The Kelly Criterion for various ITM-ROI combinations in MTT

[D0nk3y Hunt3r]

For info about Kelly criterion, check this thread: SnG HU - investigation by BelFish.
I've calculated info for most common ITM I see in MTTs: 5%, 10%, 15%, 20% and 25%. Numbers below 0, indicated in color, can be ignored, it just means that for selected combination it's advisable to go back and take CC forum crash course, again .

Here you can find gallery of charts: Kelly criterion charts

 

[BelFish]

In truth, Kelly's criterion can only be applied to games with a binomial distribution. For SnGs HU, the distribution is binomial, since it is only possible to win the buy-in or lose. And for cash games or tournaments, it will not work to apply the Kelly criterion. Some specific limit raising scheme can be used, but it will not be related to the Kelly Criterion.

It's just that in tournaments and cash games, the distribution is such that winnings can be of various sizes, and not like in the case of SnGs HU.

 

[BelFish]

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)




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.

Primedope Tournament Variance Calculator 2026

Master MTT and SNG swings. Analyze risk of ruin and bankroll management with our free MTT variance calculator. The industry gold standard since 2013.

www.primedope.com

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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.

 

[D0nk3y Hunt3r]

In truth, Kelly's criterion can only be applied to games with a binomial distribution. For SnGs HU, the distribution is binomial, since it is only possible to win the buy-in or lose. And for cash games or tournaments, it will not work to apply the Kelly criterion. Some specific limit raising scheme can be used, but it will not be related to the Kelly Criterion.

It's just that in tournaments and cash games, the distribution is such that winnings can be of various sizes, and not like in the case of SnGs HU.

In this case it works, because I calculated things for different ROI, meaning different prizes. When you consider ITM, first prize after bubble is usually double the buyin, you look for 100% ROI with your ITM, as a consequence you treat it like binomial something something (I'm not a mathematician ). So it's like for the worst case, you either lose MTT or get to first prize ITM.
Anything better would just work in plus.

 

[BelFish]

Then it is not very clear what the ROI column means for you. (If you consider the minimum prize or better as a simple return of 100%). For example, for the case of an ITM of 15%, what does an ROI of 600% mean for which a BRM of 120 buy-ins is obtained?



It just seems to me that even few professional tournament players can have an ROI of more than 100% over a long distance, and in this table the minimum value is as much as 600%...

P.S. I would still try to apply the BRM formula. You just need to find in the tracker typical winrate and variance indicators for MTTs players at a good distance and calculate what kind of bankroll is required for various indicators.

 

[BelFish]

You get such high ROIs because you are resetting all prizes greater than about 1 buy-in to the value of one buy-in. Therefore, in order to have a profit under such conditions, a person would have to have an ROI of 600% ))

In order to correctly take into account the risks of ruin and determine the correct BRM, it is impossible to count in this way, but it is necessary to take into account exactly all payments, which they are. But then the distribution will come out far from the binomial...

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Also, the risk of ruin is highly dependent on the number of participants in the tournament. To illustrate this, i have considered in the tournament calculator 3 options for playing with a $5 buy-in:

1.) Tournaments for 200 participants
2.) Tournaments for 500 participants
3.) Tournaments for 1000 participants

In all 3 cases, a typical winning player with a normal ROI of 20% and a starting bankroll of 200 buy-ins is considered.

The screenshots show how the risk of ruin changes dramatically, increasing from 4% with 200 participants to 55% (!!!) with 1000 participants. And also on the screenshots, a line is circled showing which bankroll is required so that the risk of ruin does not exceed 5%

Spoiler

Here, in order for the risk of ruin not to exceed 5%, about 168 buy-ins are required.

Spoiler

Here, in order for the risk of ruin not to exceed 5%, about 459 buy-ins are required.

Spoiler

Here, in order for the risk of ruin not to exceed 5%, about 1000 buy-ins are required.

 

[D0nk3y Hunt3r]

Then it is not very clear what the ROI column means for you. (If you consider the minimum prize or better as a simple return of 100%). For example, for the case of an ITM of 15%, what does an ROI of 600% mean for which a BRM of 120 buy-ins is obtained?

