Probability of 100BB Downswing

[PairTheBoard]

There is an interesting Problem presented on the Poker Theory Forum in the Thread "Standardly Deviating".

The simple version is, given a 1BB/100hands win rate, with a standard deviation of 10BB for 100 hands: How many hands must be played to have a 99% Probability of experiencing at least one 100BB slide sometime during the play of those hands.

http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=2032745&page=0&view=c ollapsed&sb=5&o=14&fpart=2#Post2040731

I believe this is a fairly challenging problem for the Probabilty Experts who might want to work on it.

PairTheBoard

[Lexander]

Looks like jason1990 proposes a pretty good line of attack from my perspective.

I haven't taken a time series course, or else I would try to contribute a bit more, but I would think a time series approach would be the right direction.

[gaming_mouse]

[ QUOTE ]
Looks like jason1990 proposes a pretty good line of attack from my perspective.

[/ QUOTE ]

jason,

are you saying that your bankroll fluctuations can be represented as brownian motion with drift?

[jason1990]

Not the bankroll fluctuations, but the bankroll itself. This is in the introduction to the article I linked to in the Poker Theory thread. Whether it's an accurate model or not, I'll leave for others to debate. Although I will say that I believe it is a good model when it comes to questions about the "long run." Also, such a model does generate the "usual" risk of ruin formula that folks toss around, so that may be evidence that at least some other people think it's a useful model.

I've played with this a little more. If T is the first time you see a 100 BB downswing, I want to compute

E[e^{-aT}]

for arbitrary a>0. This is the Laplace transform; it should allow us to compute moments of T and, with a computer package, may even explicitly determine the distribution of T. Unfortunately, I can only work out that

E[e^{-aT}] = E'[e^{-bT + cX*(T)}],

for some (explicit) positive constants b and c, where the notation is as in that other thread. This is no good, since I only have a formula for

E'[e^{-bT - cX*(T)}].

In other words, I think I can analyze upswings, but not downswings. I may give up on this soon, since I'm probably spending too much time on it.

Edit: If forgot the "e^"s in my E' expectations.

[gaming_mouse]

[ QUOTE ]
Not the bankroll fluctuations, but the bankroll itself.

[/ QUOTE ]

That's actually what I meant. I am not sure it is, though. Doesn't brownian motion assume equal step sizes?

If so, this is clearly not a good model for your bankroll. For example, what if the question was a 25BB downswing rather than 100? Clearly the long term normal approximation will fail, right?

I may be misunderstanding you though....

[jason1990]

Well, I'll definitely stop playing with this, since it's answered here:

@article {MR0375486,
AUTHOR = {Taylor, Howard M.},
TITLE = {A stopped {B}rownian motion formula},
JOURNAL = {Ann. Probability},
VOLUME = {3},
YEAR = {1975},
PAGES = {234--246},
MRCLASS = {60J65},
MRNUMBER = {MR0375486 (51 \#11678)},
MRREVIEWER = {P. J. Brockwell},
}

[gaming_mouse]

[ QUOTE ]
Well, I'll definitely stop playing with this, since it's answered here:

@article {MR0375486,
AUTHOR = {Taylor, Howard M.},
TITLE = {A stopped {B}rownian motion formula},
JOURNAL = {Ann. Probability},
VOLUME = {3},
YEAR = {1975},
PAGES = {234--246},
MRCLASS = {60J65},
MRNUMBER = {MR0375486 (51 \#11678)},
MRREVIEWER = {P. J. Brockwell},
}

[/ QUOTE ]

Is this available online?

Also, I'm guessing that this uses the Brownian motion model. Can you explain why my last objection would not apply (or would)?

Thanks,
gm

[jason1990]

[ QUOTE ]
That's actually what I meant. I am not sure it is, though. Doesn't brownian motion assume equal step sizes?

[/ QUOTE ]
Well, I'm not exactly sure what you mean by this. Since Brownian motion is continuous, there are no "steps," of course. But BM can be realized as the scaling limit of a random walk with equal step sizes, if that's what you mean. However, you don't have to have equal step sizes to get BM in the limit. Only independent steps with mean zero and finite variance. This is the Invariance Principle: the scaled random walk converges to BM regardless of the underlying distribution of the steps. So if the result of your j-th poker hand is m + X_j, where m is your win rate and X_j are i.i.d. mean zero random variables with finite variance, then the limit of the sums of the X_j's will be BM and the m will give you a drift. (This is very hand-wavy, of course. In particular, note that you must scale the X_j's to get BM, but you don't want to scale the m.)

[ QUOTE ]
what if the question was a 25BB downswing rather than 100? Clearly the long term normal approximation will fail, right?

