Probability model for AP cheating

[jason1990]

This post is addressed to any of the high-stakes players that are convinced cheating took place on AP. Would any of you be interested in trying to transform your thought process which leads from the HHs to your conclusion into some kind of probability model? I am a professional mathematician whose research area is probability theory. I would be willing to work with you on trying to do this.

I have my doubts about the feasibility of this idea, since there are so many assumptions to be made, but I think it is worth at least an initial attempt. See my post here for my initial comments on this subject.

If we focus for the moment on POTRIPPER, then what we need for this model are three things: (1) p_0 = the probability a randomly selected player from among all online players can see hole cards, (2) p_f = the probability that the HHs we have for POTRIPPER would look the way they do if POTRIPPER could not see hole cards, and (3) p_c = the probability that the HHs we have for POTRIPPER would look the way they do if POTRIPPER could see hole cards. According to Bayes Theorem, if these 3 quantities are given, then the probability that POTRIPPER could see hole cards, given the HHs, would be

p_0*p_c/[p_0*p_c + (1 - p_0)*p_f].

What I need from you, if you want to do this, is an estimate of p_f and p_c. That is, I would need you to walk me through the HHs -- once under the assumption there is no cheating and once under the assumption there is cheating -- and estimate all the probabilities of seeing the various actions that POTRIPPER took on the various streets.

This is the part where many assumptions would have to be made. In the end, whatever numbers we come up with will just be the opinions of the high-stakes players in this forum. But at least those opinions will be formulated numerically (possibly with some stated margin of uncertainty). If the opinions are expressed numerically, then someone who wants to raise an objection can be challenged to come up with their own numbers. They could not as easily get away with only addressing AP's lack of motive or appealing to the ubiquitousness of claims that online poker is rigged.

As for p_0, I have proposed to use p_0 = 1/(N + 2), where N is the total number of players who have ever played online poker. This is based on a well-established statistical model, so I think it is a reasonable place to start. We could, if you want, discuss other possibilities for p_0. But if we use my suggestion, then we would also need an estimate for N, and I would need your help coming up with that estimate.

If you would like to contact me privately, send email to anonymous.investigator@yahoo.com

[Analyst]

I'll have to think about this model but can give some guidance as to the number of players (N).

PokerStars announced their five millionth player in March 2006. Pre-UIGEA, Party was substantially larger; between the two sites it could have been around ten to twelve million (there would be a great deal of overlap, of course). Accounting for unique players at other sites, new signups, etc., off the top of my head I'd probably put today's number at 15-20,000,000.

Note that this includes both real and play money players. If you limit this to only people who have played for real money, the number is a lot smaller.

[jason1990]

Thanks for the input on N. Unfortunately, without p_f and p_c, it is not very useful. So far, no one has contacted me about quantifying their reasoning.

Let me expand a little on what is needed. Whatever we agree to use for p_0, it will undoubtedly be very small. So the probability of cheating, given the HHs, can be simplified to roughly

p_0*p_c/[p_0*p_c + p_f].

Dividing numerator and denominator by p_0*p_c gives

1/[1 + p_f/(p_0*p_c)].

So what we are trying to argue is that p_f/(p_0*p_c) is very small. What matters is not the exact value of this quantity, but rather its order of magnitude. If we take my suggestion for p_0 and Analyst's estimate for N, this says that p_0 is roughly 10^{-6} or 10^{-7}. Hence, we must argue that p_f/p_c is much smaller than 10^{-7}. Suppose we aim at showing it is of order 10^{-9}.

In other words, we want to argue that the likelihood of the HHs looking the way they do without cheating is a billion times smaller than the likelihood of them occurring with cheating.

Again, it is only orders of magnitude that matter. I am not asking anyone to give exact probabilities, only estimates up to order of magnitude. For example, what is the probability a non-cheater would make that 10 high call on the river? 1/10? 1/100? 1/1000? 1/10000? Or even smaller? Similarly, what is the probability that someone who could see hole cards would make that call? 1/10? 1/100? Etc... The answer to this question is not supposed to be some hard-core factual answer based on megabytes of statistical analysis. It is just supposed to be the opinion of the person answering it -- namely, the high-stakes player who believes cheating occurred. Your experience in playing these games is what gives credence to your answers. If you answer these questions for every street of every hand in the HHs, then it is certainly conceivable that the ratio p_f/p_c will come out to be smaller than 1 in a billion. Maybe much smaller.

Let me also say something about my motivations. What I see here is a number of people who are obviously not prone to making exaggerated claims about online poker. Yet here they are, completely convinced that cheating took place. Their arguments are being ignored by AP and ridiculed by other authors. I would like to see their claims given fair and full consideration, rather than being dismissed out of hand. I do not have the experience to judge their arguments for myself. But I do have the skills to help formulate their arguments in the language of probability, or at least try to do so. It is my hope that, if successful, this would increase the chances that their arguments are taken seriously.

Also, to anyone who wants me to work with them on this, we can do the analysis in private by email if you like. In the end, if you are not happy with the analysis -- if you feel it does not fairly represent your argument and your opinions on the matter -- then we will discard it completely. I am not interested in "tricking" anyone into making claims they do not intend to make. I simply want to help you, if you are interested, to translate your arguments in support of cheating into the language of probability.

[Abbaddabba]

i think this is kind of pointless.

people who dont know poker well enough to just look at the series of hand histories and KNOW what they entail will find this no more compelling than a nub-tard blurting out percentages of how unlikely it is for his aces to be beat 5 times in a row.


i think that in order to get this [censored] sorted out, we need a personal connection to a higher up at AP.

[jason1990]

[ QUOTE ]
people who dont know poker well enough to just look at the series of hand histories and KNOW what they entail will find this no more compelling than a nub-tard blurting out percentages of how unlikely it is for his aces to be beat 5 times in a row.

[/ QUOTE ]
I disagree. A "nub-tard" is very different from a professional mathematician working with a team of veteran poker players. Even people who do not know poker at all realize this.

[furyion21]

First, by no means am I a high-stakes player involved in this whole debacle. I'm just a player who's read the various posts about the AP deal and reviewed some of the HHs given as proof of cheating.

That said, any reasonable poker player (I think) can give you approximated quantities for your variables.

a. p_0 is not 100% accurate in assessing p(cheating) because higher-stakes players have more to gain in terms of monetary reward; thus, the incentive to cheat among immoral individuals is higher than that of a randomly selected poker player. I don't quite know what variable would replace it.

b. p_f is approaching zero given some HHs, and is quite high given other HHs.
e.g. p_f approaches 0 given highly improbable calldowns with T high on a scary, scary board.
p_f approaches 0 given highly improbable actions of raise/fold on the river; never calls (calling down is fairly standard on the river, so calling would be closer to a higher p_f than either raising/folding.)

The difficulty is taking every single HH since the AP patch and averaging the p_f given all of the HHs (is this even correct?). Some plays this player makes are normal-ish , assuming he is making reads (correct or incorrect) and acting on them. Sometimes HU limit play can appear to be a bit donkish/crazy, when there is a sound logic behind them.

c. Conversely, I assume p_c is 1 - p_f.

Regardless of the actual p(cheating) number generated by this application of Bayes Theorem, as we all know, events with a probability of approximately 0 occur more often than we would estimate. Even an infinitesimally small % would leave more than enough room for it to be very feasible that 1 player out of a large population could be cheating.