Hold 'em: Standard Deviation vs. Monte Carlo Simulation

[Poseidon65]

I've read that a lot of players think it's important to know what your standard deviation is, so you know whether or not your swings are within ranges that are to be expected given your variance.

However, it doesn't seem that standard deviation is very useful for this purpose. This is because the distribution of each indiviaul hand is not even close to normally distributed. I suppose you could argue (using the Central Limit Theorem) that if you play enough hands, the non-normality of the distribution isn't really relevant. I'm not sure to what degree it's relevant or not, so I thought I'd try something that might be more robust than standard deviation here.

So I've been logging the result of every Paradise hand I've played for the last few days (about 450 hands). Once I play a few thousand hands, I should enough data to determine with good accuracy the distribution of an individual hand is, for me, at least.

Once I have this, let's say I play 100 hands, I get some result for those hands (for example, down 25BB), and I want to know the probability of that happening. I use a Monte Carlo simulation, drawing 100 hands randomly from my distribution, and do so 1,000,000 times. Now, I can simply see what percentage of these 1,000,000 results lie above -25BB for the 100 hands.

I wonder if anyone's ever thought of doing this, or if it sounds like a waste of time. I definitely want to see how normal the distribution of an entire session is, and this seems like a pretty good way to find out. It also seems that this would be more accurate in finding out just how good or bad a particular session really is.

[lorinda]

I'm sure there is another flaw here, but I do sums and not maths, however one thing that springs to mind is that 1000 hands is not a huge sample and you might be doing your simulation based on 1000 particuarly good or particuarly bad hands without knowing it.

1000 hands is in the region of 25-30 live hours.

Lori

[crazy canuck]

Yes, you'd have to play a lot of hands to calculate their true earn rate/variance. A Turbo Hold em simulation might work better. The players there play differently tho. According to Abdul Jalib you can argue that your hourly rate is approximately normal based on the modified central limit theorem.

[NutJob]

Rather than calculate standard deviation, the meaning of the -25BB session can more accurately be answered by "how often did that fishy in seat 2 suck out on me" and similar qualitative analysis. The stats provided in PokerTracker (if you are an online player) are helpful in identifying leaks in strategy. Finding your SD to assess your ideal bankroll or some such theoretical result? Dont bother. The answer is always going to be "Keep your day job" if you are going to be realistic about your chances of survival as a pro. /forums/images/icons/laugh.gif

[NutJob]

BTW, to be a little less sarcastic than my other post, remember that any SD you calculate isnt "your SD" unless you are always playing against a group of players with identical strategy profiles. You will have opponents that will lead to high variance results, and opponents that will lead to ultra-high variance results. To accurately segment the data into those groups to find an underlying "personal variance" is an exercise in futility.

[crazy canuck]

If you want to improve on the normal approxiamtion for the hourly, one way would be to expand the probabilty density function around the gaussian. This is called Gram-Chalier expansion and is used to improve on the Black-Scholes model for financial options (I wrote some code for a hedge fund last year on this). Once you do this, you can calculate the bankroll by monte carlo (this isn't as easy as it seems). Anyhow, the math is kind of ugly and in my opinion it isn't worth it. But if your really want to do it this would be the easiest way i guess.

[Mason Malmuth]

Hi P65:

The problem with what you're suggesting is that the other methods, which you are skeptical towards, have now been around a fairly long time and have proven to be pretty accurate. So even if your method is better, and I haven't thought enough about it to have an opinion, it won't be better by enough to really matter.

Best wishes,
Mason