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Clayton

May
01
2012
Kelly Criterion
Posted in Poker | View Comments (19)
 

Warning: Maths

The past day or two I've spent a great deal of time learning about Kelly criterion.

Something I've struggled with the last few months is that I feel stupendously overrolled for 2/3NL, but I feel underrolled for 5/10NL. I figured with Kelly criterion, I could have a better understanding about when it's time to move up and take shots, and ultimately I'd be more confident with risk.

First I had to figure out my standard deviation from excel spreadsheets, which was kind of difficult because standard deviation tends to operate on a per hour or per hundred hands basis. If you had a bunch of sessions of varying lengths, you can't just depend on STDEV in Excel to solve everything.

Fortunately I stumbled on this article which showed me how to calculate the standard deviation. So now I have standard deviation and winrate. By the way, if anyone thinks this website is wrong, please let me know. It's sort of the backbone of this entire blog.

This lead to a formula which I got from this book (which I highly recommend, and you can get it via kindle iPad).

The formula is:

Kelly Criterion bankroll = (standard deviation)^2 divided by winrate, or
Kelly Criterion bankroll = Variance divided by winrate

Where Variance is the same thing as STDEV squared.

Plug in your variance and winrate, and Kelly Criterion will tell you the most ruthlessly precise way to take shots for a given game.

So I ran into the following for the games I play (since mid-August):

2/3 NL:
Standard deviation: 49.8 big blinds per hour
Winrate: 10.3 big blinds per hour
Hours: 360

Based on these figures, Kelly criterion thinks it's fine for me to play in this game with 240 big blinds!

A similar thing happens for 5/10NL
Standard deviation: 43.3 big blinds per hour
Winrate: 13.7 big blinds per hour
Hours: 265

In this game it's okay to take a shot with 136 big blinds!

The first thing to draw from this is that I'm running hot. I don't expect to make $135/hr (practically 30bb/100) at 5/10 forever, even if I continue to game selecting absurdly well. More reasonable is like $100/hr bumhunting and $50/hr while accepting the "good, but not great" lineups.

The second thing to draw from this is that the amount of hours is a little too small. Everyone cites 1k hours as a benchmark and I'm only two thirds the way there. The standard deviation is close'ish, though.

But let's say I keep the same standard deviation and lower my winrate at 5/10 to $50/hr. Kelly criterion still says I should take a shot at this game with 375 blinds!

So pure Kelly criterion thinks that I should be aggromonkey with my bankroll. In Newall's book, he says some gamblers adapt a "One third kelly" style where they just multiply the magical bankroll number by 3. Others go so far as to multiply the kelly bankroll by 6.

The applications are pretty necessary if you plan on jumping into bigger games with smaller edges (like, say, 40/80 limit holdem live), but I don't have reasonable samples in anything except 2/3NL and 5/10NL. Even the most cautious kelly bankroll models seem to think that A) I'm overrolled for 5/10 and B) I should begin immediately taking shots at bigger games. I would obviously have to estimate winrates and standard deviation for these bigger games before figuring out the proper bankroll, but I'm sure I could grab them from someone else with more experience.

I'm curious if anyone else has studied Kelly criterion to this effect and uses a "One third kelly" or "One sixth Kelly" that could explain their bankroll philosophy.

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Comments
05-01-2012
d2themfi is online now d2themfi
I think there's so many more factors that go into this, that Kelly basically just provides the bottom tier. Things like living expenses, taxes, tilt, opportunity cost, and a lot more make it extremely complicated. I would say, if you are thinking about sitting down at a stake and it doesn't feel right, then it's too high
05-01-2012
jupiter is offline jupiter
Pretty sure KC is covered in regard to poker near the end of MoP. Numbers you gave sound absurd and not in the area of what I recall hearing for typical cash game play. Since 2p2 is down, here's a google cache of a thread I found (there's a post near the bottom that sounds about right): 2p2 KC thread
05-02-2012
clayton is offline clayton
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they probably look absurd because i'm using per hour instead of per 100 hands. if i'm getting 35'ish hands an hour then I just multiply the winrate and the standard deviation by 2.8 or so... result is still the same for the bankroll output.

