I would like to present a new summary of the data which expands on my previous analysis and addresses several concerns and suggestions discussed ITT.

As before, the analysis was done on the suspects dooshcom, benkaremail, liverfc123 and benhamcheese. The sample size on the latter two is very small, so the statistical case applies primarily to dooshcom and benkaremail. The p-values don't change much if we exclude liverfc123 and benhamcheese from the Team.

Summary of Revisions

1. The range of 'premium' hands was broadened to determine if we still see the same pattern as before.
New Range: (44+,A2s+,K9s+,QTs+,A5o+,KTo+); top 21%
- These hands have >50% equity vs the top 60% per Pokerstove.

2. Computed 95% confidence intervals for PFR* and other metrics.

3. Computed Ratio (PFR*_vs_team) / (PFR*_vs_others) and corresponding 95% CIs. This metric quantifies the magnitude of the observed effect. (i)

4. Pooled the data from the various positions and computed overall p-values and CIs. (ii)

5. Compared the suspects' frequencies when one of them has a good hand to their overall frequencies vs each other.

Intuition for PFR*

We have discussed how filtering for shown hands biases the stats such as PFR. So what exactly is PFR*?

Let's take Opening Button PFR* vs BB as an example. We know that the BB's strong hand went to showdown since we filtered for that. There are two possibilities in this case:

(a) BU opened and ran into the BB's strong hand, or
(b) BU folded and the SB ran into the BB's strong hand

thus, PFR* represents the proportion where the Button ran into the strong hand, and (1-PFR*) would be the proportion where the SB ran into it instead. So the way to think about these proportions is approximately, "% Ran Into Strong Hand ".

Results

The chart below shows the suspect's Button Opening PFR* when the SB or BB has a strong hand. In both cases, their PFR* when another Team member holds the strong hand in the SB/BB is less than half of that when a non-Team member holds the strong hand. Both results are statistically significant at the p < 0.0001 level.

Figure 1 Button Opening PFR* when BB/SB has a Strong Hand


Thus, irrespective of whether their buddy was in the SB or BB, the Button ran into the buddy's strong hand only 27% (95% CI: [0.20, 0.36]) of the time, meaning the other player in the blinds ran into the strong hand ~73% of the time. When non-Team players held the strong hand, the Team ran into it > 50%, which is normal.

Additional details, including p-values, confidence intervals, effective stack sizes, and CO figures can be found at the tables below. The CO figures are very similar to those presented above, PFR* is ~0.10 vs Team compared to 0.26 vs Others. The sample is smaller though, p-values are 0.06 vs BB and 0.006 vs SB.

Looking at the pooled data on Table 1, the average effective stack sizes are very similar: Team vs Team is 11.3; Team vs Others 10.8. The same is true for Table 1b. Therefore, stack size is unlikely to be a significant confounding factor in the analysis.

Table 1 Pooled BU/CO Opening PFR* vs top 21% hand in SB/BB.



Table 1b Same Data as Above Split by Position (link)

If we aggregate the data from the different positions, the pooled ratio (PFR* vs Team) / (PFR* vs Others) is 0.46 [0.38, 0.55]. This value quantifies the magnitude of the observed difference.


Were the Team Members just opening tight vs each other in general?

A concern was raised to the analysis above : what if for whatever reason they were opening less against each other in general?

While I follow this line of reasoning, it is not a possible explanation for the data above. The games analyzed were mostly 3-4 handed with 2 Team members present. A Team member was in the blinds the vast majority of the time, so if they were to open tight when a buddy was in the BB and also when he was in the SB, they would just be opening tight, period.

Nevertheless, the rows highlighted in yellow in Table 2 below compare their overall opening frequencies vs Team and vs others. From the button, they actually opened more often vs Team's BB (0.50 to 0.42), but less vs Team's SB (0.43 to 0.48). From the CO they opened about the same vs Team as vs others.

The first row in Table 2 is actually PFR* when a buddy had a strong hand, taken from Table 1b above. When filtered for strong hands that saw a showdown, PFR* can be used as a conservative estimate for true PFR (iii). The values for PFR* when Team had a strong hand are still significantly lower than the overall Team vs Team PFR. Card removal effects should only account for ~0.02 of the difference (iv).

Table 2 Team Opening Frequencies vs Team and vs Others



Summary

Expanding the range of strong hands from ~9% to 21% yielded similar results. When one of the suspects had a strong hand in the blinds, the other opened 46% [0.38, 0.55] as often as he did when a non-suspect had the strong hand. This resulted in the suspects running into each other's strong hand in the blinds with only 22% [0.19, 0.26] frequency, while they ran into a non-suspect's strong hand 48% [0.45, 0.52] of the time. The result is highly statistically significant, p<0.0001.

We also investigated the possibility that the suspects were just opening tight vs each other in general, and found that to be false. This leaves only one possible explanation for the data: the suspects had information on each other's hole cards and acted on it.




