After my 1000th post on average bubble factors was well received I decided to do another number crunching one for my 2000th post.

What this is and why it matters

The Kelly Criterion is used in many forms of +EV gambling to tell us how much we should wager on one result in order to grow our bankroll as quickly as possible.

The essence of it is that we do not attempt to maximise our average expected bankroll after the current bet (otherwise we would bet our entire bankroll on anything +EV), rather we try to maximise the logarithm of our bankroll.

In addition to this fast way of growing our bankroll there are two slow ways. Playing lower than Kelly Criterion is clearly slow because you win smaller amounts relative to your bankroll. Less obviously, playing too high is also slow because the losses cut into your bankroll quickly and you end up dropping lower and for longer than you would under Kelly.

On the internet there are various Kelly calculators but these are designed for (e.g. sports) betting with only two outcomes. They are fine for HUSNGs but in MTTSNGs we need something else.

How this is calculated (TL;DR)

The calculations below are based on increasing the likelihood of a person finishing in all the money placings until their ROI reaches the required level (this also applies with a zero ROI. In a 9 man $0.90+$0.10 SNG a break even player finishes in each cash position 12.63% of the time rather than 11.11%). To a certain extent this is a limitation of this model. See 3) below. The % rake also affects these calculations (if you are maintaing a set ROI despite a high rake you have lower variance in your results) these numbers are based on rake at turbos currently found at $1.50 level on pokerstars * but without rounding prizes to the nearest cent.

As ever, there are no guarantees of accuracy but in testing my results for HU do agree with the online calculators.

Notes on the numbers

The numbers shown are the ideal fraction of your bankroll (eg. 100 = 1 percent, 50 = 2 percent, 1000 = 0.1 percent) you should risk on any particular tournament. Obviously you need to find an actual tournament that most closely matches this number (or sell action to people). On each line there are 10 numbers, representing the correct fraction for a group of 10 ROIs.

The actual numbers

9 player SNG paying 50-30-20
ROI
1-10% 200 100 67 50 40 33 28 25 22 20
11-20% 18 16 15 14 13 12 11 11 10 10

So if we have a $200 bankroll and we believe we have a 7% ROI in 9 man SNGs, we select the 7th number from the 1st row, which is 28. This means our ideal buy-in level would be $200/28 = $7.14. BTW the apparent mathematical perfection (i.e roundness) of these 9-man numbers is a coincidence and only appears with this particular rake structure and payouts - and also when we round to the nearest whole number.

18 player SNG paying 40-30-20-10
ROI
1-10% 376 188 125 94 75 62 53 46 41 37
11-20% 34 31 28 26 24 23 22 20 19 18
21-30% 17 17 16 15 14 14 13 13 12 12

So if we have a $180 bankroll and we believe we have a 12% ROI in 18 man SNGs, we select the 2nd number from the 2nd row, which is 31. This means our ideal buy-in level would be $180/31 = $5.80.

45 player SNG paying 31-21.5-16.5-12.5-9-6-3.5
1-10% 706 352 234 175 140 116 99 87 77 69
11-20% 63 57 53 49 46 43 40 38 36 34
21-30% 32 31 29 28 27 26 25 24 23 22
31-40% 21 21 20 19 19 18 18 17 17 16

180 player SNG paying 30-20-11.4-7.4-5.8-4.3-3-2.2-1.5-nine times 0.95-nine times 0.65 (rake based on the turbo $2.50)
1-10% 2455 1221 809 604 480 398 339 295 261 234
11-20% 211 193 177 163 152 141 132 124 117 111
21-30% 105 100 95 90 86 83 79 76 73 70
31-40% 68 65 63 61 59 57 55 53 52 50

So we've discovered that the 100BI rule is perfect for someone playing 180s with an ROI of 22%

1362 player MTT (based on today's $1.10 $1K guaranteed)
1-10% 7574 3748 2473 1836 1454 1199 1017 881 775 690
11-20% 621 564 515 473 437 406 378 353 331 311
21-30% 293 277 262 249 236 225 214 205 195 186
31-40% 179 172 165 158 152 146 141 136 131 126

So for MTTs we need a huge bankroll.

Things to consider when using these numbers

1) It is difficult to know your true ROI and due to regression to mean (see Kahneman's Thinking Fast and Slow) it's probably lower than you've observed to date.

2) In poker, as distinct to other forms of gambling, when "betting" smaller in lower buy-ins you should have a higher ROI. This is also a reason to play slightly more conservatively than the above.

3) As a winning player you should have a greater proportion of top cashes than the flat model used in the calculations, again a reason to play slightly more conservatively.