[/ QUOTE ]
I agree. Similarly, the usual risk of ruin formula will not be accurate if you only have a 25BB bankroll. I would only use the BM model if (a) I was asking about "big" events (big swings, long term profits, etc.) or, as in this case, (b) I just wanted to have fun playing with BM. /images/graemlins/wink.gif

[jason1990]

This is available through JSTOR, if you have access to that. If not, I can email you a copy. I don't think I'm breaking any terms and conditions of JSTOR by doing that.

[gaming_mouse]

[ QUOTE ]
This is available through JSTOR, if you have access to that. If not, I can email you a copy. I don't think I'm breaking any terms and conditions of JSTOR by doing that.

[/ QUOTE ]

Please do.

Also, do can you recommend a good article on brownian motion -- it's a been a while since i studied it and i no longer have my books.

Finally, how do you approach the problem in the 25BB case, where convergence theorems are no longer accurate. Do we need to approximate the underlying per hand distribution, perhaps using PT data, and then run a simulation?

[jason1990]

You'll have to give me your email address. I'll be happy to send you that article.

[ QUOTE ]
can you recommend a good article on brownian motion -- it's a been a while since i studied it and i no longer have my books.

[/ QUOTE ]
My standard reference is "Brownian Motion and Stochastic Caluclus" by Karatzas and Shreve. I learned out of that book originally and, interestingly, it's the first listing in Google Scholar when you search for "brownian motion." I don't know of any article off-hand, but there are some lecture notes here (http://www.math.cmu.edu/users/kramanan/oldindex.html) that look nice. She talks about the Invariance Principle of p.27 of the lectures 5-8 pdf and lecture 9 is nice as well. There are probably a lot of other course notes out there waiting to be googled, especially in finance.

[ QUOTE ]
Finally, how do you approach the problem in the 25BB case, where convergence theorems are no longer accurate. Do we need to approximate the underlying per hand distribution, perhaps using PT data, and then run a simulation?

[/ QUOTE ]
I somehow doubt one could get an accurate enough approximation to the underlying per hand distribution. Aren't 25BB downswings fairly common? It sure feels like they are. I have about 20K hands in my db. I'd have to check, but there must be many 25BB downswings. I think I would just measure the time between these downswings and take these times to be independent realizations of T. (I never tilt, of course. /images/graemlins/smile.gif) I don't know what kind of analysis you could do with that or how accurate it would be. I suppose that depends on how many of these swings you've had.

If you get any data on this, let me know. I'm curious.

[gaming_mouse]

[ QUOTE ]

I somehow doubt one could get an accurate enough approximation to the underlying per hand distribution.

[/ QUOTE ]

First off, thank you very much for the references.

As to the above, I think you may have misunderstood. We don't need enough data so that we have experienced enough 25BB downswings for an approximation. We just bootstrap. That is, make a list of of the amount won on each hand using your PT data (20K should be plenty):

0,0,0,30,0,0,-15,....

It will look sort of like the above. Now run a simulation that randomly chooses a number from that list to increment your bankroll. You can now play millions of hands to answer questions about 25BB downswings.

Do you see any problems with this approach?

[jason1990]

So this is what those applied guys mean by "bootstrap?" (They really should write a book called "Statistics for Probabilists.")

Well, that looks like a very nice idea. The first thing I would be uncomfortable with is win rate. Doesn't this bootstrap method assume that my current empirical win rate is my true win rate? Certainly, win rate will have a big effect on the frequency of downswings, so being off on the measurement of win rate will mean you're off on the frequency of swings.

I guess this is what I meant by not having an accurate estimation of the true distribution. The empirical distribution will not be close enough to the true one to give accurate enough results. At least that's my guess.

[gaming_mouse]

[ QUOTE ]
The empirical distribution will not be close enough to the true one to give accurate enough results. At least that's my guess.

[/ QUOTE ]

For 20K, maybe not. But I'd be fairly comfortable with 50K -- I could be wrong, tho.

I'm also not convinced that the 25BB downswing frequency is heavily dependent on win rate -- again, just a feeling.

[jason1990]

Incidentally, even in the BM model, you don't know the true win rate, either. So the BM model addresses a theoretical question: *if* I have X WR and Y SD, then... This is possible, of course, because a normal curve has the same general shape for different values of X and Y.

But I doubt the true per hand distribution has the same properties. Its shape is probably highly dependent on the win rate. So suppose I do this bootstrap technique and then I ask, "What if my true win rate is actually X and my true SD is actually Y?" How would I then modify the bootstrap method? I don't think you could just shift and scale all your data, because I doubt that would give you a realistic per hand distribution.

