I want to share some things I learned recently about optimal BRM and shoot taking. The First question is at what point a winning player should start to play higher games in order to maximize the rate of growth of his BR. It will depend on his WR on current stake and on higher stake as well as on variance on both limits. Formula for this is

C=((SIGMA1)^2-(SIGMA2)^2))/(2*WR1-2*WR2)

Where SIGMA1 and SIGMA2 are standard deviations in dollars on higher and lower limits, WR1 and WR2 are win rates in dollars and C is BR at which player should start playing higher limits. I'll call this critical bank roll formula. This formula is from book Mathematics of poker page 301 and it applies Kelly criteria to poker.

Example
Let's say you have 4BB WR on NL50 and you'll keep the same WR on NL100 and standard deviation is 100BB on both limits. Then for C we get C=(100^2-50^2)/(2*4-2*2)=1875$. So this player should take his first 1-2buyin shot right around 2k BR, if BR drops down below 1875$ or so he should go back to NL50.

Definition of win rate

When you use formula for critical BR, it's important to point out that the WR you should put in the formula is not exactly the same as WR you see in holdem menage. WR in that formula is all the money you put back in BR. If you don't intend to spend any money from poker winnings, then your WR is your WR from HM+ rakeback you get, if you do spend your poker winnings then you need somehow incorporate that in calculation. One way is just to subtract your average spending form WR and then use the formula.

What is better: 10 shots of 1buyin or one shot of 10buyins?

To answer this question will need one more formula. This is formula for risk of ruin in poker which is

R=e^(-2wr*b/SIGMA^2)

Where R-risk of ruin, wr is win rate, SIGMA is standard deviation. b is bank roll for a specific game(so if you are taking 5buyin shot your BR is 5BI) and e is euler's number which is 2.72.To understand whether 1bi or 10bi shots are better we get back to the example of NL50 player with WR of 4bb. Lets say he has a BR of 1k and wants to find the quickest path to NL100.

Strategy A

He'll grind up to 2900 and take 10BI shot on NL100 if loses 10BI on NL100 he'll move back down to NL50 regrind those 10BI until one of those shots stucks. First let's calculate how likely it is to have unsuccessful shot take with 10BI and 4bb wr. Using previews formula R=0.45. Now let's calculate how many hands our hero will play on average on NL50 until he moves up finally to NL100. Well first he has to win 1900$ to take his first shot with 4bb wr on average this will take him 95k hands, after that there is 45% chance shot wont stuck and hell be forced to move down and grind another 1k$ or 50k hands then there is 45% next shot wont stuck, so he moves down and so on... We get this sum
95000+0.45*50k+0.45^2*50k+0.45^3*50k+...
This is sum of the geometric series and sum of it comes as
45k+50k*(1/(1-0.45))=45k+50k*1.18=135k
So on average he will play 135k on NL50 until he finally moves up to NL100.

Strategy B

He will take a 1BI shot when he reaches 2k. Now using the formula for Risk of ruin for this kind of shot we get R=0.92. So there is only an 8% chance that this kind of shot will succeed, sounds pretty bad but let's do the math. Its similar to strategy A, but this time he needs 50k until the first shot and then he only plays on average 5k hands to regrind and take next shot, so we have
50k+0.92*5k+0.92^2*5k+0.92^3*5k...=45k+5k*1/(1-0.92)=45k+62500=107500

Surprisingly strategy B saved almost 30k hands! This is a very big difference and unintuitive result.

How to apply this to real poker?

Once you reach critical BR, my advice would be to start by introducing 1-2 tables of higher stake. Be very selective which tables you choose. This way often even if a shot fails you already will regradin some of the money back on other tables. It's important to be honest and realistic with what your WR is. We all have a tendency to overestimate what is true WR and when doing this kind of calculation it's better to be more conservative than to overestimate your EV. Also focus more on quality then quantity. Someone who has 6bb wr can move up to higher stakes much faster then 2bb winner, he needs to play less hands and he can take shots with lower BR, also 2bb winner often becomes a loser or BE player on higher stakes.

It's also important to be disciplined and move down when you have to and be prepared to handle bigger swings. This kind of strategy can take you very quickly 2 even 3 limits up if you start running hot but it also requires you to move down when its time to do so.

How quickly can you move from NL50 to NL400?

If you have 4bb WR at every stake (of course you need to work super hard to have 4bb on NL400). According to the previous calculation you'll play 107k on average until you finally move up to the next one. We need to beat 3 limits before NL400, that's NL50/100/200 so we can expect to be NL400 reg after 321k hands. If you happen to be a 6bb winner you can get there even faster within 153k hands! This shows how much more important it is to high WR over high volume when you move up the stakes.

Also here is table for critical BR for various wr



I hope this helps someone, there isn't much good information about BRM and its very important topic and it can save you a lot of time when you move up the stakes if you use a wise BRM strategy. Be sure to double check math. It should be fine, but it never hurts to check it. If you have any comments questions let me know.

Love this.

I've been saying this for a long time now. Thanks for sharing the math behind it!

The point is you have to be mega disciplined.

Which I get, a lot of players don't have the discipline.