something does feel iffy about the kelly answer in my blog, but I guess if my sample is a huge heater then it's going to be biased since the winrate tweaks everything up.
05-02-2012
JamesMa is offline JamesMa
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The thing about "win rates" is that you can never calculate it no matter how large the sample unless you are playing with the same people every time and their games remain the same over time.... I think the most obvious way to put it is every table you sit on is going to be with different people, live even more so. At some tables you will win X per hour or 100 hands and at another table you will win at Y per hour.

I think it kind of ties in w/ what d2themfi said, sit down and see how the game plays and then make a decision if you think you should stay for a sess
05-02-2012
d2themfi is online now d2themfi
Simplest calc for kelly is winrate/(std dev)^2. In mop they use a different formula because they are comparing the utility of two game stakes versus each other, which would actually apply here well. The formula iirc is
2c= (((std dev #1)^2) - ((std dev #2)^2)) / (wr #1) - (wr #2)

Where c= cut off bankroll

You'd obviously want to convert everything for the calc to dollars. So here you would have

2c= ((187,489) - (22320)) / ((137) - (39))

2c= 167,169/98

2c= 1705

C= $852.5


Way better thing to do though is use +evplus variance simulator with possible winrates, std dev, and samples and see what an average and big downswing looks like and how you would be able handle those mentally
05-02-2012
clayton is offline clayton
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"Simplest calc for kelly is winrate/(std dev)^2"

Newall's book says (st dev)^2 / winrate...

And the value for C seems awfully small. Almost feel like I'm doing something wrong with my calcs.
05-02-2012
jonstermonster is offline jonstermonster
play whatever you feel comfortable at. simple as that. play on the nitty(er) side imo.
05-02-2012
clayton is offline clayton
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I've been doing that for a long time, believe me.

There is added comfort in moving up in stakes if the math supports it.
05-02-2012
JamesMa is offline JamesMa
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didnt the guy in rounders say if you are too careful, your whole life becomes a fkn grind
05-02-2012
cntgetmedown is offline cntgetmedown
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@ Clayton

In my original notes on the Kelly Critereon I also have variance/winrate. My source was Phillip Newall when I took the notes: - http://www.twoplustwo.com/magazine/i...-selection.php, but sadly that link doesn't work anymore or currently. His book must be a typo.

If I plugin the formula from MOP into Microsoft Mathematics http://bit.ly/IWPrWM, I get an even lower number than d2thefmi. I get 776.97$.
This may seem low to some, but your empirical winrate at 5/10 is so high that when comparing it to playing 2/4 you hypothetically should be willing to play in the larger game, even when it does not support optimal bankroll growth in a vacuum. To my knowledge The Kelly Bet of 136 big blinds or 1360$ assumes that there is an infinite number of stakes smaller than 5/10, i.e. 4.99/9.99 etc. so playing 5/10 with 776.97$ would have a lower expectation then playing whatever the equivalent stake would be.
Or you could also simply say, your bankroll grows slower in the long run by playing 2/4 with 800$, then it would by playing 5/10.
05-02-2012
jupiter is offline jupiter
1. There's an example in MoP (25.3) that's pretty close to your situation. He compares 20/40 and 40/80 LHE with $35/hr and $60/hr winrates respectively. Let's say the 2/5 and 5/10 NL games you play are roughly comparable. They cutoff bankroll given was roughly $20k, where with a smaller bankroll you play 20/40 and with a greater bankroll you play 40/80.

2. I'd guess either your formula is wrong or your std estimates are way off. No way KC is that degen. Also it shouldn't matter if you do /hour or /100 hands, /54 orbits or whatever.

