Kelly Criterion bankroll question

[binions]

[Cross post on probability forum]

I understand KC is edge/odds. I have a question as to how it applies to what to buy in for at a poker game (as a percentage of bankroll).

Assume I win 60% of my sessions, and lose 40% in a given game. Is my edge 20% (60-40), or 50% (40 * 1.5 = 60) or some other number?

Assume my average win is twice my average loss, and my average session is +40% of my normal buy in. Are my odds 2:1 or 1.4:1 (assuming I only bring 1 buy in) or some other number?

What if I win 45% of sessions and lose 55%, but my average win is 3x my average loss and my average session 50% of my normal buy in? What would KC say my buy in should be as a percentage of bankroll?

Many thanks.

[HH81]

My head just exploded.

[remski]

ask phil laak..... bwahahahahahaha

[pzhon]

[ QUOTE ]

I understand KC is edge/odds.


[/ QUOTE ]
Actually, Kelly said a lot more than that. In his first paper on the subject, Kelly asked (and answered) how much to bet on each horse if you know the true odds are different from the public odds, both with and without a rake.

The formula edge/odds is a big simplification of Kelly's work designed for coinflips. Much more general (although slightly less accurate for coinflips) is to wager so that bankroll = edge/variance, or c * edge/variance where 1/c is your preferred Kelly fraction. Most people find the swings of the actual Kelly Criterion to be too severe, and prefer a Kelly fraction between 1/2 and 1/4.

This assumes that your advantage is fixed, and your wagers are scalable. In reality, this is not the case, unless you are splitting your action and are considering how much of your action to retain. That games are tougher as you move up generally means you should be overbankrolled according to your favored Kelly fraction.

[ QUOTE ]

I have a question as to how it applies to what to buy in for at a poker game (as a percentage of bankroll).


[/ QUOTE ]
In the examples you give, your typical loss is not one buy-in. To use the edge/odds formula, you want to determine the fraction of your bankroll which is represented by the typical loss, not your buy-in.

If you typically win n times as much as you lose (n:1 odds), and win with probability p, then the Kelly Criterion recommends scaling your "bet" so that your typical loss is your bankroll times

np-(1-p)
---------
<font color="white">---</font>n

or

edge
----- .
odds

To get this, I went back to first principles, which for Kelly means maximizing the expected logarithm of your bankroll. Again, most people prefer a fractional Kelly system.

[binions]

[ QUOTE ]
If you typically win n times as much as you lose (n:1 odds), and win with probability p, then the Kelly Criterion recommends scaling your "bet" so that your typical loss is your bankroll times

np-(1-p)
---------
<font color="white">---</font>n



[/ QUOTE ]

Thank you very much. Assume:

Avg winning session is +1800
Avg losing session is -800
Avg winning 55% sessions
Avg losing 45% sessions

So, (2.25*55 - 45) / 2.25 = 35%

35% means Bankroll * 35% = 800? Or Bankroll = 2285?

Did I get this right? If so, wow! No wonder people use fractions of Kelly. I assume half-Kelly would be 4570 in this example, correct?

[jay_shark]

Binions , the formula Pzhon provided is correct .

n= the odds you are receiving on your buy in . You havent specified what your buy-in is which would allow you to calculate n .

I gave an example in the probability forum .

[binions]

[ QUOTE ]
Binions , the formula Pzhon provided is correct .

n= the odds you are receiving on your buy in . You havent specified what your buy-in is which would allow you to calculate n .

I gave an example in the probability forum .

[/ QUOTE ]

Buy in is 600. But I thought n was calculated on the average loss, not the average buy in.

Here was the way I was thinking about it before Pzhon's post. I am sure it is incorrect.

Buy in is 600. 50% of the time I use 2 buy ins. So, average risk is 900. Average session is:

.55*1800 - .45*800 = +630. Odds are therefore 1530/900 or 1.67:1

(1.67 * 55 - 45)/1.67 = 28%. So, 900 is 28% of bankroll, or bankroll is 3214.

[pzhon]

[ QUOTE ]

Avg winning session is +1800
Avg losing session is -800
Avg winning 55% sessions
Avg losing 45% sessions

So, (2.25*55 - 45) / 2.25 = 35%

35% means Bankroll * 35% = 800? Or Bankroll = 2285?

Did I get this right? If so, wow! No wonder people use fractions of Kelly. I assume half-Kelly would be 4570 in this example, correct?

[/ QUOTE ]
Yes.

Those are great stats for NL, which is one of the reasons the Kelly-recommended bankroll is much smaller than typical recommendations. Also, typical recommendations are based on what someone once heard was necessary, rather than logic.

If you don't move down when you lose, or move up when you win, the Kelly Criterion bankroll has about a 1/e^2 risk of ruin, about 13%. Having c times that (1/c Kelly) gives you a risk of ruin of 1/e^2c. Another way to look at it is that your risk of ever dropping to x times your bankroll when following 1/c Kelly is about x^(2c-1). E.g., the probability you will ever fall to 1/2 of your starting bankroll is 1/2 for Kelly, 1/8 for 1/2 Kelly, and 1/128 for 1/4 Kelly.

[binions]

[ QUOTE ]
If you don't move down when you lose, or move up when you win, the Kelly Criterion bankroll has about a 1/e^2 risk of ruin, about 13%. Having c times that (1/c Kelly) gives you a risk of ruin of 1/e^2c. Another way to look at it is that your risk of ever dropping to x times your bankroll when following 1/c Kelly is about x^(2c-1). E.g., the probability you will ever fall to 1/2 of your starting bankroll is 1/2 for Kelly, 1/8 for 1/2 Kelly, and 1/128 for 1/4 Kelly.

[/ QUOTE ]

Thanks so much for clearing all this up and the link to the article. Nice to know ROR for 1/4 Kelly is 0.03%, and that 1/4 Kelly has less than 1% chance of turning into 1/2 Kelly.

[pzhon]

[ QUOTE ]
[ QUOTE ]
If you don't move down when you lose, or move up when you win, the Kelly Criterion bankroll has about a 1/e^2 risk of ruin, about 13%. Having c times that (1/c Kelly) gives you a risk of ruin of 1/e^2c. Another way to look at it is that your risk of ever dropping to x times your bankroll when following 1/c Kelly is about x^(2c-1). E.g., the probability you will ever fall to 1/2 of your starting bankroll is 1/2 for Kelly, 1/8 for 1/2 Kelly, and 1/128 for 1/4 Kelly.

[/ QUOTE ]

Thanks so much for clearing all this up and the link to the article. Nice to know ROR for 1/4 Kelly is 0.03%, and that 1/4 Kelly has less than 1% chance of turning into 1/2 Kelly.

[/ QUOTE ]
You are welcome.

To clarify, 1/4 Kelly would never turn into 1/2 Kelly, since the drawdown formulas assume that you will play in proportional games, moving down when you lose, and moving up when you win. If you stay at the same level, the risk of losing 1/2 of your bankroll if you start with 1/4 Kelly is about 1/e^4 ~ 2%.