(i) P-values use two-tailed Fisher's exact test when possible, otherwise Pearson's Chi Square. See this webpage for an online calculator and other details.

(ii) Getting a p-value and CIs for the aggregate data is more complicated. I used a common method, the Cochran-Mantel-Haenszel test, described here and here. Other methods give similar results. For an interesting technical discussion, see this paper: Sex Bias in Graduate Admissions: Data from Berkeley.
http://www.sciencemag.org/content/187/4175/398.abstract

(iii) Since it's filtered for hands that saw a showdown, PFR* doesn't count hands like when, for example, the BB has a strong hand but the BU and SB both fold preflop. Not counting those BU folds results in PFR* > True PFR.

(iv) Simulation result. A top 45% range becomes ~43% given that another player holds a hand in the top 21% range.

Overall VPIP vs Opener

Table A1 shows overall VPIP from the SB or BU vs an opener. This is the same data I presented earlier, with 95% CIs and ratios added. It also includes the aggregated data from the various positions.

The SB VPIP vs BU open is only 10% [0.08, 0.13] vs Team, and 25% [0.22, 0.28] vs others. The pooled ratio (VPIP vs Team)/(VPIP vs Others) is 0.43 [0.34, 0.55].

Table A1 Overall VPIP vs Opener


This is a case where it looks like the Team was playing tight vs each other's opens in general. Now let's take a closer look. Were they really only playing the top 10% of hands vs each other's opens?

No. The chart below are their shown hands in the SB vs a Team Member's BU open. The SB showed 41 hands out of 484 BU opens. WTSD was 93% for this data, so when they played a hand it went to SD the vast majority of the time.

Figure A1 Team SB range vs Team BU Open


That range looks closer to 25%, except for several missing hands near the top of the range. This is consistent with our previous result- what is happening is that the BU doesn't raise very often when the SB has a strong hand and that is why their VPIP after Team BU opens is only 10%.


Team vs Team Button Opening Range

Figures A2 and A3 below show all of the Team's shown hands after opening the Button when a teammate was in the BB or the SB. They support the previous conclusion that the suspects were not opening tight vs each other in general.

Figure A2 Team BU open vs Team's BB


Figure A3 Team BU open vs Team's SB


Compare the above range to their range when the teammate in the BB/SB had a strong hand:

Figure A4 Team BU open vs Team's (44+,A2s+,K9s+,QTs+,A5o+,KTo+) in BB


Figure A5 Team BU open vs Team's (44+,A2s+,K9s+,QTs+,A5o+,KTo+) in SB


I hope the above charts give you a better qualitative understanding of their ranges.

Lastly, a note to 888: I would be more than happy to share my database queries, so that all you need to do is load a bunch of hands in Holdem Manager with known hole cards, run the query, and get the Team's exact opening frequencies under various conditions.

Thanks for putting so much effort in this, the community needs people like you.

Brazzmonkey

+1 to that.

It would be a shame if nothing happened after you put that much work and effort into it.
I mean it is basically face up what's happening here and the responsibles at 888 just HAVE to investigate this further (easy to do for them with complete holecard information) if they want to keep their games integer. (or at least look like they give a ****)

It's a big ****up that some players have to do the work for the poker sites in cases like this. You still want your customers to trust you, right ?

tamiller866,

You missunderstood my post. Thsi is why people who don't read the book and just read the Cliff notes fail the test. Especially when the Cliff notes misinterpret what I said. Perhaps it is my fault, because I wanted to have a bit of fun with the example he used of Medievel time justice. So here is what I tried to convey:

1) if Dooshcom is innocent, he clearly should welcome an investigation (this was missing from your Cliff notes but was the first thing I said in my post)
2) There may be some people that might feel that an investigation by the site might be a white-wash due to his relationship with the site
3) There are some incredibly book smart people that are incredibly "street stupid" by risking huge sums of money on-line without checking out the intergrity of the games BEFORE they play. Then, when they feel they are scammed they do their due diligence after the fact. This make no sense to me. After all of the on-line scandals that have been made public (not to mention the ones that haven't ), why would anyone not be hesitant and cautious before playing on-line? It's the similar to those parents that brought their kids to Michael Jackson's Neverland Ranch, even after all of those previous allegations of molestation that were made against him. Even if he was never proven guilty, would you bring your kids there? Even live games should be checked out before you sit down to play. But to send tens of thousands of dollars to an offshore on-line site located in an Island to which you've never been, and to not meet with the owners and/or operators of the site and do your due diligence to make sure the integrity of the games are sound is utterly stupid.

On another note, I would like to add that the detailed analysis contained in in this thread is amazing. I NEVER said that if 888 doesn't find him guilty he is innocent (see item #2 above or what I wrote in the original post - I actually said something totally different).