3. http://www.bluffmagazine.com/magazine/Kelly's-Criterion-Phil-Laak-1295.htm
05-02-2012
clayton is offline clayton
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cntgetmedown,

I don't understand why newall's book would be a typo when your original notes have it as variance/winrate, because variance is the same thing as (stdev)^2.

I do think the larger winrates for NL versus limit is screwing things up though. The resources I'm using have all worked primarily with limit poker, and it even said in the book that you have to treat NL/PL the same and assume higher winrates.

I am curious though... let's say my cardroom averages 35 hands an hour. So multiply my standard deviations by 2.8 for "per 100 hands" stDEV, does that put it in the ballpark of other people who play live full ring? Wish 2p2 had a forum right now so I could ask the smallstakes live NL forum >_<



Jupiter,

I think the winrates at NL being higher than limit (in my case 10 blinds per hour versus 0.8 bets per hour) might be what is tweaking it.

Laak's article is good for people who don't know anything about it, but he doesn't delve into the math of it at all, just some practical applications he uses with it. I mean, he's treating a poker buyin like a sportsbet.
05-02-2012
cntgetmedown is offline cntgetmedown
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@ clayton

You are ofcourse right. I misread your previous post and that of d2themfi. I do believe variance/winrate it is the correct formula. If we were to apply winrate/variance on your 5/10 NL sample, then we would write: 13.7/43.3² = 0.0073. That result makes no sense in this context.

I'm not quite sure what you mean. If you assume a higher winrate for NL, then the Kelly Bet would become smaller. I could see how a Kelly Bet would underestimate how large your bankroll would need to be to grow at an optimal rate though (due to the nature of NL variance where you can drop large portions of your bankroll in one hand).

If you multiply your stdev, you would have to multiply your winrate as well, otherwise the Kelly Criterion doesn't apply. They have to be comparable. What you could do is simply assume more variance though.

As for your bankroll philosophy question in your main post, I do you use the Kelly Criterion to figure out what sort of bankroll I need to play at certain stakes. I make modest assumptions about winrates and try to figure out a variance number by looking at my empirical variance and that of other regulars in the games I'm looking to play in (assuming this data is available). Then I double the value that I receive from the formula, i.e. I play 1/2 Kelly Bet.
05-03-2012
theashman103 is offline theashman103
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Clayton, pardon my drunkenness but I think that you are over-thinking this way too much. You BR isn't close enough to rock bottom to consider playing less than 5/10 nl live. Put your effort into something else because this has no relevance to you.
05-03-2012
clayton is offline clayton
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Hey ashton,

The relevance is in the math, because if the math is correct and I can ballpark a winrate, I can plug in the same stuff to shot take a soft 25/50NL game. I figure that'll happen every now and then this summer.
05-03-2012
The_End is offline The_End
Just looked at it quickly, but intuitively it seems like your variance estimator has a large variance itself, and looks biased.

You are transforming session data into hourly, but by taking just the averages you are also averaging out some of the variance, leading to an underestimation.

It's like a stock that goes up or down 1000 every day, but at the end of the month averages 0 return. Your monthly variance will be 0 while your daily is huge.

Better estimator is to record your hourly swings. Easier to get larger sample size this way as well
05-03-2012
clayton is offline clayton
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Agree. Unfortunately I've been only estimating by session. Will have to change it up! Im gonna look silly with my phone beeping once an hour and me writing stuff into my phone
05-04-2012
oakton55 is offline oakton55
Full Kelly is theoretically the fastest way grow a given bankroll but it is super degen. When betting full Kelly with constant resizing, from any given point in time, 1/3 of the time you are going to halve your bankroll and 2/3 of the time you are going to double it. Most AP blackjack players with a medium sized non repleneshable bankroll bet 1/2 Kellly at most, and big bankolls typically go 1/4 Kelly.
05-08-2012
mootang is offline mootang
stop being a bitch and move up imo
 
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