Win Rates & Standard Deviations & Your Edge

I just finished reading Fortune's Formula. It discussed, among other things, the optimal betting strategy called the Kelly Criterion. Basically, the Kelly Criterion says to bet the amount of your bankroll that is your edge. Figuring out your edge seems a bit tricky, but if my research has been accurate, the equation is:

Winrate = Standard Dev.* (2p-1), where p is your expected probabilty of winning.

So according to my calculations, at .02/05nl, my winrate is 8.81bbs/100 and my standard deviation is 69.11 (found in HEM). Solving for p, I find that I have an edge 12.7% at these stakes.

To calculate the ideal bankroll, we use the formula:

B = SD^2/(Kelly fraction * winrate), so mine is 69.11^2/(1*8.81) = 542 bbs. That's only about $27.

Some people like to bet less than the Kelly amount to be safer due to the uncertainties of these estimates, so say a 1/2 Kelly bet would require a bankroll of 69.11^2/(1/2*8.81) = 1084 bbs. A 1/2 Kelly bet is supposed to grow at 75% the maximum growth rate, but offer 50% less risk.

The key with the Kelly formula is that your bets get bigger as you win, but smaller as you lose. Your risk of ruin is supposed to be zero as long as you can keep reducing your bets, but realistically you can lose practically everything. There is a 50% chance using Kelly that your bankroll will fall to 50% of it's current level, and a 10% chance that it will fall to 10% of current. But overall, your rate of growth will be optimized. If you bet more than the Kelly bet, say, like double, you will almost surely go broke (or close to it).

I'm not too good at math, and it took me forever to figure out this much, so if anyone sees any errors, please let me know.

Summary: The Kelly criterion is impractical for most poker players due to the uncertainty of the inputs and due to the inability of most people to stick to strict rules. However, it does allow for substantial upside so assuming you can be both realistic and conservative in your upside values and that you can stick to strict guidelines, it may be worth a flyer.

**Disclaimer: I'm not too knowledgeable in this subject but math threads generally don't get too much love in these forums so I thought I would chime in. I have never actually done the math myself so take my words with a grain of salt. **

Explanation: The Kelly criterion, while mathematically correct, has two primary issues when it comes to poker:

-The biggest reason is that the the inputs (true winrate and standard deviation) are both largely unknown and vary from table to table.
-You must be very stringent in moving down when you hit your breakpoint to move down, something harder than most people think due to common human psychology.

I think most people say that you need at least 100K hands at a limit before your winrate starts to converge to a value with reasonable confidence. By the time you play this many hands, you will generally have a roll that far exceeds the Kelly criterion or you will have moved up to a higher limit. Also, if you do some sensitivity analysis, you will find that the Kelly criterion is highly sensitive to your true winrate. For anyone looking to implement the Kelly criterion into their BRM, I would recommend that they consider the following due to the inherint risks associated:

-Run some tests with different input values (winrate and SD) and see what sort of outputs you get. This will give you an idea of how much this criterion changes as your winrate varies.
-When implementing, use a very very conservative winrate value. Even if you are trully an 8bb/100 player, you will not be an 8bb/100 player at every single table you play.
-Be conservative when establishing breakpoints to move down
-Most people (and I mean like 99.9% of people who play poker including myself) have never seen a true downswing and are unaware of the kind of tilt that this causes. Some people may think that they are tilt free but the reality is that almost everyone tilts when they go through downswings that they thought were mathematically impossible.

Thanks for the reply. I had a feeling that my post wouldn't generate much interest.

You bring up a lot of good points. I haven't figured out where the breakpoints would be to move up or down, but I know you need to continually recalculate to make sure your bet is optimal.

In practice, my guess is that you buy in for the Kelly amount at a table that is within your range, rather than always buying in for 100bb. I wouldn't want to try this with my current bankroll, since that would put me at around .25/.50nl tables and I haven't even advanced past .02/.05.

It might be fun to try at another site. I could deposit a small amount and see what happens using Kelly as my bankroll management tool.

Running my .01/.02nl stats into a variance spreadsheet, it looks like an average return over 100K hands would be somewhere around 200 buy-ins. I've lost the link to that spreadsheet, but it was mentioned in another thread here at CC.

The other side of the coin is that even when you know your variance value it will be an average value.

Your actual variance will also depend upon the variance of your opponent.

You will suffer more varience against someone who plays a high variance style than someone who plays a low-variance style.


