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/sh...=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. [img]/images/graemlins/wink.gif[/img]

[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?