I hope this isn't too unclear. I'm posting and doing my checkbook at the same time. Anyway, it seems the BM model and the bootstrap method may be suited to two different questions. The first is the general question for an arbitrary player with such-and-such WR and SD. The second is the specific question of *my* or *your* probabilities for downswings.

[jason1990]

[ QUOTE ]
I'm also not convinced that the 25BB downswing frequency is heavily dependent on win rate -- again, just a feeling.

[/ QUOTE ]
Yeah, maybe not. The dependence would drop off as the swing size decreases. But then again, I just looked at some graphs of my data. I clearly have a *lot* more 30BB upswings than 30BB downswings. If win rate was not a serious factor, I would think there would not be such an imbalance.

[gaming_mouse]

[ QUOTE ]

But I doubt the true per hand distribution has the same properties. Its shape is probably highly dependent on the win rate.

[/ QUOTE ]

I don't think it would be. I'd guess it would look similar for most 2+2 players, with just a slight shift in WR. It's shape would be different, I'd guess, for a radically different playing style, like a maniac's. Again, this is nothing more than my hunch.

[gaming_mouse]

[ QUOTE ]
I clearly have a *lot* more 30BB upswings than 30BB downswings. If win rate was not a serious factor, I would think there would not be such an imbalance.

[/ QUOTE ]

This is a good point.

[jason1990]

This is the original question from the original thread:

[ QUOTE ]
Can someone show me some sort of equation that will tell me how often (every x number of hands) I should expect to encounter a downswing of 100 BB, 200 BB, etc... Standard deviation of 12 BB/100 hands, win rate 1.3 BB/100 hands.

[/ QUOTE ]
Below is the relevant material from the paper I referenced earlier which provides the answer in the case that we use the Brownian motion with drift model.

Suppose you have a win rate of 1.3 BB/100 and a standard deviation of 12 BB/100. Then you can expect to play 13965 hands before you see a 100 BB downswing. Here's a general formula.

Let N be the number of hands you must play before you see a downswing of 100 BB. To analyze N, we model your bankroll by a Brownian motion with drift, i.e. let X(n) be your net profit in BB after n*100 hands. Then

X(t) = s*B(t) + w*t,

where s=12 and w=1.3. Let M(t) be the maximum value of X on the interval [0,t] and define T to be the first time that M(t)-X(t)=a, where a=100. Then N=100*T.

If we let g=w/s^2, then

E[T] = (e^{2ga} - 1)/(2gw) - a/w
= (e^{2aw/s^2} - 1)*s^2/(2w^2) - a/w

In our example,

E[T] = (e^{260/144} - 1)*144/(2*1.69) - 100/1.3
= 139.65,

so E[N]=13965.

If you want to answer more interesting questions such as this:

[ QUOTE ]
given a probability like 99%, say P, how many hands would have to be played to have a probability P that a 100BB slide will occur somewhere during the play of those hands?

[/ QUOTE ]
you need to know the distribution of T, i.e. you need to have a formula for the function F(t)=P(T<t). Unfortunately, I could not find such a formula. But the article does give this: for all b>0,

E[e^{-bT}]
= f(b)e^{-ga}/(f(b)cosh(f(b)a) - g sinh(f(b)a)),

where

f(b) = sqrt{g^2 + 2b/s^2}

and cosh and sinh are, respectively, the hyperbolic cosine and sine. This is the Laplace transform of T and someone with enough computing experience may be able to (at least numerically) invert this to provide us with a formula or a graph of the function F(t). However, even without an inversion, you could use the Laplace transform to compute the moments of T. For example, you could answer this question:

We know the mean number of hands needed to see a 100 BB downswing (in this example) is 13965; that is, E[N]=13965. What is the standard deviation of N?

Anyone want to take a crack at this? Anyone want to derive a general formula for E[T^2]?

[irchans]

It seems to me that if we let

h(b) = E[ e^{-bT}],

Then

h(b) = E[ 1 - b*T + (b*T)^2/2 - (b*T)^3/6 + ... ]
= 1 - b*E[T] + b^2*E[T^2]/2 - b^3*E[T^3]/6 + ....

So

h'(b) = E[T] + b*E[T^2] - b^2*E[T^3]/2 + ...
h''(b) = E[T^2] - b*E[T^3] + ...
h''(0) = E[T^2].


Jason said that

E[e^{-bT}]
= f(b)e^{-ga}/(f(b)cosh(f(b)a) - g sinh(f(b)a))

where

f(b) = sqrt{g^2 + 2b/s^2}.

I used Mathematica to do the algebra and got

E[T^2] = (-2 + e^(4*a*g) + 2*a^2*g^2 + e^(2*a*g)*(1 - 6*a*g))/(2*g^4*s^4).

Hmm.