View RSS Feed

brh

Kelly betting on high variance games

Rating: 4 votes, 5.00 average.
Quote Originally Posted by brh View Post
Quote Originally Posted by Matt21 View Post
I have actually asked a similar question on the old bjinfo but wanted to revisit this as I found some conflicting answers.

Generally speaking, when betting full Kelly you divide the edge% by the pay-off a winning bet to arrive at the % of your bankroll that you should wager on your bet.

Does this apply equally regardless whether your pay-off is 1:1, 5:1, 10:1, 50:1 or 100:1? Does this rule of thumb change if you are playing with a high edge (say 30% or 40% as an example).
I will give this a stab - please be Gentle .

Firstly, I hate this terminology, because it confuses punters and it confuses me.

I someone quotes a payoff of 1:1, I assume they actually mean if you bet one bet, if you lose you get zero and if you win you get 2 units:

That is ev(lose) = -1 and ev(win) = 1.

I will take this as my starting point, such that for a payoff of 1:m we have

ev(lose) = -1 , ev(win) = m.

Now here is my first problem, this tells me nothing about

Prob(win) = p, Prob(lose) = 1 - p.

This is how casinos rip off unsuspecting punters.

I can take a guess: for Roulette we have a supposed 1:1 payoff for Red vs Black.

For us 'lucky' Australian and UK players, there is only one '0' on the wheel and so

Prob(win) = 18/37 = 0.486, Prob(lose) = 0.514.

ev(win) = 1, ev(lose) = -1.

Total ev per spin is

ev = 1 x 0.486 + (-1) x 0.514 = -1/37 = -0.027 = -2.7%

var = 0.486 x (1)^2 + 0.514 x (-1)^2 = 1.0

This gives values of

ekb = var/ev = -37

N0 = var/(ev)^2 = (1)/(1/37)^2 = 1369 spins .

Since win rate is ev = ekb/N0 per spin we get -37/1369 = -0.027 = -2.7% CORRECT !!

Since this game has negative ev, clearly you give it a wide berth.

Now translating these formulas into general form, we have for a payoff of 1:m

ev(lose) = -1 , ev(win) = m

Prob(win) = p, Prob(lose) = 1 - p.


So

ev = ev(win) x Prob(win) + ev(lose) x Prob(lose)

= m * p +(-1) * (1-p) = mp - 1 + p = (m+1)p - 1

or

ev(m,P) = P(m+1) - 1 ........................ (1)

var(m,p) = p * (m)^2 + (1-p) * (-1)^2 = p.(m)^2 + (1-p) = p.m^2 - p + 1 = p(m^2 - 1) + 1

var(m,p) = p(m^2 - 1) + 1 .............................(2)

Time for sensibility checks:

ev(m,P) > 0 iff (m+1)P - 1 > 0

(m+1)P > 1

The domain of P is [0,1] => (m+1) > 0.

By the definition of 'Payoff' = 1:m we assume m >= 1.

=> ev(m,P) > 0 iff P > 1/(m+1) ............................... (3).

For our Roulette case above P = 0.486, m=1 : 0.486 < 1/2 and so ev(1,0.486) = 2 * 0.486 - 1 = - 0.027 < 0 CORRECT !!

-----------------------------------------------------------------------------------------------------------------------------------

Now var(m,P) > 0 always by the definition of variance.

So

var(m,P) = P(m^2 - 1) + 1 > 0 iff P(m^2 - 1) > -1 ...................... (4)

Now since the domain of P is [0,1] and m is [1,inf)

var(m,P) > 0 is self evident : CORRECT !!

Roulette check:

var(1,0.486) = 0.486(1 - 1) + 1 = 1.0 : CORRECT !!

---------------------------------------------------------------------------------------------------------------

Now

ekb(m,P) = var(m,P)/ev(m,P)

ekb(m,P) = [ P(m^2 - 1) + 1 ] / [ P(m+1) - 1 ] > 0 iff P(m^2 - 1) > -1 ...................... (5)

N0(m,P) = var(m,P)/[ev(m,P)]^2 > 0 .................................................. ....... (6).

---------------------------------------------------------------------------------------------------------------

For a Kelly bettor betting a unit $B we have

EKB(m,P) = $B x ekb(m,P)

EKB(m,P) = $B . [ P(m^2 - 1) + 1 ] / [ P(m+1) - 1 ] > 0 iff P(m^2 - 1) > -1 ...................... (7)

EV(m,P) = $B . [ P(m+1) - 1 ] iff P > 1/(m+1) ............................... (8).

VAR(m,P) = ($B)^2 . var(m,P)

VAR(m,P) = ($B)^2 . [ P(m^2 - 1) + 1 ] > 0 ................................(9).

N0(m,P) does NOT change because $B cancels out in Equations(6),(8) and (9).

