Bankroll needed on 1 table vs 2 tables.

[Tommy Angelo]

I don't give any thought to bankroll requirements or risk of ruin because the size of my bankroll has virtually no effect on my game selection. But, to the question you raise, I've given it some thought because I have on several occasions for weeks or months been on a bankroll-sharing team. We never played at the same table, but we would simply share our risk, and lower X.

Let's say we have 100 players, all with an expected earn of 1BB/h, and each with a BR of 10K. Playing alone, they each share the same risk of ruin, that we will call Y. (I don't even know the units used for risk-of-ruin.)

They pool their funds, and now they have a bankroll of 1,000,000.

It seems intuitively obvious to me that each individual's Y has now effectively hit zero. In other words, the likelyhood of the entire team going busted, when each of them has expected earn of 1BB/h, is so close to zero that each player can feel VERY secure about paying rent for a long long time. And more to the point, if their combined X were halved, their individual Y's would be essentially uneffected.

This, to me, is the same as one player playing 100 tables. The conclusion I would draw is that with the addition of each table, from 1 to 2, 2 to 3, 99 to 100, his theoretical bankroll needs, per table, go down.

Tommy

[Robk]

[ QUOTE ]
This is wrong

[/ QUOTE ]

Actually I think you're both correct. Variance is additive. So if you play two tables at once with variance x, then your hourly variance will in fact be 2x, and hence your hourly standard deviation is rt(2x) = rt(2)*one table sd. But Dynasty is also correct that this doesn't mean you need a bigger bankroll. That's because SD per hand is what really matters. Consider playing n tables, with winrate z, variance x. Your one table bankroll is 9x/4z. For n tables your variance is now nx. And your winrate is now nz. So your new bankroll needs are 9nx/4nz = 9x/4z.

[Franchise (TTT)]

[ QUOTE ]
They pool their funds, and now they have a bankroll of 1,000,000.

[/ QUOTE ]

[ QUOTE ]
This, to me, is the same as one player playing 100 tables. The conclusion I would draw is that with the addition of each table, from 1 to 2, 2 to 3, 99 to 100, his theoretical bankroll needs, per table, go down.

[/ QUOTE ]

Those two statements don't harmonize well with each other.

The pool of players' Y does not decrease because they're playing multiple tables at once, but because they pooled their cash together. The bankroll is now NX, and the hours they play is NH (X being each individual's bankroll, H being each individual's hours, and N being the number of people). Notice their bankroll has increased by a factor of N.

The risk of breaking the bankroll, Y, decreases because like Dynasty said, the number of hours played does not affect bankroll.

[SoBeDude]

[ QUOTE ]
Why would your bankroll requirements change at all? A bankroll isn't dependent on how often you play.

[/ QUOTE ]

But the probability of all (or many) players having a simultaneous losing streak is drastically diminished.

And that is what impacts risk of ruin.

-Scott

[BruceZ]

Where is Bruce Z when you need him?

Right here!

The bankroll formulas I have given here assume that you will play forever. That means an infinite number of hands. It doesn't matter if you play this infinite number of hands 1 at time, 2 at a time, or 100 at a time. It's still an infinite number of hands, and the bankroll requirement for a given risk of ruin does not change. The bankroll requirement depends on the ratio of EV to variance. Variance is standard deviation squared. When you play 2 tables at once, your EV doubles, your standard deviation goes up by sqrt(2), and the variance doubles, so EV/var remains constant, and your bankroll requirement remains constant as it must.

Now, when you play 2 tables at once, you will play twice as many hands in the same amount of time. Obviously the more hands you play, the higher your risk of going broke in that amount of time. Even if you play 1 hand at a time, your risk of going broke after playing 200 hours is higher than it is after playing 100 hours. Your risk of ruin can only go up with time, it can't go down. If you play 2 hands at a time, your risk of ruin will be as high after just 100 hours as it was after 200 hours playing 1 hand at a time. How then can the risk of ruin and bankroll requirements be the same playing 2 hands as it is with 1? That's because the risk of ruin levels off after a few hundred hours, and it never becomes higher than the amount given by the formula for infinite time. It always increases, but it does so very slowly so as never to exceed this amount. Obviously this must be true for the formula to be valid. I worked out a quick numerical example to illustrate this effect.

For a win rate of 1 bb/hr, hourly SD of 10 bb, and a bankroll of 200 bb, the risk of ruin if you play forever is 1.83%. Here is the risk of ruin as a function of the number of hours you play, playing one hand at a time.
<font class="small">Code:</font><hr /><pre>
hours tables risk of ruin tables risk of ruin

100 1 0.43% 2 1.15%
200 1 1.15% 2 1.68%
400 1 1.68% 2 1.76%
500 1 1.76% 2 1.83%
1000 1 1.83% 2 1.83%
Infinite 1 1.83% 2 1.83%</pre><hr />

This is derived from a short term risk of ruin formula, not the one I gave you earlier. It becomes the same as the formula I gave you earlier for infinite time.

Notice that the risk of ruin playing 2 tables is the same as playing 1 table for twice the number of hours. In the beginning, this is a large difference. The risk of ruin is almost 3 times as high at 100 hours. Although this difference in risk is large, a separate calculation showed that in order to have the same risk of ruin at 100 hours playing 2 tables as you had playing 1 table, you would only need to increase your bankroll to 240 bb. As time goes on, the risk of ruin levels off, and the risk for 2 tables catches up. Notice that at 500 hours on 1 table, the risk of ruin is over 95% of the value that it would be for an infinite number of hours. It levels off and increases very slowly beyond this point. This is why if you don't go broke early, in the first few hundred hours, it is unlikely that you will ever go broke as long as your EV and SD do not change. This is due to the fact that your bankroll is likely to have increased so much by this time that your risk of ruin has become very low. Remember, it started at 1.8%, and if you double your bankroll your risk of ruin will square to 1.8% of 1.8% to 0.0324%. Each time your bankroll increases a small amount, your risk of ruin decreases exponentially. Playing 1 table for 1000 hours has essentially the same risk of ruin as playing 1 table for an infinite number of hours, and this is the same risk as playing 2 tables for 1000 hours, so the results converge to the same value.

Tommy is correct that when the members of a team playing blackjack or poker pool their bankrolls, each member can play at the same limits as if he himself controlled the entire bankroll. Hence each member can play at a higher level than he could by himself, and each team member would make more money on average with the same risk of ruin as if he played at a lower level by himself.

The conclusion is that if you play multiple tables, you don't need any additional bankroll in order to achieve the same risk of ruin for playing forever. You do need additional bankroll to achieve the same risk of ruin for playing a given number of hours, but this risk of ruin for a given number of hours will always be smaller than the risk of ruin for playing forever, and after a several hundred hours, it will be approximately the same as the risk of ruin for playing forever. So if you plan to play more than several hundred hours, you need only to refer to the formulas I have given for infinite time in order to determine your bankroll requirement and risk of ruin, and these will be essentially the same even if you play 100 tables at once.

[Benman]

Dynasty is right. Adding hands played only speeds up the process of arriving at whatever is going to happen to you at a given bankroll and variance (go broke or get rich). Angelo's situation is different because in that case you are increasing your bankroll 100x, in addition to playing more hands. In that case it's not a wash--increasing your bankroll is what helps you.