These concepts make sense in theory and allow you to have some idea how variables interact with each other, but its near impossible to get values which are of use in specific situations.

I'm also thinking of moving to NL10 6 max. How does your winrate and SDev. compare to your rates at NL5 6 max?

Yeah, the variance spreadsheet shows just how huge the swings can be, especially if you have a low win rate. I guess the best you can do is just take extremely conservative guesses and average many trials in the spreadsheet to get a rough idea of your bet size. Betting 1/2 Kelly accomplishes some of this.

I jumped straight into 10NL so I don't know. A few generalized 10NL pointers that you may find useful:

-People either call too much, fold too much, or bet too large. If you can identify this and exploit it (i.e. know when to preflop raise / cbet vs weak players, value bet vs calling stations, or nut peddle against a maniac), 10NL should be a piece of cake.
-If bad players bet big on the river, they generally have it. You can generally call light on rivers against smaller bets if draws miss and their bet makes no sense.
-Do not bluff stations. No matter how believable your betting, stations will sometimes call with as little as 3rd pair (granted I'm borderline LAG so maybe I am less believable but at 10NL, i doubt image matters too much)

Original poster states "So according to my calculations, at .02/05nl, my winrate is 8.81bbs/100 and my standard deviation is 69.11 (found in HEM). Solving for p, I find that I have an edge 12.7% at these stakes."

If your edge is 12.7% then wouldn't that be the fraction of your bankroll to bet? So if you're buying in full at $5 no limit, then wouldn't you want to have a bankroll of ($5/.127) = $39.37 ?

Just my thought on the matter. I could be wrong.

I really wish I unsderstood what is being said.........

But I don't.

Yeah, I think that is right, but you don't really have to limit yourself to only buying in full. I could buy in at $3.43 at the 5nl tables and be within the Kelly formula guideline. It's been awhile since I've looked at this, but I'm about to experiment with it on Full Tilt after I get the $50 Take 2 bonus.

Ha, nice picture! That's how I felt when I first read about it. Just stubborn enough to try to understand it.

Having an edge doesn't outweigh short term variance. If you're all in pf w AA for 100bb's and get called by QQ, you've made the right play - but you're going to lose all 100bb's if no A hits the boad but a Q does... Sadly, this general scenario (got your whole stack in while a strong favorite to take the pot) can happen several times in a row.

My point is simply that if you run normally, yeah, this makes sense - but nobody runs exactly statistically normally...

That's why you have to constantly readjust your bets. Short-term negative variance will make you reduce your bet size, but your risk of ruin will remain the same. Short-term positive variance will allow you to increase your bet size as well. Overall, the goal is optimal bankroll growth given a certain (small) allowable risk of ruin. (I so don't play this way -- as you know, I'm a complete bankroll nit, but I think I'll try it on Full Tilt with my $50 bonus).

This is a mistake. Whether you're running hot or cold has zero impact on the optimal play for your current hand. Repeat after me - LUCK IS A MYTH. Everything is statistical probability, and if you flip heads 10 times in a row, it does NOT mean that the next flip has to be heads or tails (I'm ignoring the rigged coin).

Don't change your bet sizing based on how you're running.

Huh? I'm not talking about in general that you should do this. I'm trying to explain the Kelly Criterion, which is a method to optimally increase your bankroll. If you run bad, in order not to risk ruin, you have to decrease your bet size. If you run good, then you get to increase the bet. That's the way it works.

Like I said, I don't know how it will work in poker, but I think I'm going to give it a try with some bonus money, so it wouldn't be the end of the world if it doesn't work out. It's an experiment. Several people say you can't apply the Kelly Criterion to poker, but I think that you probably can.

Ah, sorry about that, I misunderstood the basic premise, I thought this was about bi's rather than bets - my bad. I looked at wikipedia for an explanation, now I think I get it. If I get it, I'm definitely in the camp that thinks this is inapplicable to poker, largely cuz I don't think poker is the same as other forms of gambling.

Luck (in the sense that you're dependent on what cards you're dealt and hit the board) is definitely a part of poker, but not in the sense that it is in blackjack (casual blackjack, not talking about professional card counting), slots, craps, roulette, horse racing, etc. If you're purely playing your cards and making hands, then yeah, it's not far from those other types of gambling. But since you're playing against people and have tools to influence the outcomes of your bets independently of your cards (i.e., by influencing how villains play), I don't see this as the same as the other activities.

