Bankroll Considerations
 

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?

Re: Bankroll Considerations
 

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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.

Re: Bankroll Considerations
 

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?

Re: Bankroll Considerations
 

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.

Re: Bankroll Considerations
 

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.

Re: Bankroll Considerations
 

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.

Re: Bankroll Considerations
 

If you want to eventually apply this to poker as opposed to coinflips that pay 1.1 to 1, you need to keep a couple of important things in mind there bro:

1)If you are not a winning player against your current set of opponents, your bankroll will never be large enough to protect you from going broke.

2)your winrate and standard deviation (from wich your risk of ruin and consequently your requisite bankroll requirements are derived) are statistics that will constantly be in flux as you go through the various stages of learning the game of poker.

Your winrate will go up as you plug various leaks and learn to value bet in the appropriate places and it will go down as you take on tougher opponents, tilt, and go though downswings.

Your SD will also move around as you adjust your game to your own knowledge, your differing opponents, your leaks, and your aggressiveness.

Since bankroll requirement calculations assume static winrate, sd, and bet sizes - your game and your opponents must become pretty consistent for this question to even relate to poker.

Here's my advice:

Start out at low limits, read, play, and learn the game until you can beat that limit and those opponents well enough to move up.

Take a few stabs at moving up before committing to a higher limit entirely.

Don't be afraid to move back down if your bankroll takes some serious hits due to better opponents/tilts/downswings.

rinse & repeat.

Remember: If you are not a winning player against your current set of opponents, your bankroll will never be large enough to protect you from going broke.

Re: Bankroll Considerations
 

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.

Re: Bankroll Considerations
 

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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.

Re: Bankroll Considerations
 

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If you want to eventually apply this to poker as opposed to coinflips that pay 1.1 to 1, you need to keep a couple of important things in mind there bro:

Remember: If you are not a winning player against your current set of opponents, your bankroll will never be large enough to protect you from going broke.

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Thank you very much for the kind advice. I'll keep it very much in mind.

But I was quite interested in the first part of your reply, where you said something about applying this to poker as opposed to coin flips.

I was wondering if you had any pertinent comment on that, besides general game play considerations. You seem to have caught the drift of the actual underlying questions.

poker sessions are not the same as coinflips that pay off certain odds. Or maybe they are. I do not know.

Just trying to learn as i go along. Just want to be aware of any patterns that might come along the way and throw a curveball.

Re: Bankroll Considerations
 

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?

Re: Bankroll Considerations
 

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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?

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Well I tried to avert this problem by saying, that we used magic and somehow came up with the numbers.

After all I never truly know my expectation, except that I can establish bounds for it.

There has to be a startinf point somewhere. Otherwise how can I proceed in the first place

Re: Bankroll Considerations
 

It's not specifically about poker....but you might find the theory discussion here interesting.

Paul

Re: Bankroll Considerations
 

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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.

Re: Bankroll Considerations
 

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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.

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The problem with this is that several winning players hee have had 50,000 hand stretched where they only broke even.

I think the most important part of analyzing your results is when you move up in limits. After you have a fair degree of certainty that you are indeed winning, the best use of your data is to make sure you still are, and to look too when times are tough.

Re: Bankroll Considerations
 

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The problem with this is that several winning players hee have had 50,000 hand stretched where they only broke even.

I think the most important part of analyzing your results is when you move up in limits. After you have a fair degree of certainty that you are indeed winning, the best use of your data is to make sure you still are, and to look too when times are tough.

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I think "several" may be an overstatement. At any rate, one should not take too much stock in any statistical analysis of one's poker results, especially when moving up in limits. There's simply too much variance. The best form of analysis is to study your specific hands and how you played them.

Aside from the specific mathematical references for the OP, my main point is that, as far as statistical analysis goes, if you assume your winrate is a function of time, you will not gain enough to compensate for the added complexity and potential inaccuracies. But if you wish to do this, there are standard models in place.

