized.