SnG HU - investigation of the game according to the Kelly criterion.

[BelFish]

The Kelly Criterion is commonly used in betting. In this case, it is possible to place any bets. In SnG HU, we initially cannot place bets of any amount, but only bets equal to the buy-in values. But just as in the case of any bets, there must also be an optimal betting system, in which the bankroll growth rate is the most optimal (quite fast and at the same time with a relatively small risk of ruin). Compared to any other betting systems, on average, the biggest profit should be obtained.

Gradually, in this topic i will write what i received in this study.

P.S. There are no final results yet (the optimal bankroll values for transitions to limits are higher), but the data obtained is very close to optimal). So i will be glad to any ideas and discussions on this issue. The results are quite interesting...

 

[BelFish]

If we approach the question purely from the point of view of mathematics, then we get the following problem:

We will consider limits 2$/5$/10$/25$/50$

And there are game statistics at these limits, that is, a set of ROI values. Or, for the case of SnG HU, it is better to use the frequency of wins, and not ROI.
For example, the winning frequencies at these limits can be as follows:

61.5%/58.4%/56.3%/54.7%/53.6%

And you need to determine at what bankroll values it is best to move to higher limits.

The start is supposed to be at the lower limit with a relatively small bankroll.

 

[BelFish]

To make it more interesting, i'll ask this question:

For example, a person has the following winning frequencies:

64.7%/61%/58.4%/56.3%/55.2%,

which is equivalent to ROI on limits 2$/5$/10$/25$/50$ (including rake): 21%/16%/12%/8%/6%

He start on limit 2$ SnG HU with a $10 bankroll, but in case of losses in unsuccessful streaks, he later can make mini-deposits (10$) 9 more times.

In your opinion, which of the 3 schemes for raising the limits proposed below will be better?

1.)

5$ - 100$
10$ - 200$
25$ - 500$
50$ - 1000$

2.)

5$ - 30$
10$ - 80$
25$ - 300$
50$ - 800$

3.)

5$ - 15$
10$ - 30$
25$ - 100$
50$ - 200$

Here, everywhere the first number is the limit, and the second is the bankroll value for moving to this limit.

 

[BelFish]

He start on limit 2$ SnG HU with a $10 bankroll, but in case of losses in unsuccessful streaks, he later can make mini-deposits (10$) 9 more times.

Well, or there may be such a variant of the game: start with a bankroll of $80 on a limit of $10 SnG HU with the possibility of 2 mini-deposits of $10 each in case of ruin. This version of the game will be closer to the scheme of the game according to the Kelly criterion, since it is possible to go down to the limits lower during streaks. And the transitions between different limits are the same as indicated in the 3 schemes above.

 

[BelFish]

Here are 2 calculators i use to research this problem:

Kelly Criterion Calculator: Calculate what your stake should be

Make your sports betting experience easier with our Kelly Criterion calculator, learn about the Kelly Criterion and how much you should wager to maximise your profits!

kellycriterioncalculator.com

Tournament Variance Calculator

Poker Tournament Variance Simulator – calculates variance for poker tournaments, MTTs and SNGs.

www.primedope.com

I also use a program in Excel, which, according to a given algorithm, makes transitions between SnGs HU limits

 

[BelFish]

By the way, i found a bug in the "primedope" calculator. Using the formula for BRM, you can calculate the exact risk of ruin. You can also get it by simulations. There is also a classic task from the probability theory about the ruin of a player, there is then its own formula, but without the rake. So, the calculator considers the risk of ruin for the case as if we had exactly 1 buy-in more. This is shown in my screenshot above. There should be just such a risk of ruin with 2 buy-ins, and not with one, i.e. if starting from 4$, not from 2$. This is where i have it indicated by an arrow on the screenshot.

For cash games, there are a lot of BBs in the bankroll and this error has almost no effect, but for SnG HU, the calculator gives errors... We must remember that the real number of buy-ins is 1 more than you write!

Too lazy to write to them, maybe they will fix it someday ))

 

[BelFish]

During this time, i gradually performed many simulations with schemes with different win rates.

For example, a scheme in which we start with a bankroll of $80 on limit of $10 SnG HU:

















Average with confidence intervals. 500 "runs" for 2000 SnGs HU according to 2 schemes. A mega-fast scheme with 3BI of each limit for go higher is very likely is better:



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Also created schemes for new, lower win rates. And in the program i made it to show ROI for win frequencies. I'll collect statistics later.