Since the size of your bets is one of the basic tools you have to influence outcomes, basing bet sizing on the Kelly Criterion seems to me to be a bad idea, it decreases your ability to influence the other players on the one hand. On the other hand, if you have a 60/40 scenario in your favor, making your bet size smaller than all in because of your perceived edge also seems to me to be a mistake.

The Kelly Criterion has nothing to do with bet sizing other than the fact that it affects the two inputs (win rate and standard deviation).

The Kelly Criterion is a bankroll management tool that gives you the optimal bankroll size to maximize bankroll growth for any limit given that you have a true win rate and standard deviation. Thus, you can change nothing about the way you currently play, and the Kelly Criterion will spit out how much you need in your bankroll to play a certain limit given that you can freely move up and down.

The Kelly Criterion has no problems with it on a theoretical level but it's relatively impractical for the reasons I outlined in the previous post I made in this thread.

I think Kelly's whole question was; given your edge, your current bankroll, and your odds how much should you bet for optimal bankroll growth?

The "bet" here is how much you sit down at the table with, not how much you "bet" on given streets.

Edit: Actually my statement is lol... Give me a second to organize my thoughts.

I think Nassim Taleb might have a bone to pick with you about how "optimal" it is to use incremental statistical knowledge to calculate risk exposure.

OK, I'm just confused. If this is how big your br needs to be to play a specific limit, it makes sense to me, but the word "bet" is confusing the heck out of me.

I'm not familiar with him, but after a quick Google, I think I just discovered my next read. Thanks!

Buy-in amount is how I intend to try to apply it. I'll need to recalculate based on changes in overall win rate and standard deviation, and I'll have to accept the uncertainties if I'm able to move up quickly (without win rate evidence). At some point, I'll reach a level where I have no or a negative edge; then the answer is not to play.

I understand exactly what you're doing, and I know that bankroll management is the crux of this thread. However, my statement about how much to sit down with, while probably the best way to think about what the "bet" in the Kelly Criterion is, is technically incorrect in referring to the "bet" as outlined in the Kelly Criterion (if you've read enough of my posts, you've probably realized how much I dislike incorrect statements, especially making them ). I explain this below.

Also, this didn't occur to me before because I just gave you brief conclusions on what I recalled from several months earlier, but you need to consider that there are not infinite limits below the limit you are playing which is what would make the Kelly Criterion optimal (or the probability of going busto = 0). As a corollary to this, in determining the optimal bankroll size, you do not factor in your win rate and standard deviation at 2nl which would most likely further reduce the size of your bankroll needed. The value you give in your OP gives a value assuming your 2nl win rate and standard deviation is equivalent to that of your 5nl win rate. However, keep in mind that there are no limits below 2nl so you have to put a very very very conservative number on your 2nl win rate (which may very well be your 5nl win rate).

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Okay, I needed to recollect some thoughts and review a thread (another forum so I can't link) because it's been about 5 months since I last thought about this topic in depth (post I made two months ago was based on general conclusions from before).

Very quickly, the Kelly criterion is discussed as a bankroll management tool ("how big your br needs to be to play a specific limit") in order to maximize the growth rate of your bankroll.

It is not how much to "bet" on any given street in a poker hand because you do not have a probability distribution of winning and losing money on any given street (except the river) because the hand may or may not be over. It is assuming that for any given hand, there is some probability distribution of how much you win or lose in any given hand.

Win rate and standard deviation are used to characterize this probability distribution. The reason why win rate and standard deviation is an appropriate model is based on sample size and through the use of the Central Limit Theorem (I'm not 100% convinced of this because of an independence of observations issue but apparently it's widely accepted so I'll follow the crowd). If anyone is interested and has a fairly good understanding of statistical theorems, I can explain but I'm sure 99.9% of this forum won't care.

The statement I made about "bet" being how much you sit down with is incorrect but it is probably the easiest way to think about what the "bet" is if you want to characterize it in the Kelly Criterion. Technically, it is the sum of all bets made in any given hand, but the Kelly Criterion as applied here makes no effort to determine bet sizing on individual streets so it is best to think of how much you "risk" as the amount you sit down with which is how much you "risk" in any given hand.

Actually, I used 5nl numbers throughout, but 5nl derived from PokerStars, so it will be interesting to see how that compares to Full Tilt. Seems like FT players are a bit worse at the same levels.

Incidently, my edge at 2nl calculates out to 19.5%. Coincidentally, I've played almost exactly 60,000 hands at each of the three levels I've tried in my short career , 2nl, 5nl, and 10nl. At 10nl, my edge is practically nothing, so it's only useful to play for bonus points or rakeback (or to try to get better -- lots of room for that, hehe).

