Monday, February 1, 2010

Variance in internet and big bet poker(an article I wrote in Feb 08)

Most people don't know what they're talking about when they speak the words "variance" or attempt to do analysis. There are a lot of statistical Platonists out there, doing math based upon faulty assumptions, rendering their conclusions of little more relevance than if you were to go and ask a priest about your downswing(or upswing). Perhaps they should set up a confessional.
Is poker normal? For the last 20 years poker has been explained by classical ideas of statistics, the normal(or Gaussian) distribution, statistical mean and variance, based upon flipping coins. Yet there are things in poker which traditional theories have never quite been able to explain. Once winning players go on larger(and longer) than expected losing streaks, players win thousands and even millions from unexpected rich fish. These would have been labeled “outliers” and anomalies to the traditional theories, but upon closer inspection it appears that some of the applications of normality are fundamentally flawed.
I have been delving into statistical inference for awhile now; I have not shared what I have discovered from writers like Taleb, Hume, Popper, and most importantly the mathematics of Benoit Mandelbrot because I felt I had not yet understood it. Unlike many mathematicians, I worry about false precision.
The truth I have discovered is this. Many of the ideas of the normal distribution as they are applied to thinking about swings and risk in big-bet and internet poker are incorrect. True, the card distributions you receive will be normally distributed, however your poker results will be dependent, "fat-tailed", and may exhibit skew. The latter two I leave for a later explanation.

Dependence is an important idea, and it is one that can be analyzed. Over 50 years ago the Hydrologist Harold Edwin Hurst set about to build the dams along the Nile River, which were used to regulate water flow and irrigate the cotton fields of Egypt. The Engineers of Hurst’s day made the same assumption many mathematicians today would about the year to year deviations of rainfall. Independence. The amount of rainfall from year to year is simply random Brownian motion. They turned to the mathematics of coin-flipping to construct the height of their dam. They wanted to be able to store enough water to protect against drought, but also to hold water in a series of wet years. They gathered data on the past years average rainfall and jumbled it up, and found a bell curve. Luckily for the Egyptians, Hurst was an empirical scientist.
He examined the actual data and found that the year to year accumulations did not follow a normal distribution. The distinction was this; that if you took the predictions of an assumption of a normal distribution to the data, it would not predict correctly the highest accumulations, nor the lowest troughs. Jumbling up the data was philosophically corrupt, as the order of occurrence was paramount. The engineer’s assumptions would leave the dam height far too low. What Hurst found was a clustering effect, that wet years and dry years tended to occur together at a rate that wouldn’t be predicted by independence. This wasn’t exactly an entirely new observation.

“What God is about to do he showeth unto Pharoah. Behold, there come seven years of great plenty throughout all the land or Egypt; and there shall arise after them seven years of famine; and all the plenty shall be forgotten in the land of Egypt; and the famine shall consume the land.” Gen 41: 28-40

Enter Benoit Mandelbrot, and what he calls “the Joseph Effect”, after the biblical character . It is the idea of long dependence and can be measured by a test called a rescaled range statistic and the Hurst exponent(H). H is a measure between 0 and 1 that assumes no underlying distribution of data. Rather it is test that asks the question, so what is the distribution of data?
Hurst gathered data, lots of it. He gathered records of rainfall from wherever he could find them. He examined not only the deviations of rainfall, but also the order of the occurrence of the rainfall and found the same thing over and over. Dependence.
So the obvious question is what relevance does this have for poker? Fast forward to the present day, with the internet poker boom, Poker Tracker, and an abundance of data and you will shortly see. When we play poker we build up bankrolls to play in certain games, and that Hurst’s task of building of a dam is similar to our building of bankrolls.
We can use the same equation Hurst did to test the distribution of our poker results. H = log (Range/standard deviation)/log(N/2). The range is the value difference between the highest and lowest accumulations and N is the number of points in the unit you measure your results in.
If normality was true the Hurst exponent for the data would be .5, indicating normal Brownian motion. A Hurst exponent of less than .5 would indicate that the data tend to keep doubling back on themselves. Yet he found and approximated that deviations in annual rainfall had an exponent of .74, which indicated the persistence of long runs.
Each hand movement today is likely to be recorded; both the dollar deviation and the order of occurrence. In all of the data I have analyzed on internet and big-bet poker I have found dependence(and an absence of normality), however in order falsify the earlier assumptions I need more data, which is why I am asking for it from 2+2ers.

