# How SICK variance REALLY is - Probability & Simulations Blog

• Bronze
Joined: 24.05.2008
Hello Guys,

As a SNG player, I was really interested in what strategies to use, how to play etc, and that is where PokerStrategy.com learned me quite a lot .. As a probability and mathematical statistics student, I have different set of questions to which it is really hard to find answers .. So I decided with a friend of mine (also a SNG player and student) to try answering some of those questions.

This blog will be sort of a log of research I am about to do on expected values, variance and other things considering SNGs from a probability/statistical point of view and I am hoping to produce very practical results using simulations to get a big sample size needed for such a thing.

I hope a few of you will be interested in such results and maybe also comment and critise the methods/calculations/simulations if necessary Also, if any of you is interested in joining me and help, I would only he happy about it
• 50 replies
• Bronze
Joined: 18.09.2008
Cool! Will read with interest - from a MTT SNG point of view anyway.

I have no relevant skillset or decent HH to contribute so will primarily be a lurker.
• Bronze
Joined: 01.06.2008
i am thinking about doing something related to stats in university so this should be interesting.
• Bronze
Joined: 24.05.2008
Hello guys,

Thanks for the interest, I am definitely looking forward to any of your comments

So the first point of my research is what  really interests me as a SNG player – How much can my current ROI be different from the expected ROI I would get by playing a very very big amount (theoretically infinite) of SNGs ..

In other words, say I have a x% total ROI over n SNGs, what variance can I expect ? This is a question I would like to explore and try to give some reliable answers to. So in the beginning a bit of math about ROI (the concept of ROI in SNGs is basically well known and described for example here : http://www.investopedia.com/terms/r/returnoninvestment.asp) .. For our needs we get :

which can be rewritten in this way

where:
• r - rake in absolute numbers (dollars)
• %r - rake relative to buy in ( %r = r / BI )
• paid - number of places paid
• wi - payout for i-th place in buyins
• pi - probability of finish in i-th place

This formula may look a little scary, so here comes an example to make it easy : Lets take the PokerStars \$3+0.4\$ SNG with 10 entrants and payout structure 50%-30%-20%, and say our player wins 13% of tourneys, finishes 2nd in 10% and 3rd in 15% of tournies:
• BI =3
• r = 0.4
• %r = 0.1333 (or 13.33%)
• paid = 3
• w1 = 5 ; w2 = 3 ; w3 = 2
• p1=0.13, p2=0.1, p3=0.15

This leaves us with a 10.2% ROI:

Now lets look at the formula in a closer way :
• In the second line 3.866p1 means that a player will win 3.8666 net buyin with probabilty p1 (meaning if he wins the tourney)
• The last part on line 2 means he will lose 1.1333 buyins with probability 1-p1-p2-p3, meaning he will not finish in the money

It is also notable that

It is very important to realise that the ROI calculated here uses probabilities of winning and is what we may call Expected ROI. We may state that if our player played infinite amout of SNGs he would have 10.29% ROI under the assumptions of winning with probability p1, finishing 2nd with probability p2 and 3rd with probability p3.

This much for the post, I will continue tomorrow with some practical applications and assumptions
• Bronze
Joined: 30.06.2009
i love you Augustus.

+ subscribe
• Bronze
Joined: 01.06.2008
Originally posted by AugustusCaesar
It is very important to realise that the ROI calculated here uses probabilities of winning and is what we may call Expected ROI. We may state that if our player played infinite amout of SNGs he would have 10.29% ROI under the assumptions of winning with probability p1, finishing 2nd with probability p2 and 3rd with probability p3.
it would be impossible to determine p1, p2, and p3 without a huge sample size in the first place
• Bronze
Joined: 24.05.2008
Also these would vary in time both as the player is changing and the opponents are changing so we can't do any specific actual measurements unless we assume some kind of average which we cant rely on for future calculations anyway

Also, nice to see some more people with same interests. Subscribed.
• Bronze
Joined: 24.05.2008
@lessthanthree : thanks
@fun101rockets : more about that in this one
@z4tz : thats what we use simulations for

Calculating %ITM

Now to the point itself –We will focus on 10 player SNGs with 50%-30%-20% payout structure and 13.33% rake first. We have an idea about about ROI, what variance can we expect ? To estimate this we will make an assumption that our chances of finishing 1st, 2nd and 3rd are the same, which will make it a little bit easier (if you have a better idea about the chances please comment on them, if you have some nice sample size with results please post). Either way, it will only create very small to none changes in our results. Our formula from the previous post now changes into a much simpler one :

So for %ITM we get

As for the question „how did number of entrants get there ?“ – The answer is the number of entrants = the number of buyins in the tourney (as we are now calculating in buyins and not dollars, if we were calculating in dollars the number would be entrants*buyin = total pricepool, which hopefully makes sense ).