Re: Bankroll Considerations
 

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Thank you very much for the kind advice. I'll keep it very much in mind.

But I was quite interested in the first part of your reply, where you said something about applying this to poker as opposed to coin flips.

poker sessions are not the same as coinflips that pay off certain odds. Or maybe they are. I do not know.

Just trying to learn as i go along. Just want to be aware of any patterns that might come along the way and throw a curveball.

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The primary difference between a poker game and a coinflip that pays 1.1 to 1 is that with the coinflip you have all of the information in the game available to you, and you know that you can make a certain play (to make the bet as opposed to passing on it) every time with positive EV.

In poker there are several facets of the game that you do not know when you make your decisions, and therefore you cannot be certain about the specific EV of your play in any concrete sense.

Ex:you hold AKs on a flop of 9TQ rainbow.

The only things you can calclulate for sure are
a)the probability of hitting a J for the straight on the next card or the river
b)the probability of hitting an A or K on the next card or the river
c)the requisite pot odds required to chase each card

Things you cannot calculate precisely that affect your EV in the hand include
a)your opponent's cards
c)what your opponent thinks you have
d)the size of a bet your opponent will call
e)the size of a bet your opponent will make if checked to
f)the size of a bet your opponent will fold to
g)what your opponent's next move will be

The most you can do is estimate any of these factors, and your ability to do so will almost entirely determine whether or not your bankroll is ever going to be large enough.

Solid mathematical play is an important foundation for any poker player, but once you've done the math in any hand you have to play the art in order to execute your move.

Bankroll requirements are not hard and fast rules that will ruin a player if violated. Have an appropriate bankroll will not guarantee that you won't go broke. Having a short bankroll will not guarantee that you will.

They are guidelines in place to show winning players how much to have available when attempting specific stakes.

Re: Bankroll Considerations
 

Thanks for your reply.

The real reason I posted that topic, was just to get random thoughts on the topic. The question does not really describe accurately the intentions behind it.

Not surprisingly, yours has been the most insightful reply.

Now let me try and explain the meat of the problem.

As you said yourself, it might be better to take last 50 hands. ( and if you will allow me to replace hands with sessions or a suitable unit that is just n consecutive hands) I replace it with saying Lets take last 50 hands of play at a certain game or one tournament game, or 12 tournaments. Anything meaningful and relevant and call it a session.

Also time analysis is not really meaningful, so we replace time with x which stands for the xth session.

for example, now i can describe my S(x) as a prime function or a Möbius function (of course this might be far from the truth) BUt its a luxury I can well afford theoretically just to illustrate the point.
http://www.2dcurves.com/discrete/h12mobix.gif
http://www.2dcurves.com/discrete/h12prim+.gif


similarly D(x) can be defined too.

At a time that I change from one stake to the other, I could see large changes in expectation and deviation, which almost have no relation to the mean and deviation of the last session. However in our normal way of analysing this situation, we do not take that into consideration.

Additionally, if the suggestion was to just play consistently at a level, until we managed the bankroll requirement for a higher stake, and then decided to move on up, the inherent dilemna we are faced with is that we do not really know whats going to happen at a higher level.
And noone really knows how long to stick at a level before moving on. So it is quite plausible that several players (myself included) try and experiment. Mix and match. And then settle down at a level for a fair bit of time. And then repeat this process. Some make several premature abortive attempts at jumping to higher stakes. Some might jump to lower stakes after a prolonged bad run.

Now I would imagine that something interesting might happen to the rik of ruin, rate of growth of your bankroll in such situations

Exactly what I dont know, but I'm glad my differential equations text will finally found some use afer a few years of gathering dust.

ps: Even though this might turn out to be an overkill for poker subject, but the fun is well worth it

I did try and look up the key words you mentioned, but given the lack of my familiarity with the state of affairs, I'm still googling and prolly will be for quite a while. haha

A mistake I made in several of my replies above was in using independence and IID interchangeably sometimes. Hope you guys forgive me for that