--------------------------------------------------------------------------------------------------------------

Now a Kelly bettor determines the unit bet $B by the relationship between Bankroll, ekb(m,P) and the Kelly fraction 'k':

$B= Bankroll/( k . ekb(m,P)) where k is the Kelly fraction

So

$B = [Bankroll / k] / [ P(m^2 - 1) + 1 ] / [ P(m+1) - 1 ]

$B = [Bankroll / k] * [ P(m+1) - 1 ] / [ P(m^2 - 1) + 1 ] iff P(m^2 - 1) > -1 ...................... (10).

Again as a consistency check we have:

EKB(m,P) = $B . ekb(m,P)

EKB(m,P) = Bankroll / k ................................................(1 1)

since ekb(m,P) cancels out as it should - remember this is a much simpler game than Blackjack.

--------------------------------------------------------------------------------------------------------------

I think we are now done and your questions can be answered.

<Generally speaking, when betting full Kelly you divide the edge% by the pay-off a winning bet to arrive at the % of your bankroll that you should wager on your bet.>

Let's not assume you are playing full Kelly just yet.

[Payoff on winning bet] = m ............................................ (12)

NOT (m+1) since the +1 is your own money.

edge% = ev(m,P) = ev(m,P) = P(m+1) - 1 from Equation(1).

edge% / m = [ P(m+1) - 1 ] / m ........................................ (13).

Bugger: I am afraid I do not see any reference to variance in Equation(13) - it cannot be used to determine a Kelly bet !!!!

I think this is the correct expression:

$B= Bankroll/( k . ekb(m,P)) where k is the Kelly fraction from above ...... (14),

where from Equation(5)

ekb(m,P) = var(m,P)/ev(m,P) .................................................. ........... (15),

now we see the unit variance - nice.

%bankroll to bet = $B/Bankroll x 100% .................................................. ..... (16).

Substituting in the above terms we get

%bankroll to bet = [Bankroll / k] * [ P(m+1) - 1 ] / [ P(m^2 - 1) + 1 ] / Bankroll * 100%

-----------------------------------------------------------------------------------------------------------------

%bankroll to bet(m,P,k) = [ P(m+1) - 1 ] / [ P(m^2 - 1) + 1 ] / k * 100% ........... (17)

and we are FINALLY done !!!!!

-----------------------------------------------------------------------------------------------------------------

Just for fun, assume you are completely mad and playing full Kelly k=1.

Say the payoff is 1:5, that is m=5.

Assume your probability of winning is 51%, so P=0.51.

From equation(1) we have unit ev of

ev(m,P) = P(m+1) - 1 = (0.51)(6) - 1 = 2.06 units per round.

From equation(2) we have unit var of

var(m,P) = P(m^2 - 1) + 1 = (0.51)(25 - 1) + 1 = 13.24 (units)^2 per round.

From equation(5) we have unit ekb of

ekb(m,P) = var(m,P)/ev(m,P) = 13.24 / 2.06 = 6.43 units

Then from equation (17) the percentage of your bankroll to bet is

%bankroll to bet(m,P,k=1) = [2.06] / [ 13.24 ] = 0.156 = 15.6% of your bankroll.

N0(m,P) = var(m,P)/[ev(m,P)]^2

N0(m,P) = [13.24 ] /[ 2.06]^2 = 3.12 rounds.

Normally this gives a Kelly doubling time of (k=1) from Patrick Sileo:

T = 0.693 * N0(m,P)/(k - k*k/2) .................................................. .. (17),

unfortunarely this is under the assumption that N0 > 1000 rounds, so it won't work.

So just assume for now your Kelly doubling time is N0 plus a bit = 4 rounds.

With a bankroll of $10000, your unit bet will be $B = $10000 x 15.6 = $1560, call it an even $1500.

So even if you wait to double your bankroll before resize, you should expect to win about $10000 in the first 4 rounds,
$20000 in the next four rounds, $40000 in the next four rounds... get the idea :-)

I hope you have underpants of steel and your name is Walter White - or else whatever casino you try this in,
your life expectancy is about 10 rounds - good luck with that.

But it is only my juicy example - but for this example, I suggest you play with a Kelly factor of k=1/100 if you want to live.

Brett.

Oops - forgot about ROR - in the example above with k=1 you will continually have an ROR of 13.6%, way over betting.

One of two things will happen, you may be ruined after 4 rounds, or you may die - whichever is preferable I guess.

Submit "Kelly betting on high variance games" to Google Submit "Kelly betting on high variance games" to Digg Submit "Kelly betting on high variance games" to del.icio.us Submit "Kelly betting on high variance games" to StumbleUpon

Tags: 1'"1????%2527%2522ror Add / Edit Tags
Categories
Uncategorized

Comments

About Blackjack: The Forum

BJTF is an advantage player site based on the principles of comity. That is, civil and considerate behavior for the mutual benefit of all involved. The goal of advantage play is the legal extraction of funds from gaming establishments by gaining a mathematic advantage and developing the skills required to use that advantage. To maximize our success, it is important to understand that we are all on the same side. Personal conflicts simply get in the way of our goals.