Bankroll Considerations

[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?

[uphigh_downlow]

[ QUOTE ]

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?

[/ QUOTE ]

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

[paulnortonyoung]

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

Paul

[jason1990]

[ QUOTE ]
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.

[/ QUOTE ]
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.

[Alex/Mugaaz]

[ QUOTE ]
[ QUOTE ]
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.

[/ QUOTE ]
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.

[/ QUOTE ]

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.

[jason1990]

[ QUOTE ]
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.

[/ QUOTE ]
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.

[Guruman]

[ QUOTE ]

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.

[/ QUOTE ]

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.

[uphigh_downlow]

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