As long as we're talking about Bankroll and ROR...

[AleoMagus]

It seems like we get a fair amount of Bankroll questions on here and I've been thinking about our 'usual' responses lately.

Typically, 30 buy-ins is the advice given and for many, this makes good sense.

There is a problem with this advice however, and I believe it may be somewhat misleading.

30 Buy-ins is adequate to give a 40% ROI player an extremely small risk of ruin, but for players with a smaller ROI (and I think that there are many), even 30 buy-ins may be far too small if securing a low risk of ruin is important.

Just as an example, if a player is only getting a 2.5% ROI (virtually break even) then a bankroll of about 150 buy-ins will be needed to give less than a 5% ROR - And 5% is actually not all that good!

Granted, a ROI of 2.5% is probably not even enough to be confident that you are a winning player and if you are not, no bankroll will be enough. Still, the point remains.

Despite the numerous claims on here about big 30%+ ROIs, Over very large samples, many of you are probably not getting that high. That is fine. Any positive ROI is great, but you should be careful assuming that 30 buy-ins will be sufficient in those cases.

Now, of course I still find the whole issue somewhat unimportant because for most of us it's not too hard to replace a 20-30 buy-in Bankroll anyways. Many do find busting psychologically difficult though and if it is important to you, factoring your actual win rate into your bankroll requirements is very important. If your win rate is not a stellar 30%+, you might want to adjust upwards and try as many as 50 or more buy-ins instead.

If you only have a small sample to determine win rates from, or you are just starting out, 30 is always a good 'starter' value, but if you find that you struggle at all, don't be shocked if your bankroll is not as secure as you might have thought

Any thoughts?

Regards
Brad S

[Jurollo]

I agree that 30 buy-ins is just simply the "easy" answer to this question. I think it is mainly targeted at new players who simply do not know how they will fair in either a) a new level, or b) sng's in general. 30 is a quick and easy response and is for all intents and purposes fairly safe. Players new to SNGs simply just wouldn't throw 150 buy-ins to be sure and most likely those that fly through a 30 buy in stake initially probably won't come back to SNG's too soon. I think the only time someone can really get a decent guess into what they need is those moving up levels after a long stay at a given level.

[durron597]

Wow, I just threw some numbers into the RoR calculator, and with a 27% ROI, and a 20 buyin bankroll, your ROI is still 5%. My ROI is significantly higher than that right now, but it's just a little reality check for when I decide to move up, and can't expect a 50% ROI anymore. I guess I won't be cashing out for awhile.

[Yawkey]

Let me start by thanking you Aleo and the other guys who modded your spreadsheet and made it available to us. I'm really just starting out with taking SnG's seriously, so please bear with me through some of these questions. Just to give you a better idea of where I stand, I'm playing at the $10+1 SnG's with at 49% ROI, $100 bankroll(all profit I took the initial investment out), 10% RoR, and a 57% Winning Confidence. Now from my reading I do realize that this is considered a risky situation, however it is a risk I am willing to take right now. Also you can probably see from my winning confidence that I am quoting a small sample set.

Now my question is what makes for a comfortable RoR? Can a player with a 40% ROI comfortably play with a 5% RoR or should they shoot for a much lower RoR like 1%? Next, should winning confidence have any impact on a players risk comfort level? For instance, if my winning confidence went up to 95% should I be more comfortable playing with a 10% RoR, or should the two just remain seperate stats?

[AleoMagus]

With a 49% ROI and a 57% 'winning' confidence, you must be using a REALLY small sample

In a manner of speaking, your confidence and ROR are related simply because your confidence in your win rate rises as your winning confidence rises. Your win rate is a key part of the ROR calculation and you want to be fairly sure about it before you start trusting the ROR calculations.

Given that your ROI of 49% is on the high side of what is normally considered possible I'd be inclined to think that your win rate is actually lower. With less than a 100 tourney sample (which I guarantee you have considering that ROI and low confidence), it's very possible that you are a losing player (43% possible actually)

this is like saying there is an almost 43% possibility that you will definitely lose your bankroll. So much for the 10% ROR huh?

It is of course more likely that you are a winning player, but just not a long term 49% ROI player. Even in this case, your ROR becomes a lot higher if your 'actual' win rate is not as good as you are assuming.

As far as practical ROR values, most experts recommend going to three standard deviations which is about a 1% ROR. It is important also to remember that even a 1% ROR is not as good as it sounds if you intend to keep cashing off large portions of your bankroll.