My interpretation: if, on average you are getting 6 buyins of positive return (meaning winning 7 buyins), from 15% ITM, you should use 1/120th (0,83%) of bankroll to get best results. If you are not getting at least 6 buyins on average from 15% ITM, you are losing long term. Actually this is almost break even situation.
If you are getting better results (better ROI), you can increase percentage of bankroll as a stake.

 

[D0nk3y Hunt3r]

You get such high ROIs because you are resetting all prizes greater than about 1 buy-in to the value of one buy-in. Therefore, in order to have a profit under such conditions, a person would have to have an ROI of 600% ))

I may be in wrong, because I transformed on my own equations from wikipedia , though my interpretation is that you need to get 600% ROI from 15% of MTTs. My intuitive understanding of this binomial thing is that every 15%*600%-85%*100% would be kindofish EV? Of course it's the transformation to "binomial" of advanced case, because you are not getting exactly 600% every 15% but you accumulate it from series of MTTs getting different ROI through the series.

The screenshots show how the risk of ruin changes dramatically, increasing from 4% with 200 participants to 55% (!!!) with 1000 participants. And also on the screenshots, a line is circled showing which bankroll is required so that the risk of ruin does not exceed 5%

Yes, I have seen this before, I wonder how they calculated in number of participants.

 

[BelFish]

Yes, I have seen this before, I wonder how they calculated in number of participants.

When there are many participants, it is more difficult to break through to good prize payouts, and without them it is difficult to maintain this ROI. In order for the ROI to remain the same, it is necessary to get into good prizes more often.

P.S. And it's pretty easy to calculate using simulations. There ordinary normal distribution. Even in excel, you can create a mini program using the function of generating normally distributed random variables.

 

[BelFish]

My intuitive understanding of this binomial thing is that every 15%*600%-85%*100% would be kindofish EV?

The equation is true, but it means that 85% of cases you will lose buy-in, and the remaining 15% of cases should win an average of at least 6 buy-ins. That is, each hit in ITM must be on average with profit of 6 buy-ins throughout the distance and as a result there will be a ROI of only 5%.

You can make some models for various types of winning players. I will show on the example of tournaments with 200 participants and a prize structure of 15%.
If absolutely all the players of the tournament would have an equal level of the game (skill), then at the distance everyone would take an equal number of different places in the tournament. So, at the expense of the rake, everyone would play with a negative ROI. So that the ROI become positive, the number of prizes occupied should be a little more than the number of prize places in the table of payments from the screenshot. Consider the case when this amount increases evenly for any place by exactly 10%



5.5 — +7$
5.5 — +8$
5.5 — +$9
6.6 — +$13
1.1 — +19$
1.1 — +23$
1.1 — +31$
1.1 — +40$
1.1 — +59$
1.1 — +68$
1.1 — +85$
1.1 — +134$
1.1 — +231$
167 — (-5 $)

That is, then it turns out that out of 200 tournaments, the player will pass into prizes not 30 times, but 33 times, which corresponds not to 15% ITM, but 16.5% ITM personally for this winning player. It turns out that he "takes" part of the places from players who play worse, for which the number of hits in ITM decreases, for example, to 13.5%
To make there are no fractional values for the number of different final places, we multiply the number of tournaments by 100. So we get this balance for profit:

16700*(-5$)+550*7$+550*8$+550*9$+660*13$+110*(19$+23$+31$+40$+59$+68$+85$+134$+231$) = -83500$+97680$ = 14180$

14180$/20000 = 0.71$

ROI = (0.71$/5$)*100% = 14.2%

97680$/3300 = 29.6$ = 5.92bi

It turns out that for such a model, the player should on average win 5.92 buy-ins, when he goes into prizes.

I think that in reality, good players hit not in equal proportion into all prize places, but occupy fewer places with small payments and more places with large payments, by playing at the right moments with the justified risk. Later i will try to make a more realistic distribution model in the prizes occupied.

 

[BelFish]

By the way, i made a mistake in the above with spoilers, because i did not notice that there the number of prizes is indicated not in percentage, but in the number of participants.

It turned out that there for tournaments for 200, 500 and 1000 participants chosen only 15 prize places, and not 15% of prize places )))

So the risk of ruin increases not so sharply. Not from 4% to 54%, but from 0.7% to 19%. And the number of buy-ins required is less than indicated in that post.