Effectively, we're talking about capping exposure in any given pot then by capping the bi amount?

This doesn't guide us in br size to play a limit, does it? In other words, for any given pot, I shouldn't exceed value x for the bi when I sit down at a table (x being calculated according to the formulas above), but it doesn't tell me whether I should sit down at the table at all, instead of a lower limit table? I mean, in theory, I can sit down w $5.87 at a 50nl table if that's the fraction of my bankroll that's optimal to "bet" given my calculated edge and current br size at the table, even if I'm better off sitting at a 10nl table fully stacked?

I'm still thinking that this whole idea works against playing good poker. Capping exposure by capping bi's affects SPR, playable hands, and the ability to run high variance plays like semi-bluffing.

Intellectually interesting discussion nonetheless.

In general, I would say you want to play at a level that is close to a full buy-in, and also at a level that you have evidence of a good win rate. Can you buy-in for more than 100bbs at FT? I think that would be advantageous for me since I'll have around $65 after the bonus. 12% of my bankroll would be $7.80. Or I could play a fractional Kelly amount and buy-in for $5.00. That helps ease the uncertainty of the assumptions.

I don't know what to do if I increase my bankroll to a level that I can't beat right off. I suppose then I can withdraw the excess and stay at a level I can beat, or use the extra to take shots at higher levels and try to learn to improve against better players.

FT does have some deepstack tables, not sure about all limits.

Playing 12% of your br on a single table is very aggressive acc to most recommended br strategies, fwiw.

I think the way Rogue is trying to pose the problem is that he is assuming he is buying in full at every single limit and wondering which limit he can play according to the Kelly Criterion. This is a much more practical application of the Kelly Criterion because we can isolate our inputs to our win rates and standard deviation of the limits that are available for play. This application methodology does nothing to stunt poker growth which the Kelly Criterion does not take into consideration because it assumes our inputs (win rate and standard deviation) are constant.

While there is technical merit to analyzing the scenario you provide, it is largely impractical due to the lack of availability of input data. While it is technically possible to apply the Kelly Criterion for very specific buy ins (not full stacked), we would need to know our win rate and standard deviation for every buy in amount which is largely impractical. You are always weighing risk vs reward so if you have a decision between playing $5.87 at a 50nl table and $10 at a 10nl table, you would be making a comparison between your win rates and standard deviations at the respective limits for those buy in amounts. Also, as I stated before, it does not take into consideration your growth of poker skills. It is purely a bankroll maximization technique given your current skill sets.

Btw Rogue, I have no idea what sort of circumstances you have, but maybe spend more time studying the game rather than grinding infinite hours or thinking about brm? It really doesn't take too many hands to move up in limits at the micro stakes with a decent win rate even with relatively conservative brm rules. If you're think about bankroll maximization from a broad perspective, the fastest way to improve your bankroll is to improve your poker skills.

Kelly is over complicated ;
http://en.wikipedia.org/wiki/Gamblin...rmation_theory

Keep it simple and stupid
play with %1 and upsize when u reach it,downsize when u lose it.

* dont change the bet size as u are running good that doesnt mean u should take risks.
> also when losing dont change it unless u broke the %1 bankroll,sometimes u play good but can run bad.

You'll enjoy it. You also might want to read Kelly's paper "A New Interpretation Of Information Rate".

Rogue, I have some issues with the whole thought process that is going on here.

First of all, Kelly's formula (I believe) is based on a game with known probabilities and odds. Take a ten sided die (yes I used to play AD&D and I'm a loser) with six X's and four O's. You can bet on X or O and get even money on your wager. Kelly wanted to know, if we had $1000 then how much should we bet on X to maximize our bankrolls growth rate. The environment that Kelly tested his Criterion in is controlled. Poker is not that type of game.

Secondly, you can't reverse the Kelly to find an optimal bankroll. Or, I should say that you don't need to. The optimal bankroll is infinite. But, it doesn't work anyway because the Kelly assumes that you'll be able to continue making bets at a fraction of a penny. You can't. If you use the Kelly to determine your bankroll size, and lose your first bet, then you'll have to move down in stakes. That move will only require half of your current bankroll (or less {$25NLHE to $10NLHE}) and you won't be playing "optimally" any more.