I know $100 is a lot to some people, but with the amount of money that you are talking about I'd suggest you not worry so much about it. $100 is a pretty replaceable BR so it's really just a psychological issue. Bankroll and ROR is actually a pretty unimportant subject unless you are talking about really big bankrolls which can be hard to replace.

You should read this thread (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=565423&page=&view=&sb =5&o=)

It's a tough read in some places, but be sure to check out Guy and Bozeman's comments.

Regards
Brad S

[Yawkey]

Thanks. One point that really struck home was on my 'winning confidence'. If I'm confident that I am going to win 57% of the time what's happens the other 43% of the time. That's the loosing confidence not the break even confidence. Also I appreciate your help even though it's quite obvious I havn't come close to a statistically sound sample set (11 tournies to be exact), mostly I'm just interested in learning this stuff and if I get to really apply it later on that will be a bonus.

Now a new question, if your loosing confidence(100% - Winning confidence) is greater than your RoR, is the RoR a valid stat? I know that I'm looking at an extreme example, but eventually my loosing confidence should become less than my RoR. At that point is my RoR calc. valid?

[AleoMagus]

'winning' confidence is just something I made up.

Really it is just the confidence that you actually make between 0 and 2x your derived win rate

this means that your 'losing' confidence would not actually be 100-Winning confidence as you could conceivable get even better results than 2X your win rate which would not be taken into consideration by your 'winning' confidence.

When I originally devised the 'winning' confidence I thought about adding 1/2 of the excess possibility but opted instead to leave it as is because despite the normal distribution's assumptions about possibility, I KNOW that nobody is getting long term 100% ROI values, etc...
Besides, I prefer to estimate conservatively.

Still, you might consider your 'losing confidence' to actually be (100-Winning confidence)/2

I know it all sounds highly technical but it's actually all just a big approximation. Those formulas assume that sng results conform to the standard normal distribution which they do not. It's a good approximation, don't get me wrong, but it's not perfect.

Truth be told, I'm not actually a math guy and learned almost everything I know about stats on these forums. Perhaps some math guru will comment in this thread with more precise answers.

As far as the thought that ROR only becomes reliable when your 'losing' confidence is less, I can't say with any degree of expertise, but that seems like a great rough and ready suggestion . Another rough and ready suggestion is just that you need to play a significant sample before it becomes valid. 100 is probably a good start.

Still, these are actually good questions (though they will do nothing to improve your sng game /images/graemlins/grin.gif), and not actually newbie questions at all

Regards
Brad S

[dethgrind]

[ QUOTE ]
Just as an example, if a player is only getting a 2.5% ROI (virtually break even) then a bankroll of about 150 buy-ins will be needed to give less than a 5% ROR - And 5% is actually not all that good!


[/ QUOTE ]

How did you come up with this? As Bozeman pointed out in an earlier thread, the small number of tournaments required to bust you in SNGs makes the normal distribution a bad approximation for RoR calculations for SNGs. So Malmuth's method in GTAOT won't work for this. Did you use that other formula?

I read that earlier thread a while ago and decided that running thousands of simulations and averaging the results is probably as good or better than any formula. Another poster also did this (Eastbay I think?). I got some pretty interesting results.

An interesting point is that if you don't let your initial bankroll grow (you always take out your profits) you will necessarily go broke over an infinite timespan. Also, with huge bankrolls (like 150 buyins), it will take a very long time to go bust those times that you do. So an important parameter in the simulations is how many tournaments will you play over your career? For your scenario of 2.5% ROI, I got the following results for a career of 500 tournaments, without taking out profits:
0.12 first
0.11 second
0.1 third
0.33 ITM
0.0272727 ROI
30 stakes
500000 number of simulation runs
500 number of tourneys per run
4950 bankroll
999999 cutoff

risk of ruin: 0

I.e. over 500000 runs, the simulation didn't go broke a single time if it played only 500 tourneys per run.

