99.99890581% chance that cheaters were operating on Absolute?

[Josem]

Dear Probability Gurus,

I'm just a lowly internet gambling/stt/bbv poster. i have no qualifications in statistics, so could someone please check my reasoning at http://forumserver.twoplustwo.com/sh...umber=12202343 and http://forumserver.twoplustwo.com/sh...umber=12202371 ?

Essentially, it is my contention that there is a logically provable 99.99890581% chance that players on Absolute Poker were able to see the cards of others. Am I right? Am I wrong?

I've also reproduced my posts below:

[ QUOTE ]
Assumption: cheater plays with 90% VPIP.

In the 25 hands analysed by Adanthar (the first ones listed in the cliff notes) the cheater folds whenever another player - and only when another player - has AA, KK, QQ, JJ. There is another hand analysed, but it involves a re-raise squeeze, which is a slightly wider range, and we can ignore it for this analysis.


So - what are the chances that a player who plays 90% of hands will randomly fold the 4 precise hands that someone else has AA, KK, QQ, or JJ, and none others?

I think the mathematics can be written out like this:

.1^4 * .9^21

ie, it is .1 * .1 * .1 * .1 * .9* .9* .9* .9* .9* .9* .9 etc.
^--these are the four premium hands ^--these are the rest

chucking that into excel provides the following answer:

0.00109419% chance of occuring.

in other words, a 1 in 100,000 chance of occuring randomly.

hopefully this methodology is right. can someone who is smarter than me (ie, almost anyone) confirm or correct me?

does this then mean that we can say with 99.99890581% certainty that the cheater was cheating in these hands? the more i think about it, i think not, but i feel that there is a calculation somewhere here that would give us the probability that this was cheating - can someone who knows mathematics/statistics well chime in?

[i]incidentally, the figures vary a bit depending on what the cheater's VPIP is. if the VPIP is .95, then there is a 0.00021285% and a 99.99978715% chance respectively. if the VPIP is .8, 0.00147574% and 99.99852426%)

[/ QUOTE ]

[ QUOTE ]
in answer to my own question in bold, i think the following analogy is fair and reasonable. i've been thinking about this, and I am increasingly confident that if the cheater had a VPIP of 90%, then there is a 99.99890581% chance that he is cheating.


Let's say, hypothetically speaking, someone had a secret way to win the lottery (a 1 in 100,000 chance in this particular lottery). You say to this person, "prove it."

So, this person goes away, picks their numbers, and wins the 1 in 100,000 lottery.

Thus, either they won the lottery randomly (ie, it really was a 1 in 100,000 chance) or they cheated. 99.99890581% of the time they will have cheated.


I think the same thing applies here - because the cheater was accused before the data became available.

Obviously, it is not reasonable to accuse someone of winning the lottery of being a cheater after they have won the suspicious - at that time, the lottery win is in the past, and thus has a 100% chance of having occurred (it already did).

However, because the cheater was accused before the data became available - and we then tested the data on our existing hypothesis, I'm now confident, with a 99.99890581% certainty, that the accused cheater was actually cheating

[/ QUOTE ]

[DrVanNostrin]

It's not fair to only analize 25 hands because you've observed him for more than 25 hands. It is much more likely than you claim that would have a streak of 25 hands where he played very well (FTOP terms) preflop since he's played more than 25 hands.

Also you're assuming that 90% is his true VPIP. 90% is a statistic in this case, not a parameter.

edit: you also should look at whether he was the SB or BB when he folded and how his VPIP differs in the SB vs. the BB.

[Josem]

good point.

i see if anyone has HHs of playing vs the cheater HU so we can see the opponent's hole cards on every hand for an extended session.

[Siegmund]

More information needed - but my first take is that the chance is a LOT smaller than 99.998% that this is cheating.

If (for instance) a player has a strategy of limping every hand and folding to any big raise - it just so happens there are a lot of people who make big raises very frequently with the top pocket pairs (and perhaps with AK), but not with lesser holdings. Against such people, yeah, he's going to fold whenever he's up against a big pocket pair and going to play the rest of the time.

Now, if you could show this smarty always folding to big raises from pocket pairs, AND always calling equally-sized raises when the same raiser has a big ace or a bluff, that might be evidence.

[knappis]

Here is my 2 cents.

If villain folds 4 hands out of 25, what are the odds that he is doing it exactly the 4 times somebody had JJ+?

4/25)*(3/24)*(2/23)*(1/22) = 1:12650

If we also factor in the likelihood that folding 4 hands of 25 is part of villains distribution of bet/fold patterns we would know the odds of this to happen by chance. The problem is that we don't kow his distribution of bet/fold patterns but it is likely pretty high (hes at approx 80-90% vpip), maybe 33%ish. That would give odds in the 50k range indicating that similar "coincidences" probably happens every day in online poker.

If there are other cues that would give villain a hint when he is upp against a big pair like Siegmund suggested the odds will fall drastically ofc.