Third, you're treating your bi's LIKE bets. Essentially betting on yourself to have a "winning" session. I won't even start...

Fourth, what advice would you give your best friend if he told you that he was moving to Las Vegas to become a professional poker player with $700?

I agree that putting up 12% of your bankroll on 1 table is pretty risky. I mean, you had better be very confident that your edge is in fact 12%. I know that Chris "Jesus" Ferguson ran $1 he earned from a freeroll tournament at Full Tilt up to some $20k using a 5% bankroll management strategy. If a world class professional like Chris Ferguson feels he has no better than a 5% edge, do you really think you know something that he doesn't? Not trying to be rude, just trying to keep your goals realistic. Again, I could be wrong.

***Disclaimer: I'm going to come off as supporting the Kelly Criterion but I'm actually on the side that says this is impractical for most people. I'm just debating that it is practical on a theoretical level.***

A similar environment is present assuming the accuracy of the inputs to model your return distribution. In the example you provide, our bankroll is $1000, we have a bernoulli distribution of, bet size*[.6*(return on x)+.4*(return on o)]. In poker, on any given hand, we can have a $1000 bankroll, a normal distribution characterized by mean (winrate) and standard deviation based on how much we buy in for.

The major difference is that if we isolate ourselves to full buy ins, the increments that we can reduce our bet size turns into the difference of buying in full stacked at the different limits available. If we knew our win rate and standard deviation for every single buy in amount, we could actually reduce our buy in amounts all the way down to the min buy in at the lowest limit but this is impractical and it has a conflict of interest with growth of poker skills.

The reason why this shouldn't be used for poker is that the accuracy of the model is highly dependent on the accuracy of your inputs where your win rate is much more important than standard deviation. The accuracy of your inputs doesn't converge until you have a huge sample size and by then, any winning player should have moved up. The math is there.

We are not finding an optimal bankroll. We are finding the minimum amount necessary to be able to play a given limit with minimal risk of ruin. Thus, when deciding between two limits to play, we play the higher one we can safely play according to the Kelly Criterion assuming our risk/reward is greater at the higher limit. As stated above, we can not reduce the amount with risk on a continuous basis but instead, we reduce it on an incremental basis. Obviously our risk of ruin is not 0, but Kelly was originally trying to apply this on a monetary basis meaning the reduction was on a $.01 or more practically $1 basis and there is a minimum amount we can risk at casinos also so this isn't that far off from what Kelly was trying to do. This is why our win rate and standard deviation at the lowest possible limit needs to be extremely conservative to minimize our risk of ruin.

Umm... maybe I'm not getting what you're saying but isn't this what all poker players that are risk averse tend to do? I always expect to make money whenever I play poker. In the Kelly Criterion, a fundamental assumption is that on any given hand (as opposed to session which is a series of bets), our expectation is +EV. If it isn't, the Kelly Criterion would tell us not to bet (or buy in).

I like this question. It's a good way of objectively thinking about the use of this methodology because there are other factors (such as playing scared money and tilting) that don't enter the equation. There is some argument that says the Central Limit Theorem washes this away but I disagree with this and I'll leave this at that.

I don't think the environments are similar at all... Your basing your inputs off of past data and assuming that things will stay that way. In my die example, we don't need any past data. We know what the results are going to look like.

If someone handed us a spreadsheet with results from previous tosses of my die then we'd be better off lighting it on fire.

If we can not make approximations and assumptions in statistics, a huge portion of the field would be useless (sampling, central limit theorem, bayes theorem, etc)... It's like putting people on ranges in poker. A large portion of statistics is making educated guesses based on a set of assumptions and incomplete information and then finding the best possible solution.

To put your example into the context of this situation, imagine that you do not know what the probability of rolling each possible outcome is (for example, you have a six sided die where each side has a random number between 1-3). Can you not roll the die a lot of times to find out what the "true" distribution of the numbers 1-3 on the dice are? For example, if we roll it 100 times and we get the following distribution: 1: 22, 2: 62, 3: 16, we can be fairly confident that our six sided die has one side with a 1, four sides with a 2, and one side with a 3. In your example, we know the "true" distribution. The whole point of sampling (playing hands) is that we can make an educated guess as to what our "true" distribution is.

The environments themselves are similar because there is a true win rate and standard deviation for any given hand. The only difference is that we do not have perfect information when applying this methodology to bankroll management in poker. Note that I have pointed out several times before that the primary reason which I find the Kelly Criterion to be impractical is due to the lack of accuracy of input data...