top bet problem

1 kelly for bj bet is x.76

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.

.76 implies a variance of 1/.76 = 1.32.

Hi Colin,

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)

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.

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.

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).

for instance, if u have $100k, one n the only 1 Br.,
u bet at EV 2%, 1 spot, how much?
3%, 4%,... 10%.... how much?

Rather than betting your advantage, multiply the advantage by .76.
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).

That's the number Thorp used in BTD. zg

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?

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

100k x 2% x 0.76
.
.
.
.
100k x 10% x 0.76

no , two spot , its .76x .7

three spot , its .76x .6

i may suggest to play full kelly

if your bankroll is below 300K US

for 300k to 700k , play 1/2 kelly

700k to 1.2 mil play 1/3 kelly

1.2 mil or above play 1/4 kelly

any one have your real experience ,

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.

Risk of ruin of full kelly = ?
personal qn, ur bankroll can be replaced/ refilled fr other income?
apology if u feel bothered.

ror is zero , if u resizing your bet each time

the most concern problem is the table limit actually

how much u have won in your gambling years ??

LOL! good one.

i bet 3 hands x 2000 US , for a TC 5+

my bankroll is 230k US



:devil:

u really did it in reality ?
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.

i resize my bet once I win/loss about 15000 US