Bankroll Considerations

[uphigh_downlow]

I'm just trying to seek confirmation on this.

Wonder if there is more to the picture. I think an example might illustrate my question better.

Lets say I place a bet 'b' on a game repeatedly.

And as per my average returns and variance(unknown at this stage), I have somehow magically determined that an appropriate bankroll for this kind of venture is 'BR'

However I dont have enough money to meet the bankroll requirement.(Again meeting the bankroll requirement is a pretty hazy term in itself). But hopefully we can take it at face value. And assume that it is a meaningful statement. Lets say, its something that gives me a 99% chance of not going broke ever.


Anyway my dilemna is that I only have an amount X, which is below BR

However I decide to take a chance that I wont go broke before I can turn a profit and decide to gamble anyway.

A few fine days later, I have managed to turn a profit and I now have the exact amount to meet the bankroll requirement.

Can I now feel better, assuming that I'm safe, as if I never played that game before and am making a fresh start with the appropriate bankroll.

Does the fact that I have had good days not mean that I'm probably going to have bad days soon, which would mean that I cannot really feel safe?

Now if we replace this "game" with
a) SNGs
a) Ring game at $blah buyin

etc etc

Does the independence of the result of each bet still hold??
Is independence a safe assumption, or are there reasons to believe there might be corelation.

Also assuming that I started with X, what is the probability that I bust out before I reach BR?

[fnord_too]

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Can I now feel better, assuming that I'm safe, as if I never played that game before and am making a fresh start with the appropriate bankroll.

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Yes

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Does the fact that I have had good days not mean that I'm probably going to have bad days soon, which would mean that I cannot really feel safe?

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No. Past results do not influence future ones (except for psychological implication in games with decisions played against other humans. e.g. the fact that you have been running hot may cause your opponents to fear you and play worse.)

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Does the independence of the result of each bet still hold??
Is independence a safe assumption, or are there reasons to believe there might be corelation.



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Yes. Again, the only reason past results will influence the future is from a psychological standpoint. Another for instance, you may loosen up if you have been winning or an opponent may adjust to your style of play once he has played several hands against you. But as far as chance goes, every hand is independant.

[uphigh_downlow]

Thank You for the confirmation.

Now for the underlying question. When we try to fit the slightly more complex games, where external factors, including psychological and many others like people ajusting to you, or you adjusting to people bla bla
to this Normal distribution model, how far are we from the truth.

Have there been any empirical studies that suggest that the events are indeed IID?

[Girchuck]

The safe bankroll calculations assume averaged values for win rate and standard deviation and have no other parameters (besides bankroll safety factor), they cannot explicitly take into account an event such as people starting to play better against you as time goes on. One could probably guess a winrate as a more interesting function of time than a flat average, and do some complicated analysis. If one has a very very large database, one could analyze winrate vs average table VP$IP, table AF, table players to date number of hands against you, and many others. One could possibly extract some correlations, but it would be very complicated.
Certainly not a one line equation popular on these forums.

[uphigh_downlow]

[ QUOTE ]
The safe bankroll calculations assume averaged values for win rate and standard deviation and have no other parameters (besides bankroll safety factor), they cannot explicitly take into account an event such as people starting to play better against you as time goes on. One could probably guess a winrate as a more interesting function of time than a flat average, and do some complicated analysis. If one has a very very large database, one could analyze winrate vs average table VP$IP, table AF, table players to date number of hands against you, and many others. One could possibly extract some correlations, but it would be very complicated.
Certainly not a one line equation popular on these forums.

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Thank you for the reply.

I am not very familiar with the body of literature out there, and was hoping someone could point to another situation which is similar to the one we face here.

for example that we assume that the average expected pay off on the bet is a function of time t. Now is there a similar situation, possibly somewhere in economics.

I'm not looking for closed form solutions to the problem(which is admittedly complex), but certainly interesting insights that can be drawn by comparing two different situations.