For a career of 5000 tourneys:
0.12 first
0.11 second
0.1 third
0.33 ITM
0.0272727 ROI
30 stakes
200000 number of simulation runs
5000 number of tourneys per run
4950 bankroll
999999 cutoff

risk of ruin: 0.02247

Career 500 tourneys, but removing profits from the initial bankroll:
0.12 first
0.11 second
0.1 third
0.33 ITM
0.0272727 ROI
30 stakes
500000 number of simulation runs
500 number of tourneys per run
4950 bankroll
4950 cutoff

risk of ruin: 2e-06 (essentially zero)

Career 5000 tourneys, removing profits:
0.12 first
0.11 second
0.1 third
0.33 ITM
0.0272727 ROI
30 stakes
200000 number of simulation runs
5000 number of tourneys per run
4950 bankroll
4950 cutoff

risk of ruin: 0.086555


This simulation method is pretty damn powerful. I'd recommend it for solving most bankroll-related problems. Check out my work at http://www.bol.ucla.edu/~sharnett/poker/

[AleoMagus]

VERY interesting stuff.

I used the formula Bozeman suggests in that other thread

R=EXP(-2HB/S^2) where,
R=Risk of ruin
B=Bankroll
H=Hourly (or tourney) win rate
S=Standard deviation

I have been pretty unsatisfied with all of my sats calculations for all the reasons that you suggest. Normal disribution may not do a good enough job.

What also has me thinking is the possibility that in simulations we can model factors like a tendency to play worse after losses. I have long thought that given the small number of sngs required to bust a player with only 20-30 buy-ins, tilt could make ROR go through the roof if a player was bad at dealing with it.

I'd be interested in a calculation which assumed - ROI goes up after cashes and down after losses.

I'm also curious what you think a simulation has to say about my other thoughts on ROI in the 'Poll about ROI' thread. Is it difficult for you to do this stuff? I'd love it if you could run a few million similated sngs off and tell me what kinds of ROI fluctuations a sng player can expect from each 100, 1000, 2000... sngs to the next.

Sorry if this is asking a lot. I think it's time I started learning how to do this stuff myself

Regards
Brad S

[dethgrind]

[ QUOTE ]
I'd be interested in a calculation which assumed - ROI goes up after cashes and down after losses.


[/ QUOTE ]

This one would be tricky. Deciding exactly how much and under what circumstances the "real ROI" fluctuates based on results will be important. The most obvious and simplest idea is something like this:
start with initial ROI 30%
if lose 4 in a row drop to 10%, stay at 10% until next cash, then return to 30%
if win 4 in a row jump to 45%, stay at 45% until lose 2 in a row, then return to 30%

Even in that simple model there are so many parameters that it'll be hard to interpret the results. Is that basically the idea you have in mind?

[ QUOTE ]
I'd love it if you could run a few million similated sngs off and tell me what kinds of ROI fluctuations a sng player can expect from each 100, 1000, 2000... sngs to the next.


[/ QUOTE ]

This should be fairly easy. I'm guessing you want to see a distribution of the "results ROI" after several stages (100, 1000, 2000 tourneys) given a "real ROI". And I guess the standard deviation will completely describe the distribution since we know it will be normal and that the mean will be the "real ROI". Or did you have something slightly different in mind? This is actually a really interesting question. Have you seen a good analysis comparing the swings of SNGs vs limit cash games?

Lately I've been questioning whether SNGs are actually the right way to go. I've always suspected that they'd have smaller swings, requiring a smaller bankroll than cash games. If that's the case, wouldn't ultimate bet's six player SNGs and heads-up matches have even smaller swings?

And there's the problem of requiring a lot of focus on the bubble and in the money, focus that you rarely need in limit cash games. If I got some laidback computer job I could probably be running a few limit cash games at work, while I could never play SNGs since my work could need my attention at the same time I'm in the money. Then again, that part of the game is the most fun for me. Poker at it's finest.

So I'm wondering if I should make the switch to limit cash, start studying those. Any thoughts?

[LinusKS]

[ QUOTE ]
Despite the numerous claims on here about big 30%+ ROIs, Over very large samples, many of you are probably not getting that high. That is fine. Any positive ROI is great, but you should be careful assuming that 30 buy-ins will be sufficient in those cases.

[/ QUOTE ]

I'm no expert - and I haven't even been here that long - but I have a hard time believing all the claims of forty and fifty percent ROI's can be true.

In the first place, my intuitive understanding of sng's tells me the variance (luck) involved in placing would make it hard for even perfect players to do that well.

In the second place, it seems like you'd need tons of people losing virtually all the time to support those kinds of win rates.

In my experience, even the worst players get to the money from time to time.

I'm wondering if you have any insight into this.