4) Unless you 1-table, you already enter further tournaments before you know your result in the present one. If you are close to the border between two buy-in levels this could be a reason to make the subsequent tournaments a lower buy-in

5) Somewhat counterbalancing 1)-4) is the idea that you learn faster when playing higher buy-ins.

6) You probably have some kind of hidden benefits such as rakeback or FPPs not included in your observed ROI but worth a few percentage points.

Overall I would knock only 5% of your observed ROI and no more, but that's just personal preference.

And finally:

Like all (sensible) forms of BRM. You also need to be able to move down when you no longer meet the criteria to play higher.



* For a 1% ROI in the 9 man, the rake makes a difference between 200 (based on $1.32+$0.18) compared to 241 (no rake at all). For a 20% ROI in the same tournament the number with no rake would be 12 rather than 10.

Very interesting post, thanks!

However, what would you recommend a person who doesn't yet have a good sample to estimate ROI? Taking 1-2% ROI as maximum and re-evaluating after let's say 500 games?

Yes, something like that.

It also depends what other games you can beat. I'm moving over from reg speed 45s to turbo 180s at the moment (actually playing a lot of cash too) so I'm guessing at an ROI based on the 45s.

How many 180s would it take to get a good estimate of my true ROI?
Currently my ROI is 58.3% over 554 180s, which is obviously unsustainable. What would be a reasonable guess at my true ROI given those figures?

Mana do a search for this. There are a few very detailed threads about this that are quite interesting. But it's a really crazy number definitely above 5k games possibly even 10k games.

^ Assuming those are turbo 180s it's unsustainable. TBH I would cap it at a max 25% (others would say even lower) at least till you have 2000 games and just keep adjusting as more information comes in showing a win rate lower than 25% (which will involve dropping down stakes perhaps but that's what you have to do).

I'd say at this stage it's most important to get using a proper BRM system even if you don't actually know your ROI very accurately.

Also - the Bankroll/ROI containment thread would be the best place to ask.

Thanks for the help, I've been sticking to a 100BI BRM system for the 180s and dropping down immediately if I fall below that. The Kelly criterion maths you have done gives me peace of mind that this should be enough for now/

Hi there, thanks for posting this. I'm having some trouble working out the optimal amount to sell and the optimal amount to buy for HS live tournaments - there's very little info out there i could find for applying the kelly criterion on MTTs - could you post a little more about your methodology/ link us to the spreadsheet you used to come up with these numbers?

Could do a few models from 1% thru 5% and -1% thru -5%. Beyond that, Kelly's got a pretty light touch and your dilemmas solve themselves without calculations.

Are you selling for markup? I don't think I've done one like that before.

i think the principle is the same though.

Make a column with all the possible scores (including failing to cash),

Next to it make another with the probability of you hitting that score. They should add up to 1 - I just upped the probability of all the cashing places by a fixed amount to get a particular ROI and reduced the no-cash one to make them add up to 1. Potential pitfalls here are forgetting that "equal chances" give you a negative ROI because of the rake, so to simulate a given ROI you need to increase these probabilities to cover the rake first. Another pitfall is remembering that some scores are paid out the same to more than one person so are much more likely than a score paid to only one person.

Next make another column with your bankroll after that score (you need to be consistent about whether you count the entry fee into the score or whether you are adding it now).

Next to it calculate the logarithm of your bankroll after the score.

Next to it calculate the log of the bankroll multiplied by the probability. Sum this column to get the expected log of your bankroll.

Now paste the whole thing somewhere (e.g. just to to the right) and with the new one, edit the bankroll after score column to reflect selling action at a particular markup then hitting a particular score - expected log figure at the bottom should change.

The "correct" Kelly strategy is the one with the highest expected logarithm of bankroll.

Look up regs in the 180s with much bigger samples and average them out.

If I recall, the floating mean is ~25% although closer to 35-40% is attainable for players with optimally tuned games.

Still, I would cut smaller sample ROI's by a third and go from there. In your case, 20%. That way as you accumulate sample size, you can adjust your Kelly ROI.

"playing too high is also slow because the losses cut into your bankroll quickly and you end up dropping lower and for longer than you would under Kelly."

Is this true ? For example if you are rolled for $10 games but can beat the $100 games isn't it quicker to take shots when you can afford it until you win ?

Is strikes me that the Kelly criterion is more relevant to sports betting etc where the EV doesn't change when you bet more. You can bet $10 or $100 on Brazil to win the World Cup it doesn't change the EV. Poker isn't like that. When you move up EV and ROI tend to go down. The Kelly criterion also isn't useful for telling you what bankroll you need to play at a particular level eg in the OP 9 man stt at 10% roi shows 20 buy ins. You will soon go busto playing with 20 buy ins ! You probably need more like 200 buy ins as a bankroll with 10% roi in those games. The Kelly Criterion calculates how to maximise profits not how to avoid going busto.