It is quite easy these days to gather data and analyze, and so I understand why it may have been difficult to see some of these ideas past, however the continuance to adhere to an idea in the face of contrary evidence is against the principles of falsification and I believe of science, as outlined by the philosopher Karl Popper.
The ancient Greeks and Romans believed in the beauty of symmetry and the perfection of nature. They believed in ideals. For example they thought the height of the highest mountain had to equal the trough of the deepest depth of the ocean. This is not a view that corresponds to reality. The way that many fit a bell shaped curve to their view of reality, and poker results, has this same ideal at heart.

Some of the swings you go through have an independent and normal aspect to them, and these can be large; however those will all be cancelled out by the law of large numbers. If you look at the measure of your standard deviation, if these were the biggest cause of your downswings, this should converge relatively quickly, say a few thousand hands. However, everyone I have analyzed sees big fluctuations in their standard deviations and it never seems to settle down to anything. So why do you go through a variation of results?
I offer the following explanations as some alternative hypothesis for the reasons you experience deviations, apart from those of the normal distribution of cards.
The first is the way in which you deal with swings, and the way others respond to you. Poker players know this as tilt. If you play worse after losing a hand in which you were a statistical favorite, there will be a heavy dependency to your results. Hours and days later may still be thinking and making plays because of the path dependence of that hand. Your results in this case are not linked to the normal distribution, rather your response to it. This is also similar to the idea that a single butterfly flapping its wings in South America, can cause a hurricane in Florida, while the same butterfly wing flap a day later might not do anything. In this case the normal deviations were a cause of your downswing, but it was not the biggest cause.

The second factor is ecological luck, and I would stress this point. The opponents you are playing against, the amount of money they have, your access to games, are all aspects of luck that are far under emphasized by people that delve into "uncertainty" and then tell you about the deviations of the normal distribution.
Consider the following thought experiment. You are a winning Pot Limit Omaha player over the last few years and join a new site and employ your strategy. You are in luck as its quite effective against the majority of the players on the site. You win 20 buy-ins in the span of a month. Now think about this from your opponents’ perspective. The people who you are best against may now avoid you, they may adapt or they may go broke. The people and strategies you are now playing against are different than the ones you were playing against before. The tough part is that it’s so difficult to notice. You do not notice the big pots you would have won the previous month in the exact same "normal" situation if the other player was playing. You start losing, maybe due to some slight normal randomness of cards, maybe because you use the wrong strategy against someone. You make many of the same bets you did before, however this time maybe you get played back at, or your better hand reading opponents fold their losers and call your bluffs.

Before you know it you've lost 10+ buy-ins. You are stuck as to why. You go to the confessional of your local Gaussianist minister. He tells you about sample size, and the swings of the normal distribution, and although he has no evidence that this was the biggest cause of your losses, or even a cause, he assumes that is why, because that’s his belief. He does some math that you don't quite get, but he assures you, you'll be better in no time and regression to the mean will take care of everything. You feel a little bit better.
You go to poker tracker and find some "mistakes" that you have been making. Basically you adapt, or if you don't you'll go broke. You experiment with some new plays and think about what you are doing wrong. "Wrongness" and "rightness" can be such fickle phrases. Maybe the previous opponents you won so much against come into some money and start playing again and when you go back to the tables and employ your old strategy you win. Maybe you adapt, get better and win. If you do the process starts all over again. If you don’t you go broke.
Over time you seem to go through some rather large deviations and changes that your minister never predicted. One time you even have to switch game types, as no one seems to want to play that old game anymore. Everyone knows how to play it, and there’s a much more exciting game now.

What your minister didn't tell and what is the truth, is that you are involved in an evolving dynamic system. One in which past success does not guarantee future success, like effects may not produce like causes, and you can not make accurate long term predictions. The idea that your risks can be managed by, and that the area of luck you must be most concerned about is bell curve card deviations is a delusion. You must continually adapt your strategies to a changing environment and there can be a great amount of ecological luck and variance in your results.

By LA Price Dec. 5 2007

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