If you feel this guy is doing what is alleged, and if the site does nothing, then don't give your business to that site.

Be Well.

Would you trust 888's response on this issue, if they respond, given that they haven't said a word for weeks now?

It's my feeling that when companies operate their businesses this way, they're generally incompetent in many areas.

Short of them providing all the hole card information along with a conclusion they reach, it wouldn't carry any weight with me.

When would be an appropriate time to cross link this to their site's thread, I haven't checked whether this has already been done. At the very least it could spur them to come out with SOME sort of response. Saying nothing is about the nut worst option imo.

Jaytorr's recent post has made it very difficult to have a "reasonable" doubt about the cheating. Most certainly on a preponderance of the evidence standard (and in consideration of Doosh's replies and non-replies) he is guilty of collusion.

888, the ball is in your court as to deciding what you value more.

Just as a sidebar, it is for that reason that if the US ever legalized and regulated online poker, sites like MGMpoker.com or CaesarsPalacepoker.com that were connected to actual regulated B&M casinos would dominate the entire market. It really should be only a matter of time.

p.s.- A note from those that play on 888 to 888 that they will not play on the site until this matter is resolved and details equivalent to what Jaytorr put out are provided (either way, I should add) might be in order.

This, great post that I think addressed pretty much all that needed addressing and still essentially proved that either doosh is one of the unluckiest people on the planet or else he is a cheater.

It was clear that Darren was cheating already after the first post Jaytor did. I still find it it odd that 888 does not say anything. What kind of shady company is just being quiet in a matter like this? Time to link this thread to the official site thread.

I don't disagree, but perhaps with enough people complaining, it could be arranged for a private third party to review all relevant hhs.

Curious to know when & what 888 comes with a reply.

Do people really find it odd that 888 is:
-Likely doing nothing
-Likely incapable of doing any type of reasonable analysis
-Likely completely indifferent towards this thread and cheating in general

really?

With the exception of Poker Stars, it seems to be the industry standard.

Poker companies do not really hire or have any need for poker experts so it is not really surprising since they are business people and not poker players.

They should however have strong programmers who should be able to build-in collusion flagging measures into their back-end however I am not sure how common this really is in practice.

Nice work, jaytorr. I'll give it another read or two later to make sure, but I'm convinced.

Thanks Noah, our chat a couple of weeks ago motivated me to learn more about the stats and improve the analysis.


On the subject of 888, it wouldn't have been easy to catch the collusion in the first place just by watching the games. It's not like they were folding 88 when their partner had 99; they were careful about opening Ax and pocket pairs on the Button. The collusion involved folding hands like K9o, T9s, etc. To a non-expert that might not look too suspicious.

Let's hope that 888 is just dotting their i's and crossing their t's before finalizing the investigation.

Like I mentioned earlier in the thread; I do have some formal training in statistics and econometrics. I have a PhD in applied econometrics and currently work at a European central bank using statistical software daily. I've corresponded with Jaytorr and after looking at this more thoroughly I'm convinced by his methodology. Obviously I can't go as far as to confirm his results since I haven't looked at the data myself. But it looks like he's been asking the right questions and employed a proper methodolgy to answer them.

Great work, Jaytorr!

Darren Woods The Cheater approves your post, jaytorr.



(Just posting an image so people who google him can be sure it's proper Darren Woods)

What a dumbass since this guy is a huge winner. Lol

Hi all,

The investigations into the allegations of collusion on our poker platform are still underway. As the allegations concern VIP players, there is a huge amount of hands to review, and an in depth expert review is required before any conclusion can be reached. The review also needs to compare the play of these members to other players of a similar stature, on similar high stakes tables. Further to this, we are extending our investigation to include third party reviews by independent expert sources. Only once we have completed our investigations internally, and once our findings have been corroborated by an independent impartial source, will we be able to release our conclusion. This in depth inquiry is testament to our deep commitment to detect and root out any suspected cases of collusion, including by our VIP players. Once concluded, we will take all necessary steps as appropriate in coordination with the authorities. In the interim, we have suspended all players allegedly involved from activity on our sites.

I am pleased to read this. The only way online poker will gain any credibility is by demonstrating that it is deserving of it. Identifying and punishing cheaters ASAP is a good start (and then doing everything the opposite of what FTilt did is probably a good second step....)

Hi Gareth,

Thanks for the response.

The main question about investigation is - when do you expect it to be finished?

We are waiting for a month now and nothing happened.

I think it's a reasonable time to get at least the preliminary results.

And if you are looking for experts you wont find people better suited for the task than Jaytorr and NoahSD.

I hope you and the investigation team at least read the big well explained analysis made by Jaytorr.

While this is a good thing the fact that it took what happened in this thread to convince them just to even launch an investigation is a shame. With access to hole cards it should be extremely easy for sites to have built in detections that would be able to catch collusion long before it even dawned on the players in question to do this study.