Here, the schemes start not from the middle, but from the bottom (limit $2), from a deposit of $25, as i myself started playing.


P.S. And even later, most likely, i will make game schemes for very low ROI values. For example, what can be in hyperturbo SnGs HU. And i'll make an even faster scheme with 2BI for transitions to each next limit. It seems that if there is an "airbag" (a sufficient supply of buy-ins at the lowest limit) with which the risk of ruin is minimal, the higher the speed of transitions, the faster the bankroll builds up, although not much faster. But on the other hand, it is much faster than playing according to the "clean" Kelly scheme. You can even take a seemingly ludomaniac scheme for the game - and it will give a better result than playing according to the Kelly criterion! The main thing is that a person can endure "corrections" (jumps on the profit chart up/down for a long time in the form of long horizontal sections), not to "break" psychologically.

P.S.2. But in general, even though such charts with "corrections" come out quite often, but not always, there are many different types of charts.
The best chart that has turned out of all time is at the very bottom for scheme with start of 80$. There, everything went like clockwork, more than double the average profit rate turned out

 

[D0nk3y Hunt3r]

One thing to point out, HU SNGs are a different beasts, for example on GGpoker there is no such sngs format available. The visible combinations force to play shortstacked in some parts of tournaments, which in consequence create shovefest requiring bankroll to make a volume.

Thank you for your analysis, very helpful. My take on this: Kelly criterion is fully dependent on given probability, so to make it work you need to keep it at minimum same level. So we are coming back to same old story.

 

[BelFish]

Well, in general, the Kelly Criterion was developed for betting, where you can choose the amount of the bet yourself and constantly reduce it with an unsuccessful streak to almost zero bet values ))

Under such betting conditions, you get a game with almost zero chance of ruin.

But in SnGs HU, we cannot choose the amount of the bet ourselves, but we can only bet values equal to the buy-ins. Also, we cannot go below the minimum limit.

The point of the study is that simulations show that it is possible to play much more aggressive betting schemes than based on pure Kelly criterion, and it will be more profitable.

If you look in Kelly's calculator what bankroll is required to move to a higher limit, then it will show much higher values for the same win frequencies than i have in the schemes. And if you look in the book "The Mathematics of Poker" by Bill Chen and other authors, then in the chapter devoted to the Kelly criterion, an even more conservative scheme of play is proposed than in accordance with the Kelly calculator. If you play SnGs HU with such transitions between limits as the calculator or book advises, then this will greatly slow down the growth of the bankroll compared to using a truly optimal scheme...

 

[BelFish]

And in the program, you can choose any other frequency of wins (ROI) and the size of the rake in SnGs, even for hyperturbo SNGs. For any variants of SnGs HU from various rooms, you can calculate and run simulations.

For example, for such ROI values on similar limits:

5%/3.5%/2.5%/1.5%/1%

 

[BelFish]

Indeed, there are probably a lot of players playing at an overly conservative BRM, using a pure Kelly criterion to determine levels on the Kelly calculator for moving to higher stakes. If you look closely, this calculator also has fractional bets in increments of 10% of the optimal bet, which are used for an even tighter BRM.

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The whole idea is that since in SnGs HU we cannot go below the minimum limit by reducing the bet, as it can be done in betting, where the risk of ruin is virtually zero, then in order to compensate for this in the case of playing SnGs HU, we must take such a reserve of buy-ins of the lower limit, so that here, too, our risk of ruin tends to zero.

To find this number of buy-ins where the risk of ruin is less than 0.5%-1%, we use the "primedope" variance calculator. For each specific frequency of wins at the lowest limit, which differs for different players, there will be a corresponding risk of ruin. The resulting number of buy-ins i called the "airbag" above. In the screenshot below, such a number of buy-ins of the lower limit (in this case, the $2 limit) is selected for the case of a player with a ROI of 15%, in which the risk of ruin is less than 1%. For this value of ROI, this number of buy-ins is ~ 15

 

[BelFish]

And with so many buy-ins on starting limit, in the beginning we're playing tighter than in pure Kelly criterion. But this happens only at the lowest limit! And all the moves to higher limits are already based on a very aggressive BRM, which leads to the scheme of moving up to limits much more aggressive than in the pure Kelly criterion! And i have not seen this anywhere in the literature, which is why i started this study.

In fact, we have a mathematical problem with its own boundary conditions, and each boundary condition greatly affects the result! As an example, we can cite such an experiment, as it is written in the screenshots below.