I wouldnt mind reading a bunch of papers, only if someone can point the right resources.

[Bad Lobster]

The biggest hidden variable in these bankroll calculations is the likelihood that you aren't as good a poker player as you think you are. One of the inputs to the equation is the number of big-size bets you can win per hour. Suppose you're a relative novice who's just decided to start playing poker seriously, and then you go on a losing tear--do you think your original calculations are still valid?

[jason1990]

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I am not very familiar with the body of literature out there, and was hoping someone could point to another situation which is similar to the one we face here.

for example that we assume that the average expected pay off on the bet is a function of time t. Now is there a similar situation, possibly somewhere in economics.

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The "Invariance Principle" or "Functional Central Limit Theorem" tells us that, after a "long time," the graph of your bankroll vs. the number of hands played will look like a Brownian motion. (Actually, you have to subtract the mean for it to look like a Brownian motion, so the actual graph looks like a Brownian motion with drift.) One way to derive the risk of ruin formula is to assume your bankroll is a Brownian motion with drift. The risk of ruin is then given in terms of the time it takes this process to hit the line -b (b is your bankroll). This is called the "hitting time."

The specific model is this: if X(t) is your bankroll after 100*t hands, then

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

where B is Brownian motion, s is your standard deviation and m is your win rate (in BB/100). Written in differential notation, this is

dX = s dB + m dt.

If you want to assume that your winrate and/or standard deviation are (possibly random) functions of time, then you could write

dX = s(t) dB + m(t) dt.

This is what is called a stochastic differential equation. Depending on s and m, it may or may not have an explicit solution in terms of t and Brownian motion. You could analyze the hitting time of this solution to obtain a generalized risk of ruin. Modelling random phenomena with such processes is a common tool in mathematical finance. You can find several references if you just look up the key words Brownian motion, stochastic calculus, and mathematical finance.

However, I think it would largely be a waste of time to use this approach to analyze your poker results. It's interesting stuff and you could learn a lot by studying it, but for poker, I think it is overkill. The i.i.d. assumption about your poker results is a pretty good one, in my opinion. Where it breaks down would be when you move up or down in levels, or you switch to a different site. Your game will probably improve as you play, so that your winrate might go up within a single level at a single site. But I don't think you could observe this change without a very large sample size. And even if the changing winrate is a big factor, I think the most practical way to deal with it is to simply ignore your old results. For example, if you've played 100k hands, you may want to analyze the last 50k and the first 50k separately. Doing something like this would be much better, in my opinion, than trying something as complicated as the above.

[Girchuck]

When your chance of going broke is calculated, an event that your bankroll will immediately go down and never reach a higher value has a certain probability. If this event happens, your chance of going broke increases. If the opposite happens, and your bankroll goes up, your chance of getting broke decreases.

[wadea]

This OP is amusing in the way that he uses terms and jargon to imply that he has at least some minimal understanding of statistics and probability, then proceeds to type this:

"Does the fact that I have had good days not mean that I'm probably going to have bad days soon, which would mean that I cannot really feel safe?"

Yeah, man, you're DUE for some poor results.

I mean, if you need to ask a question like that, you'd better not be playing poker until you KNOW the answer. And not just think you know it, but really UNDERSTAND the answer. I hate to flame a guy for asking a question, but there's just something about the way this one was asked that struck me as ridiculous.

-w.a.

[uphigh_downlow]

Well that might have been uncalled for, but I'll accept your criticism.

I understand independence. I'm a CS buy with a decent mathematical background
What I was really trying to question was the independence assumption, and if it really holds. I wish I had used less jargon to come off as a rookie, and atleast get your sympathy [img]/images/graemlins/smile.gif[/img]

The question is just play with words to pose the question to a layman. I think you took it to literally. And if your answer is a blanket, it doesnt matter, you have blindly assumed the independence of events.

I'm just trying to question that, and was wondering if there is any similar situation, where such literature was already available, and could lead to more insightful answer to the puzzle.