Great info to have, thanks for taking the time to share! It’s good that you discussed the difference between taking money out of your winnings or not. I think it’s also important to factor in money that can be replaced as well. You can be a much more aggressive shot taker if you can reload easy.

hard to define how much worse and for how long (until u get used to it) u play when moving up or down in stakes and it might be very subtle and in different ways whether going up or down

nice that quality over quantity is solver approved

Great post!

Mathematics of poker was truly a revolutionary book. I never made it to page 301 and stopped around 150. But it's amazing that book was published in 2006.

That's good to know that a 1buy shot is more efficient than the 10 buyin shot.

Psychologically, it's a lot easier for me to stomach losing 4 buyins at a stake if I have a big bankroll behind me. I also feel I play better and go for thinner EV play's

If you don’t have the discipline risk of ruin is almost guaranteed. Which is why aggressive br management is better for those who can manage, “yeah I just folded 3 spots and now have to move down fml”. But it works if you’re a clear winner.

I disagree that smaller shots is unintuitive. If the math says you should be playing stake X, then continuing to play stake Y until you have 10 buyins won't maximize your growth because you're not playing the stake you're "supposed" to be.

There are practical considerations which make 1 buyin shots unreasonable though, IMO. What are you going to do if you lose 25BB? Top off? Now you're risking more than your 1BI. Play at 75BB? Now you're playing with less of an edge. Move down immediately? Ok but then maybe you win the money back at the lower stake in the next few minutes. This just adds too much micromanaging of tables.

I agree that 1bi shots might not be super practical, the main point is that you players should take small shots very often. Numbers you get are not super precise because we dont know true WR for every table we play, so you should take them more like guidelines. So losing 125 or 150bb wont hurt you in significant way imo. Simplest way is to take 1-2BI shot, so you play higher limit until you either get stacked or you lose 100bb in smaller pots, that way you always lose less then 200bb. You can also as you said not top off, wr dose not drop down that much with stack size, it dose a little bit but so dose your standard deviation.

In that case you just play highest EV tables almost always.

It's definitely important how BRM will impact your mental game.There are some mental game benefits to aggressive shot taking imo. You get more familiar with higher limits quickly, you learn how to handle more pressure, it's easier to motivate yourself to play/study because there almost always the next shot take just around the corner.

If you are willing to move up and down limits when you supposed to, your BR is much bigger than it looks. Let's say you have 1K and play NL50 that's only 20bi, but if you move down to NL25 if your BR drops below 500, and to NL10 if goes down to 200$ and so on, you need to lose 44BI to lose the entire BR chances of that are like 3%.

Excellent. Someone has read The Mathematics of Poker so I don't have to.

Nice to see someone else employ Kelly to bankroll management. I think a lot of the science behind this hasn't been well studied.

For anyone looking for a more intuitive explanation, I'd recommend reading this post. Essentially, you can think of each 100 hands as a loaded coin flip in your favor, with the standard deviation being the size of the bet.

https://forumserver.twoplustwo.com/s...36&postcount=5

Kelly translates most directly to formats like HU SnG:


https://www.instagram.com/p/Cbl2X45v1pg/

Something else to point out here is that Haizemberg's formula is presumably based on full-Kelly. This is like redlining your engine; it's the fastest you can go without imploding, but you're really flirting with the line. There's a very real risk of blowing up your engine/bankroll if you overestimate your WR or underestimate the variance. Most investors and financial advisers recommend half-Kelly or even quarter-kelly. A half-Kelly strategy is a lot more conservative while still capturing like 75% of the growth rate.

In the link you posted he doesn't take into consideration that you could lose more than 100bb over the course of 100 hands. It is still a good way to think about it but math is a bit more complicated.

Yeah full Kelly is quite aggressive. I think whether you should use it or not depends on how devastating it is for you to go down the stakes compared to going up. For Kelly dubbing your bankroll has the same utility in positive direction as losing half of it has in negative. So if your goal is to get to higher limits quickly then Kelly is the way to go, if you are already playing some decent limits and you don't want to risk going down, be more conservative. In theory Kelly investing has Risk of ruin equals to zero, because you constantly scaling down when you start losing.

You definitely should go with conservative approximation of your WR!

lol this is amazing

First of all, great post!

... but this seems wrong, no?

Let's clarify by changing risk of ruin to only 10%.

50k+0.1*5k+0.1^2*5k+0.1^3*5k... = 50 000 + 556 = 50 556 hands

But your formula of 50k+5k*1/(1-0.1) = 50 000 + 5556 = 55 556 hands

It seems we need to change this formula to: 50 000 + 0.1*5000*1/(1-0.1) = 50 556 hands?
Or even simpler put: 50 000 + 0.1*5000/(1-0.1) = 50 556 hands

(Also, I assume 45k is a typo?)

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In the end I actually get the same numbers as you, although I am not sure where you get some of your numbers from.
Like 95k turning into 45k in strategy A and 50k also turning into 45k in strategy B.