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Here, the boundary conditions greatly influence what the optimal betting system should be: it turns out that instead of betting at 20%, which would be optimal, the boundary conditions reduce this value and lead to the fact that bet of 12% of the bankroll became optimal.

 

[BelFish]

In general, i wonder if anyone from the forum plays SnGs HU at all?
Or is this poker format played by a very small percentage of all players?
It seems that there is even a special forum about SnGs HU, and it's quite big...

 

[CheezeWiz]

BelFish,

I just happened to have some time tonight and I stumble upon your thread, and since I see you have put a lot of time into this thread, my main reason for posting is to let you know that I find it interesting. Thank You!

Disclaimers are 1) I do not play much heads-up sng's 2) I am just a recreational player and do not play to strict "Bank Roll Management" but then again, I try not to play stupid/careless with my Poker/ Wagering $'s either, 3) I am only very generally familiar with the Kelly Criterion, but I get it... I almost consider it to be like a mathematical version of common sense / realistic / responsible play/wagering, but then again, I am not trying to make a living at it. For me, it is more like just trying to be semi-responsible with my entertainment dollars, and 4) I did have some difficulty understanding the details of some of the charts and graphs, but I think I got the general idea.

Having said that, I will say that I enjoy math, recreational poker and sports betting. So I will make a few comments, and you can correct me where I am off base, mainly for purposes of my understanding and to just engage in this conversation. So I think I see where you are going with this... it seems you are thinking that you can play more aggressive than the Kelly Criterion would dictate, and be more profitable... but you give yourself the opportunity of being able to re-load or possibly add to the bankroll. I think this is where the difference comes in... The Kelly Criterion, I believe, is based on avoidance of ruin of a fixed bankroll. So it is not surprising to me, that if you allow yourself some opportunity to re-load, if necessary, or possibly do some add-ons to bankroll, if necessary, that you could post more favorable results.

So if I am following up to this point, I get that if you are playing responsibly within your means, and having good results at various levels of progression in stakes. In this case, I think this is a very reasonable and interesting topic to explore. I think the next question becomes, at what point do you "take your profits" and reset, or move back in level of play? Otherwise, without this reset/profit taking, I would think you do ultimately face the risk of ruin as a result of being more aggressive than the Kelly Criterion dictates. This is probably where I would need some more elaboration on the direction of the experiment, or just a better understanding of the Kelly Criterion, lol. And of course, that is assuming that I am following where you are going with this. I very likely could be off-course in my interpretation, and feel free to correct me.

Thank You Again For The Work You Have Put Into This Topic, Belfish!

 

[BelFish]

The main point is that even though there is the possibility of additional deposits, in contrast to the pure Kelly criterion with a fixed bankroll, in total, the starting bankroll is much less than if you start playing with a conservative BRM.

In SnGs HU, many players play with raising limits according to the Kelly Criterion chapter in Bill Chen's The Math of Poker, and many even use fractional bets up to 0.1 or 10% of the suggested Kelly Criterion bet, which gives much more slow bankroll growth. Why should we move up the stakes slowly when we can move up much faster, with virtually zero risk of ruin, and starting with a small bankroll (even with the possible of additional deposits).

And the profit (for cashouts) will be after reaching the upper limit. On almost all charts, at the chosen win rates, fixing at the upper limit occurs much earlier than at the end of the 2000 SnGs distance chosen in the simulations.

And there is always the possibility to break through to the upper limit with lightning speed and gain a foothold there in the event of a successful streak, which will happen much more slowly if we play with more conservative BRM.

P.S. But it is very important to go down the stakes when necessary, and not to continue playing in the hope of luck, and also to correctly evaluate your win rates at various limits, so that the risk of ruin remains really minimal!

 

[BelFish]

I think a lot of players play according to the chapter on the Kelly Criterion from Bill Chen's book, and this results in slower bankroll growth.

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It would be possible to perform simulations with the game without additional deposits. Then the risk of ruin would be as in the pure Kelly criterion and some of the simulations would lead to ruin. But at the same time, the final charts for cases where there was a ruin would give approximately the same profit as on average: the chart would first cross the minus mark by a small value, and then go up - and we must assume that there was a ruin, that is, instead of these final bankroll values, we would have to add zeros to find the average profit for a long distance. This is inconvenient for comparison with conservative schemes, since there are almost no cases of ruin there.

Therefore, it is more convenient to make such a normalization so that here too there is no significant percentage of ruins, as in the case of a game according to the pure Kelly criterion. The main thing is that the starting bankroll at the same time remains very small, and we move on hihger limits at bankroll values that are much lower than in the case of conservative BRM options.