[AaronBrown]

In addition to the factors already mentioned, there is another problem with this argument. A cheater is not going to do most of his folding against top pairs.

Suppose he holds something like Q2. If someone holds Q with a higher kicker, he's in bad shape. There are 132 hands of Q with higher kicker, versus only 21 top pairs. Also, seeing a Q in someone's hand drastically reduces his chances of winning against other hands, even of the Q folds. Most important, if he's playing Q2 against Q7, it's virtually impossible for him to win when the other player has a good hand; the only time you make enough money to consider coming in with a disadvantage. Q2 against AA can win big if Q2 shows up on the board.

While a cheater may fold to a top pair, most of the time he folds, it will be to something other than a top pair.

[Josem]

1) the cheaters seem to have little knowledge of poker, and don't understand the point about kickers

2) we're particularly relating it to a tournament, where the implied odds don't exist for Q2 to be played against AA.

3) a large chunk of the villain's hand equity (thanks to viewing his opponent's hands) is from forcing people to fold hands like AJ on a Q95 board (for example). JJ+ have a lot fewer boards to do this with.

[AaronBrown]

(1) I suppose that's possible, but it seems like a stretch to assume someone went through the trouble of cheating, but didn't bother to learn much poker; even to the point of having an odds calculator running. I don't say it's unlikely, it's just that it's not my first assumption of how people cheat.

(2) I don't say you should play Q2 against AA, just that most of the hands you shouldn't play Q2 against aren't top pairs.

(3) It's true you'll have trouble getting a top pair to fold on a Q95 board, but you'll have even more trouble getting Q with a higher kicker to fold. At least playing against JJ+, you have a chance of getting a good hand and getting called. But the main point is the same as (2); you don't want to play top pairs, but you also don't want to play the much more numerous hands that share cards with you and are stronger.

[jason1990]

As DrVanNostrin said, you are generating a misleading analysis by not utilizing all of your observations. You must not only use all the hands you observed, but also all aspects of those hands (not just the preflop anomaly). I think you want to do that anyway. The preflop play is not what convinced the high-stakes players that cheating occurred. It was a combination of all the evidence that convinced them. I do not think they would have been convinced on the basis of this preflop play only. For that reason, your analysis does not accurately capture their reasoning process.

I also do not think you can avoid invoking some kind of prior probability. Before this scandal erupted, POTRIPPER was just another random online poker player. At that point (when there was no evidence) what was the probability he could see hole cards? I think I saw in the main thread that you were disputing the need for such a prior probability. But to illustrate its importance, suppose it was 0%. That is, imagine that somehow we had absolute certainty through whatever means that it is 100% impossible for anyone ever to see hole cards in online poker. In that case, the only explanation for the HHs that were presented is that POTRIPPER is stupid and lucky.

There is a fairly common statistical method for getting a prior probability like this. Put yourself back in time before the scandal erupted. At that point, we had witnessed N people log in and play online poker since online poker was invented. (I do not know what N is, but maybe you can find some estimate of it.) Assume that none of them have ever been able to see hole cards. What is the probability that the next new online player we see will be able to view hole cards? A standard model says that the probability is 1/(N+2). The model I am referring to is described here. Note that 1/(N+2), under this model, is the probability that POTRIPPER can see hole cards, calculated only on the basis that he is a random online poker player, without factoring in any evidence. The next step would be to factor in (all) the evidence in order to increase this probability.

There are two serious problems I see, which make me think that a statistical analysis might be a bit useless. First, I think it may be an intractable problem to incorporate ALL the evidence. Many assumptions would have to be made, and you run the risk of either making bad assumptions, or tailoring your assumptions to specifically generate the conclusion you want. Second, no matter what you do, someone will be able to question your methodology, your assumptions, your prior probabilities, etc. You will not be able to create a "bulletproof" analysis based only on statistics, which I assume is what you want.

From the main thread (link):

[ QUOTE ]
I'll be honest, I haven't read every post in this newest thread, but what i've seen there hasn't been an attempt to unify the data. People are getting mathematically longwinded here with calculations that don't factor in EVERY PT stat. It takes an understanding of poker, not statistics, to grasp this. Anybody that knows pokertracker well can see something isn't right with even a cursory glance at the data.

[/ QUOTE ]
Many of the high-stakes players are convinced beyond a reasonable doubt that cheating occurred. Their thought process which leads from the HHs to the conclusion of cheating is an intuitive one. I think the best you can hope to do with statistics is to (try to) model this intuitive thought process with a Bayesian analysis. I repeat that I think it may be intractable to do this, but if it were possible, and if it were done honestly, then it would capture the true arguments of these particular high-stakes players. Anyone who disputed the model would be directly contesting those arguments. Such an approach would probably be the one least likely to derail the discussion. However, I really think it would be very difficult if not impossible to do this.

[Josem]

thanks - that makes a lot of sense, and was explained really, really well.

the more i learn abotu this, the more i realise that there is no way to mathematically "prove" that there was cheating here in any way that would be accessible to someone like me.