In any case, for Strategy A I get:

95 000 + 0.45*50000/(1-0.45) = 95 000 + 40 909 = 135 909 hands

For Strategy B I get:
50 000 + 0.92*5000/(1-0.92) = 50 000 + 57 500 = 107 500 hands

I think we just used different formulas for geometric sum.

In my calculations first term is 5k (that's why 50k turns into 45k) and in yours it's 5k*(0.92)^1. Both methods are correct.

when calculating win rate, Would we include all factors that make up ones win rate? table winnings and rakeback. Or should we only be assuming the post rake win rate?

All of that, but be conservative with your estimate if you don't have huge sample.

Hahaha, love it!

To be frank, my Quant fellah!

1.) Is Kelly's criterion actually still relevant? If so, how so?

2.) What is "correct" Bankroll Management? Is 200bb correct? 20bb? Or 2000bb?
Why? Because of Kelly's criterion?

Jeez, but I'm really disturbed by the consistent lack of articles.

Ok, let me read it again with the "articles in my mind".

Let's go back to the premise: What's even the premise of the Kelly formula?
Or in Russian English:
What is premise for Kelly formula?

The premise is some level of Risk of Ruin you don't want to overextend.

This premise, however, is up to debate. And with it, the whole formula.

Because: You might want to believe that you have more financial freedom than the Kelly formula suggests.
Or... you might even have less financial freedom.

Basically: It's useless.

If I do a 200 buyin BR management is vital if I live off poker.

If I don't... I can as well perform a 20 buyin BR management. Or even less.

Like right now, where I am losing 10bb/100 over 5k hands. 15 stacks. I don't care.
And I pay more, because I'm a ****ing losing reg.

So what exactly does the Kelly Criterion tell us?

That we need to be tighter?
Or less tight?

Or whatever?

I've been thinking about this. If we start by introducing just 1-2 tables, can't we move up even faster?

Say we introduce 1 table of NL100 and play 3 tables of NL50.
Then winrate and standard deviation increase by 25% and we get:
Kelly = (62.5^-50^2)/(2*(2.5-2)) = $1406

2 tables of NL100 and 2 tables of NL50:
Kelly = (75^2-50^2)/(2*(3-2)) = $1563

Kelly just gives you the fastest way to grow you roll. If you have enough roll for nl100 you play that, if not you play lower. This turn if you have same WR on all tables, ofc this is not the case so for very soft tables you need smaller roll. In practice there is usually 0-2 soft tables, so you just start with those.

I dont think you cant put average WR and stnd dev in formula if you are mixing limits. Playing 2k in one session where half of them is on nl50 and other half is on nl100 is same like playing 2 sessions of 1k hands first on nl100 and second on nl50.

I think Zamadi's idea is valid.

There comes a point where the Kelly Optimal stake is between the current one and the next one up. Mixing tables allows you to achieve that.

After reading chapter 25 of MoP, I think I was wrong.

Expected value obviously still increases with 25% in my example above.

However, I don't think you can just add 25% to the standard deviation.

Instead, to calculate the standard deviation of a "portfolio" of uncorrelated bets it seems we have to raise the standard deviation of
every bet to the power of two and then take the square root of that sum.

So if we have a std.dev of 10bb/hand at both NL50 and NL100.

Simultaneously playing 1 hand of NL100 and 3 hands of NL50 gives us a std.dev of: SQRT(3 * $5^2 + $10^2) = $13.2 per 4 hands.
$13.2 per 4 hands = $13.2/SQRT(4) = $6.61 per hand

$6.61/$5 = 1.32

So our std.dev increases not by 25%, but by 32% when playing 1 table of NL100 and 3 tables of NL50, compared to playing only NL50.

This gives us a kelly value of: (6.61^2-5^2)/(2*(0.025-0.02)) = $1875 = exactly the same as if we changed all the tables at once

But introducing just 1 or 2 tables to begin with still makes practical sense, for the sake of table selection and mindset.

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However, if we had to choose between playing 2 tables of NL50 or 1 table of NL100 for the rest of our life, we would prefer 2 tables of NL50, because it allows to achieve the same $/hour, but at a lower standard deviation.

1 table of NL100: 100 hands per hour at a std.dev of $10/hand = $10 * SQRT(100) = $100/hour std.dev
2 tables of NL50: 200 hands per hour at a std.dev of $5/hand = $5*SQRT(200) = $71/hour std.dev

I think you have to take the weighted average of the sample size for each group, like so:



This formula doesn't work for one hand. But we can model it as playing 75 hands at NL50 and 25 hands at NL100:


Converting back to a single hand: $55.32 / sqrt(sample size) = $5.532

So the average standard deviation when mixing stakes like this is $5.532 per hand, or $55.32 per 100 hands.

I haven't studied math at this level and I don't really I understand the formula... but the answer feels wrong.

Unless I am missing something, this would give us a kelly value of: (5.532^-5^2)/(2*(0.025-0.02)) = $560

That's lower than the kelly of going from NL25 to NL50 ($938).

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Also, I am able to replicate the answer in the sales example in the link, but I am not able to replicate your answer.

Trying to do use that formula I get: SQRT((74*43.3^2 + 24*50^2)/98) = $45 per 100 hands

Not sure where I went wrong...