Variance & Standard deviation

Now a few words about variance and standard deviation. It would take too long to describe variance as a mathematical term and it would be kind of tough to fully understand. To get in touch with a simple explanation please read the example here http://en.wikipedia.org/wiki/Variance or watch HasenBraten‘s video (silver+) : http://www.pokerstrategy.com/video/9152 .

For what we need, a large standard deviation indicates that the results are far from the mean and a small standard deviation indicates that they are close to the mean. We will also apply a very important fact about standard deviation: For normally distributed data, at least 99% of the data are within 2.57*standard deviation from the mean.

Simulation Methodology
1. We will take a ROI level , such as 5%
2. Calculate %ITM (in the case of 10 players, 50-30-20 and 13% rake) = 35.7%
3. Take a few sample sizes like: 100 , 500, 1000, 2000 etc
4. Simulate each occurence 100 000 times, which simulates having 100 000 online players playing exactly the same game with 5% ROI
5. Look how can their ROI move within the set sample sizes .. I will post some results as soon as I can get them into some readable and understantable form

• Black
Joined: 02.11.2008
• Bronze
Joined: 24.05.2008
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How sick variance really is - SNGs with 5% ROI

Hey guys .. So after some time I made the simulations and the results were pretty shocking even for me, and I think that my background has a good impact on how can I think about variance and randomness itself .. As I mentioned before, the simulations were on 100 000 different players playing exactly the same way

The results show the interval, in which you ROI will be with 95% probability after playing x SNGs if your real ROI is 5%:

And the graph, that shows it much more better :

Conclusions , if your real ROI is 5%:
• If you play 3000 SNGs and you are under 0, means loosing money it does not mean anything (except you are unlucky)
• With 200 SNGs sample you should be up by 5.67BI, but you will be with 95% between -29BI and +40BI
• With 10 000 SNGs your ROI can be 3% different then the real 5% ROI very easily

Thats how sick variance really is .. I felt a little scared when I first saw the results .. I will continue with reseatching with different ROI% (next will be 10%) and also with more players (next will be 18 player SNGs)
• Bronze
Joined: 18.09.2008
I guess the other thing to take from the simulation is that being b/e after 4k SnGs is reasonable even as a winning player.
• Bronze
Joined: 11.04.2009
Nice econometrics skill. I'll try my own tests sometime in the future.
• Silver
Joined: 18.03.2008
first of all: amazing blog .

could you try do the simulations for HU sngs? i would guess the calculations would be fairly simple compared to 10 man sngs.

may variance be with you
• Bronze
Joined: 15.10.2008
Originally posted by AugustusCaesar
Conclusions , if your real ROI is 5%:
If you play 3000 SNGs and you are under 0, means loosing money it does not mean anything (except you are unlucky)
Could you take a look from other side and find answer what is probability that 5% is your real ROI if you are under 0 after 3000 games?
• Bronze
Joined: 24.05.2008
Hey guys, thank you all for feedback :

@Waiboy
Exactly .. Kind of scary, right ?

@ihufa
Thanks You are right, it would be easier .. I can also look at that some time, if you have some ROI% (or ITM%, in headsup ROI% is actually determined just by ITM% and rake I guess) that you would like to see specifically, just write

@ChoChikun
Hello, yes .. The answer to your question is 3.93% (the chance of having 5% real ROI if you have less than 0% ROI after 3000 SNGs with 10 players and 13.33% rake) .. I will also make this comparison as it seems like an interesting question to me
• Bronze
Joined: 11.04.2009
Advice: if you know programming you can make a really wonderfull program that will let you choose what roi you have, what's itm%, whatever.

I got bored about writing like 5 lines and felt like playing poker is more usefull unfortunately
• Bronze
Joined: 11.02.2008

Thought I would post you what a 2% ROi for \$1 DoNs looks like. I think this is a good example of how games toughening up can affect ROi. The guy in question 120 tables so I think he implements a very simplistic strategy that rarely changes.
• Black
Joined: 15.07.2007
@ augustus:
great blog. very interesting, even though im more a cg player.

i would also be interested in hu sng calculations:

- 6% roi, 9% rake
- 4% roi, 5% rake

@ richard:
omg, what a robot! unbelievable! he should learn to play poker imo, there are easier ways to make 4k\$. rly crazy guy lol.
do u know in how many years he played such a sick amount of sngs?
did he also move up or does he still just play 1\$?
• Silver
Joined: 18.03.2008
i'd like to see a simulation for 7,5% ROI, 4% rake
• Bronze
Joined: 17.01.2007
Originally posted by TwiceT
@ augustus:
great blog. very interesting, even though im more a cg player.

i would also be interested in hu sng calculations:

- 6% roi, 9% rake
- 4% roi, 5% rake

@ richard:
omg, what a robot! unbelievable! he should learn to play poker imo, there are easier ways to make 4k\$. rly crazy guy lol.
do u know in how many years he played such a sick amount of sngs?
did he also move up or does he still just play 1\$?

he moved up to \$2

j/k