Swings in NLCASH

[MasterLJ]

[ QUOTE ]
According to my simulations even a 50 buy-in downswing is possible with a stable winrate of 8 PTBB/100 (no tilting/other psychological defects allowed) although it'll be extremely rare. That is for a player whose standard deviation per 100 hands is about 75PTBB/100. 20BI downswings seem to be quite common.

I have been researching variance in heads up poker lately and I'll probably share my simulator with you once it is easier to use.

[/ QUOTE ]

Given my ridiculous first response, I must correct what I said. I totally agree with the quoted post (BTW, please share your simulation that would be awesome!).

20+ BI downswings will happen to a full time player probably monthly.

It really depends on game selection. If you stick with passive players I believe your downswings will be much much more manageable.

[TNixon]

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That is for a player whose standard deviation per 100 hands is about 75PTBB/100.

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Where does this number come from? Is that your personal std. dev, or a guesstimate?

I would be *very* interested in seeing some real player winrates and standard deviations. The data point of exactly one that I have seen so far is 55.04BB/100, but I don't know what the associated winrate is.

[ QUOTE ]
20BI downswings seem to be quite common.

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And what does "quite common" mean here? Just a ballpark would be good, like an average number of 20BI downswings in 10k hands (or 100k or 500k hands or whatever if 10k isn't enough).

[ QUOTE ]
20+ BI downswings will happen to a full time player probably monthly.

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About how many hands does a month full-time represent? Again, ballpark figures are fine.

Inquiring minds want to know!

[jay_shark]

A heads up player will typically experience about 50 to 60 ptbb/100 hands or about 100-120 big blinds /100 hands .

The probability that a player with a win rate of 8ptbb/100 hands will experience a 20 buy-in downswing with s.d = 120 big blinds is :

B=-s.d^2/(2*win-rate)*lnr
Solve for r and we get the risk of ruin for this player .

e^(-B*2*win-rate/s.d^2)= r

B = 2000 big blinds
win-rate = 16 big blinds
s.d = 120 big blinds
r=1.1 %

So with that being said , there is only a 1.1% risk of ever busting from this game if we were to play indefinitely .
Contrary to some of the arguments already expressed in this thread .

[TNixon]

Er, correct me if I'm wrong, but it looks like you've calculated a 1.1% risk of ruin when starting with 20 buyins, but said absolutely nothing about how likely a 20 buy-in downswing is over any given X number of hands.

[jay_shark]

Fyi , I think having a s.d of 150 ptbb/100 to be much higher than the norm .

In any case , I computed the ror of a player with a win-rate of 8ptbb/100 hands and a s.d of 150 ptbb/100 hands to be 5.81% .

The probability of losing 20 buy-ins in a month must be lower than this number .

[jay_shark]

[ QUOTE ]
Er, correct me if I'm wrong, but it looks like you've calculated a 1.1% risk of ruin when starting with 20 buyins, but said absolutely nothing about how likely a 20 buy-in downswing is over any given X number of hands.

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No , this formula tells you the probability of busting if you play this game indefinitely . There is no simple formula to tell you the probability of busting in x hands .

[TNixon]

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The probability of losing 20 buy-ins in a month must be lower than this number .

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Why?

You could lose 20 buyins after winning 1, and that situation would still not be part of the "ruin" percentage. In fact, I would think the chance of having a 20BI downswing over X hands (for a sufficiently large X) would have to be *higher* than the risk of ruin percentage, and pretty significantly so.

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No , this formula tells you the probability of busting if you play this game indefinitely . There is no simple formula to tell you the probability of busting in x hands .

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We don't care about the probability of busting in X hands. The question is the probability of losing 20 buyins in X hands.

[jay_shark]

It should be obvious that it has to be lower .

P(bust in one month) + P(bust after first month ) = P( busting out)

Notice that the two probabilities in the lhs are mutually exclusive . So the probability you bust in one month has to be lower than the probability you bust out if you play forever .

The formula I gave above is more useful for knowing your risk of busting out .

[TNixon]

You keep talking about busting out, but we don't care about that.

Risk of ruin calculations include the fact that you're gaining over time, which reduces your risk of ruin as you get deeper.

If after the first month, you've gained X buyins, then you're less likely to bust out entirely (which is certainly a factor in the RoR calculation), but your chances of having a 20BI downswing are exactly the same as they were for the previous month.

[ QUOTE ]
The formula I gave above is more useful for knowing your risk of busting out .

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And not at all useful for answering the question that's actually being asked, which is why I'm confused.

[TNixon]

In fact, given this:

[ QUOTE ]
P(bust in one month) + P(bust after first month ) = P( busting out)

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For some number of hands, P(having a 20BI drop) is certainly going to be greater than P(bust in one month). That number might be 10k, it might be 40 million, I don't know.

Whatever that number is is irrelevant, because it is quite obvious that the probability of dropping 20 buyins *can* be greater than the probability of going broke in the first month.

And if P(Drop 20BI) is greater than P(bust first month), then it follows that the chance of dropping 20BI over an infinite number of hands *has* to be greater than the chance of going broke when starting with 20BI. Because the chance of going broke per month has to decrease on average, but the chance of dropping 20 buyins per month can remain the same.

In fact, I'll go even a little further with this.

If our RoR is 1%, there is unquestionably *some* number X, where the chance of having a single 20BI downswing is greater than 1%. That 20BI downswing might happen after we've already won a thousand buyins, but if we play enough hands, there will eventually be a point where we're down 20 buyins from a previous point.

But since the chance of being down 20 buyins in a single month can possibly be higher than the chance of going broke sometime between now and the end of time (if we can just play enough hands in that first month), then this:

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It should be obvious that it has to be lower .

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Is not "obvious" at all, since it's quite clear that it *can* be higher, for a sufficiently large number of hands.

The real question is how large "sufficiently large" is.