Variance, winrates, and expectation (long)

What evidence to the contrary? Very few people who are true 2bb winners will have had 20k hand downswings. The math doesn't lie. I see a lot of people who THINK they are 2bb winners who have downswings that "defy" the math. But it is the estimate of their ability that is suspect, not the math. And if you look at the graph, a 20k hand downswing should happen more than 1% of the time. That means that a given 2bb winner should see at least 1 such downswing every 2 million hands. OR, 50 such winners who each play 40k hands should expect to have at least one of them hitting such a downswing. When you have a forum with hundreds of good players posting, it is not unlikely that you will see several such "1% events".

-v

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I see a lot of people who THINK they are 2bb winners who have downswings that "defy" the math. But it is the estimate of their ability that is suspect, not the math.

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Amen. The math gods are pleased.

[joseki]

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Also, dude, these graphs address the probibilty of future outcomes starting "now".

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This is the only way to calculate future outcomes.

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Nice post.

It just seems to me that people think of the 99% CI as a bound, when it clearly is not. There is no need for alarm if your results fall outside these lines on occasion.

Isn't the whole point of the OP to discount the likelyhood of large downswings? If so, I thought I could clarify how these downswings easily fit into this CI exercise.

[RunDownHouse]

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Very few people who are true 2bb winners will have had 20k hand downswings.

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OK, I agree completely and think we just got mixed up on semantics. When you said "big" downswings, I misunderstood and took it as magnitude, not length, as you meant it.

[cartman]

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these graphs address the probibilty of future outcomes starting "now". Every time you're delt cards that's one "now".

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This is true and crucial to the assessment of this type of situation. I think many are overlooking this. The fact that the -99% line in the bottom graph intersects 0 just short of the 40,000 hand mark doesn't mean that there is only a 1% chance that a 2BB/100 winner will have a breakeven stretch of that length at some point. It only means that he only has a 1% chance of breaking even on his VERY NEXT 40,000 HAND STRETCH STARTING RIGHT NOW! That is entirely different than having a 1% chance of ever breaking even for 40,000 hands.

Cartman

[MrBig30]

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Great job.

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I agree. I'd also like to see the graphs for 17 and 19 STDV.

Krishan

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Me too. Like Surf said 15 is really low and those beautiful graphs would be more interesting to consider with a more normal shorthanded SD (FWIW mine seems to be about 16.5-17)

Other than that, great post.

[tallstack]

Very nice post. I do think, though, that you should not use this calculation method to address downswing severity.

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Take a look at the 2 BB/100 graph. If a player played at a solid 2 BB skill level for a sample size of let's say 20,000 hands, he really shouldn't have a very big downswing. If the player was able to maintain this quality of play even through lots of bad beats and cold cards, only about 5% of the time should he experience a downswing of greater than 110 BB's.


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I think that there is a problem with reaching this conclusion from your data. The results show only end points for a given hand count, not the path taken to arrive at this end point. By this I mean at 20,000 hands it shows you the likelihood of being up or down a given amount on the 20,000th hand. It does not give you any information on the path taken to get there. Any poker player is going to quote his downswing based on how low he is from his highest peak. Thus, downswings are entirely path dependent and the information regarding the path taken to reach a given end point is not extractible from your type of calculation.

If we try and calculate maximum downswings for very large hand samples we can see more clearly that this type of calculation is not suitable. If we choose 100,000 hands for the 2BB/100 player for example, what would the maximum downswing with a xx% probability be? We can't look at the 100,000 hand total, because this will include all the winning periods before and after the maximum downswing. We also can't look at the maximum dip prior to 100,000 hands because this would show no larger dip for 100,000 hands than it would for say 50,000 hands or 500,000 hands. This cannot be correct. Intuitively, as sample sizes get larger the probability of a certain sized downswing must increase as well. Again, the information of the path taken to reach a hand count must be considered.

We could simulate a large sample of hands and run the simulation many times to get a measure of the downswings expected. I know that this has been done by several posters in the STT forum, and it likely has been done by someone in a limit forum, but I don't have a 2+2 link that I know of. Outside of simulating the path taken through a given number of hands, I do not think that we can conclude what an expected downswing should be.

Dave S

[Luzion]

Nice post sir. Well written, organized, informative, and detailed.

[beachbum]

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Take a look at the 2 BB/100 graph. If a player played at a solid 2 BB skill level for a sample size of let's say 20,000 hands, he really shouldn't have a very big downswing.


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How do you explain all of the empirical evidence to the contrary? Tilt? How does a player running ultra-hot for 1k hands, then ultracold for 3k hands fit into this?

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Well, although SD converges fairly quickly for a player's style of play, in the short term it can increase quite a bit. This could be a result of a bit higher VPIP and WtSD in the short term. Also, short term theoretical winrates can be higher or lower than normal.

For instance, my hottest stretch was being up about 120 BB in ~300 hands. For these 300 hands, my SD was probably much higher than normal. Also, my theorectical winrate was maybe closer to 3 or 4 BB. However, I wouldn't be surprised if I never won this many BB's in such a small amount of hands ever again since it was <<1% chance.

[beachbum]

I think I can explain your concerns. Let's consider looking at a 20,000 hand sample within a 100,000 hand sample. Are there 5 20,000 samples we can choose from? Not really, there are 80,000 20,000 hand samples. 1-20,000, 2-20,001, 3-20,002, 4-20,003,...,79,999-99,998, and 80,000-99,999.

For illustration purposes, if we want to get exact, a 200 BB downswing will occur on the 2 BB graph at a 99.25% confidence interval right around 10,000 hands. Since there are 90,000 10,000 hand samples in total, there should be 0.75% of these samples where a downswing of that magnitude has occured. This results in 675 (0.75% x 90,000) ranges of 10,000 hands. Like you mentioned this is looking at some highest point in the graph and calculating from there.

Let's say a player goes on a rush and wins 100 BB's in 1000 hands. Then he cools off alot. There's a decent chance if you look at hand 1001-11,000, there will be a 200 BB downswing. Then look at immediately succeeding hand ranges like 1002-11,001, 1003-11,002, etc. and you might get hundreds of ranges where this downswing is 200BB.

However, if you just look at your results in consecutive 10,000 hand intervals, there's a very small chance you're going to have a downswing of this severity. I guess my point is, instead of blaming variance, turn the focus inward and see where our play has been suboptimal.