Well it depends what games exist in a given place. Obviously Kelly assumes you can choose any buy-in you like, that's probably true at the highest levels where people can sell any % of themselves they like but not for us.

But the general principle applies - if you are on a Cowboy planet with just a $100, $10 and $1 game running, and you need to run your bankroll up to $1000 to get a ticket on the Millennium Falcon out of there as quick as possible, then if you jump up to the $100 dollar game too quickly (e.g. when you have $150, even for just one shot) there is a significant risk that a short downswing can send you down through the $10 level into the $1 level - so entering the $100 game when you have $150 is a "slow" way to do it though there is of course variance in that. So you take shots with money you have above the amount you need to be secure at the $10 level.

That's true (and is note number 2 in the OP) although not to the extent people seem to think - after two years of playing I only started really properly beating the 25 cent 45 mans one year ago, and now at the end of my third year I beat NL25 euro which is more than 100 times higher stakes - I was (IMHO) already better than every stake I played on the way up when I moved into it. Obviously I've continued to improve during the year - I'm not saying NL25E is the same level as $0.25 45-mans or that I could have beaten it one year ago - but ultimately a particular stake runs at all because there are bad players who buy into it direct, and they don't have to go through any kind of qualification to do it.

I also think a counterbalance to the lower ROI is that you learn faster when you play at a higher level. You don't pick up so many bad habits and BS moves that aren't going to work in a stronger player pool. Also I think for some people, just playing as high as possible without repeated deposits is their actual goal in poker.

But yeah, I think someone should read this, then go and read about Jennifear BRM (and try to understand why and how it works, not just read some random numbers relating to game conditions a decade ago) and come up with a BRM strategy that makes sense for them and what they want out of poker and the particular stakes available for the game type they like.

The point is that if you have lost half your roll in the $4 game, you should now be playing the $2 game, and still have 20 buy ins. At least theoretically, when your bankroll drops down to $1 you should be playing the $0.10 cent SNGs on iPoker and selling half your action to make it a $0.05 SNG. You never go bust.

If you use this only as a guide to when to move up and never as a guide to when to move down then you end up busto as explained here:

https://en.wikipedia.org/wiki/Gambler%27s_ruin

"So we've discovered that the 100BI rule is perfect for someone playing 180s with an ROI of 22%"

Thanks for your reply. I think the above is misleading. 100 buy-ins is the MINIMUM to play the 180s using the Kelly criterion. If you drop to 99 (likely after the first game lol) then you should drop down a level. Players are generally more interested in the bankroll needed to play the games continuously without going bust. That is more like 400 buy-ins if you want a low risk of ruin.

That's not right. It depends what the next level is.

You compare which tournament increases the expected logarithm of your bankroll by the most. I've found it's generally the one closest to your BR/(number from tables above) not the next one down. Having said that, going for the next one down might nicely compensate for the effect of what we assume about ROIs being higher in the lower buy-in tournaments.

Then lol @ players.

If you have $100 dollars in your account and a given ROI, the correct tournament to play isn't affected by whether your bankroll is on the way down from $400 or on the way up from $25.

As shown here https://en.wikipedia.org/wiki/Gambler%27s_ruin if they aren't willing to move down when they run bad, the risk of ruin is always 100%

"The original meaning is that a gambler who raises his bet to a fixed fraction of bankroll when he wins, but does not reduce it when he loses, will eventually go broke, even if he has a positive expected value on each bet."

Yet on the other hand, when running well it would be mind-numbing for someone with an ROI of 22% and a $400 bankroll to be playing $1 games for a $0.22 EV each time.

That's true if you play millions of games but with 400 buy-ins the risk of ruin is less than 5% for 20,000 games (it might be less than this I've forgotten the precise numbers). Players often ask what roll they need to play the $15 180s etc. The answer is 400 buy-ins or at least 300 buy-ins. They will go bust quite quickly if they keep a fixed bankroll of 100 buy-ins. I watched someone play $2.50 180s with 100 buy-ins on twitch. He went bust after a few hundred games and he was a very good 180 player.

More regs, more variance.

I'm sure there's a chapter that could be written about the various elements and et effing w/e, but that's basically it.

Thinner slices off depleting fish pools.

You mean that he played a fixed stake level and let the number of buy-ins he had vary, eventually down to zero.

If someone isn't willing to move down ever then starting at 1/4 of the stakes one would normally be able to be playing is probably the best plan for that person, though on the other hand I'd also say they have issues they should deal with before even sitting down at the table.

He was a top notch player doing it for a challenge. The $250 didn't mean anything to him but it illustrates that 100 buy-ins doesn't last long playing the 180s. The $15 players regularly drop $1500 or more in their downswings.