P.S. It would be better if the topic was called not a game according to the Kelly criterion, but a game based on the Kelly criterion, and these are different things!

 

[wavetune]

I roughly understand what I'm talking about, firstly, how do you determine the probability, and secondly, if you get 6 cons in a row, how much interest will you have from your bankroll in this case?

 

[BelFish]

If by probability you mean the risk of ruin (probability of ruin), then it can be calculated using the BRM formula:

R=exp(-2*m*Br/D)




R - risk of ruin
m - win rate
D - dispersion
Br - the required bankroll at which the risk of ruin will be set, for example 1% or 5%

But it's easier to calculate in the "primedope" calculator, since it calculates based on the same formula.

Tournament Variance Calculator

Poker Tournament Variance Simulator – calculates variance for poker tournaments, MTTs and SNGs.

www.primedope.com

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If we lose 6 times in a row, then we will have 6 less buyins )))

After all, we are looking for the total risk of ruin, and not after we lose or win a certain number of times...

 

[BelFish]

Let me clarify that the last scheme with 5-6 starting buy-ins was taken for a case fairly high ROI (the frequency of wins at the lower limit is 62%). With this frequency of wins, the probability of losing 6 buy-ins is very small. And also, in accordance with the scheme, this is not a complete bankroll, but there is some reserve of buy-ins calculated on the calculator. And in a significant number of cases, this additional bankroll will not be needed, i.e. he may not even be there at the beginning, but for example, he will gradually play enough in freerolls or, if such a rare loss happens (6 or more times in a row), then it will be possible to return bottles worth several additional buy-ins at the glass container collection point )))

And for a scheme with other ROIs, the starting bankroll will be different ... For lower ROIs, the starting bankroll will be larger.

 

[wavetune]

I will try this scheme on sports betting, but for this I need to first raise my bankroll and not withdraw money
in any case, it's interesting

 

[BelFish]

I will try this scheme on sports betting, but for this I need to first raise my bankroll and not withdraw money
in any case, it's interesting

But only in sports betting, the ROI is usually quite small compared to the possible ROI in SnGs HU, probably rarely anyone can have an ROI > 5% at a distance, most likely the ROI will be about 1%-3% for winning cappers, and a thicker one is needed there "airbag" (stock of minimal bets). Everything must be calculated on the "primedope" calculator, choosing 10% in the rake column. You also need to split the bets into conditional limits that are convenient for you, for example, the minimum bet is $1, then $2, $4, and so on. Then it will be possible to apply a similar increase scheme at these rates.

P.S. If you calculate everything wrong, then the risk of ruin can be quite high, so if you apply scheme, it is better to be very careful when calculating!

 

[TeUnit]

I tend to look at the optimal mix as whatever allows you to maximize your hourly earning rate (assuming you are properly or over rolled for the stakes). Sometimes the higher stakes games are harder to fill and regs may seek out higher volume stakes. Also fish may migrate between stakes.

 

[BelFish]

Right! It is necessary to look at the circumstances for various types of SnGs HU.

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I also decided to write in more detail about the BRM formula and its application.

In general, to determine the correct BRM in any game, it is best to use the BRM formula, because it is universal, that is, it is suitable for calculations for absolutely all games, and not just for varieties of poker formats.

R=exp(-2*m*Br/D)

To apply this formula, all you need to know is your win rate and variance for the selected poker format, taken from any poker tracker.

For clarity, i will show how this is calculated for cash games. For example, in a tracker at a long distance, a person has a win rate of 7.5BB/100 and a standard deviation (std.dev. in the tracker) of 100BB/100hands.

For finding bankroll from the formula:

R=exp(-2*m*Br/D)
Ln(R)= -2*m*Br/D
Br=(D/2m)*Ln(1/R)

We choose the risk of ruin ourselves and usually for cash games it is chosen around 5% or 0.05

Then Br=(100^2/{2*7.5})*Ln(1/0.05)=(10000/15)*Ln(20) ~ (10000/15)*3=2000

2000BB is 20 cash game stacks. I think that this is how the standard well-known rule was obtained that you need to have at least 20 stacks to play at the limit you want to play.

For case with higher win rate, you will get a more aggressive BRM, for example, 10 stacks will be enough for someone.

Also, someone may want to play with a risk of ruin of less than 5%, for example 2%, or even 1% - then the formula will give a more conservative BRM with a larger number of stacks.