[JNash]

You simulation work was very interesting - thanks.

On your thought abut what to do when you get out of school...

[ QUOTE ]
If I got some laidback computer job I could probably be running a few limit cash games at work, while I could never play SNGs since my work could need my attention at the same time I'm in the money.

[/ QUOTE ]

I hate to bust your bubble, but...

Ethical questions aside about playing poker while you're supposed to be working, in most corporate environments there will be filters that prevent you from accessing gambling-related sites. (Unless your IT job enables you to circumvent those, which would create even bigger ethical questions.)

The good news is that as a smart ucla grad you'll be raking in enough dough that you'll be able to just enjoy poker "after hours". /images/graemlins/smile.gif

[dethgrind]

yeah, I bet it'd be pretty hard to pull of poker at a "real" job. I was thinking more along the lines of the bs campus jobs. they have all these computer labs on campus, and they pay these people to basically sit on their asses all day. technically, these lab assistants are supposed to help students with the computers, but who needs help using MS Word?

[dethgrind]

After some thinking and checking with my simulated data, I've come to the conclusion that ROI over 200+ tournaments is distributed normally with mean "real" ROI and standard deviation s/sqrt(n), where s is the standard deviation of one SNG and n is the number of tournaments. This is straight from the central limit theorem. (the reason you can't use the normal approximation for RoR calculations doesn't apply here. here, we assume that even if you go broke, you keep playing)

The ROI given 1st, 2nd, 3rd probabilities is:
ROI = (1st*3.9 + 2nd*1.9 + 3rd*0.9 - OTM*1.1) / 1.1
where OTM = 1 - (1st + 2nd + 3rd)

To find the standard deviation of ROI for one tournament, use the following formula:
var = 1st*(3.9/1.1-ROI)^2 + 2nd*(1.9/1.1-ROI)^2 + 3rd*(0.9/1.1-ROI)^2 + OTM*(-1-ROI)^2

1st, 2nd, 3rd, OTM, are the probabilities of those events. For ROI 30%, use .15, .14, .13, .58. Variance is E[(X-EX)^2]. Standard deviation is the square root of variance, so take the square root of that number. For those values of 1st, 2nd, etc, the single tournament standard deviation is 1.697.

Once you know the single tournament ROI standard deviation, you can use the probability density function to answer all your questions. Here is the PDF for a normal random variable:

1/[S*sqrt(2*pi)] * exp[-(x-ROI)^2 / (2*S^2)]

S is the standard deviation of n tournaments, i.e. s/sqrt(n) from the first paragraph. For 500 tournaments and the ROI 30% numbers from above, this is 1.697/sqrt(500) = .0759. exp[?] is the exponential function, e^(?). Here's a decent link for a good explanation: http://www.riskglossary.com/articles/normal_distribution.htm.

To find, for instance, the probability after 500 tournaments of having an ROI between 15% and 25% for the real ROI 30% numbers from above, integrate the PDF from .15 to .25. The answer is .231. Here is a site that'll do integration: http://www.geocities.com/SiliconValley/2902/simpson.htm. It's pretty anal about syntax, so I recommend copy/pasting this and just changing the parameters:
1/(.0759*Sqrt(2*Pi)) * Exp(-Power((z-.3),2)/(2*Power(.0759,2)))

(note: if you want the upper bound to be +inf, use 3 instead. There isn't much probability over 300% ROI so this won't change the results, but the calculator seems to screw up if you use like 9999999)

Better yet, find a better calculator. Better still, figure out how to convert this problem into a standard normal and use a table (this would be tedious to explain).

With this information, you can answer all questions about what ROI your results will show given a "real" ROI. The more interesting question is what sort of real ROI you can expect given your results. I know close to nothing about statistics so I can't shed much light on that question.

Here are some example questions and answers:

1) Suppose my ITM finishes are distributed as follows: 1st - 13.5%, 2nd - 12%, 3rd - 11.5%. So ROI = 15% and OTM = 63%. What is the probability of getting an ROI over 35% after 300 SNGs?
s = 1.655, S = .09556, answer: 1.8%
What about over 25%?
answer: 14.8%

2) If my actual ROI is 30% (distributed .15,.14,.13), after 1000 tournaments, what's the probability of getting an ROI between 25% and 35%?
s = 1.697, S = .05366, answer: 64.9%
What about over 40%?
answer: 3.1%