OK, thanks, forgot about that. That's the same as looking at their stats Vs other shortstackers, which someone suggested earlier. Pretty clear really, my bad.

However - now this is scraping the barrel, but I guess we should be thorough in our defence: Could we define two types of shorties - A) old-style 1/2 street and B) uber postflop skilled shortie. Could it be correct to play a different way against type A and type B?

Furthermore, could some type of implicit collusion be correct? E.g, if a new player came along, and they identified him as independently playing "their strategy" would they then adapt by e.g. 3-betting less? I understand that we are now getting into something of a "grey area", but from a quasi-legal standpoint, it raises interesting questions...

Owned.

why would stox do this? he mad ea lot o fmoney playing online before, and he has a published book, and his own trianing company.. so why risk so much to gain so little???

greed

the reverse conspiracy theory

nh wp sir

Is it possible to use the hand history data to determine how much total money kinetica and stoxtrader made while playing at the same tables compared to when stoxtrader in particular was by himself? Would this add to proof to support the accusation of deliberate collusion?

Sample is too small for such thing.

Even if the sample weren't too small, I still couldn't do it because of the long list of other factors that are correlated with who's at table with you.

The correct way to figure out what edge they gained is to figure out exactly how they cheated and to run simulations to figure out what edge that gave them. Since I've already proven that they cheated in at least one way and run simulations to show that they gained at least some edge, I figure I'll leave it to the sites who have hole cards and teams of professionals to figure out exactly how they cheated and exactly what edge they got from it and then to pay players back.

Actually, what you said is roughly the same as "the sample is small". If the sample was big, the effect of who's with you at the table will be eliminated.

I don't think this can be done, as modeling the strategy of everyone else on the table is impossible, and that would be needed to run such simulation.

Why should sites do anything else than standard disribution of money to all players that were at the table, or in anyway how they do it when ban someone. I don't think any "edge calculation" is meaningful.

BTW I just briefly read OP again: You work/display all the time with confidence interval 1 (68%) and just the last graph with 96% (std2), but even there there are already other guys falling in the space of stoxtrader?

Sorry, not nagging, but where exactly is there something that even gets remotely close to something like a proof? What we have is something like 90% or 95% that he cheated (expressed dummy, staticians don't jump on me).

That seems to me wrong (using "x" in both places) but no issue anyway, it's clear what you mean.

Hm, from that you can say confidently that he 3bet with 2% against him, but not that he played differently.

Disection of this other 4.4% player:
Here you again stress on the less relevant (IMO) fact, or at least you need to revise the expose of the whole thing.
You leave the impression of something dramatic - hinting the odds of not cheating is 1/3.5m, while it is only (the irrelevant fact) just repeating the above thing that you simply have enough hands of these two guys and the 3bet range is 2% with high confidence.

Now this is the first I would consider some evidence, comparing with the average, although I don't understand what means "weighted average". If you took #of hands into account and have few players with lots of hands and rest with small, that's a big skew. The guys with the most hands will determine the mean, and then std can easily be big.

And now getting back, to what you did not discuss:

Now if you do that:
If you conclude from your data and expose in this chapter that these two guys (LittleZen and knockstiff) colluded, then this other 4% 3bet guy colluded as well with 5% probability, which is not ignorable probability. OK - the hands played for this other guy are much less (I would have expected that you post the #of hands played, btw), but I don't know how to techically bring that into the picture and come to a better number.

Then you go to a discussion that there might have been a valid reason to play differently (by having most amount of hands on each other or knowing their play, etc), but for me the above is a weaker part (that needed to be improved first, I cna then read the rest) rather the theory for the possible different play.

In summary, again from my very fast reading of the first pages, based on the data I would certainly bet a lot of money that they colluded - but for me it isn't sufficient to persuade me it is a certain event.
To put it dumb, it looks to me that there is 5% chance they didn't colluded (In fact it's less, but I don't know how to techically get to the lesser number. And you certainly did not prove, even point to, a better/smaller number, or I don't get how you do that)?

Maybe another way to improve is somehow combine probabilities for the various behaviours (in the different types of actions).

Only think I want to point is none of the numbers like 16stds, 1/35 million etc. apply to the probability of cheating/not-cheating. Would be good to get to such number ...

Hey Indy,
Please nag away. If I messed something up, I want to know it.

I think I explained this in my OP, but it's obviously pretty long and hard to read, so no harm in getting me to explain it again.

I showed 1 SD confidence intervals because that's the standard for displaying this kind of data. You can just double the bars to get a roughly 96% confidence interval. As I acknowledged in a few spots, the normal approximation isn't very accurate for very small samples or very small %ages (with sample * 3-bet % < 5 or so). I originally included graphs of 68% and 96% convidence intervals in all the spots where some of the data fell into this extreme, but I decided to replace them with a quick parenthetical note because they didn't change anything and the post was long as hell already. I included it in that one spot because the normal approximation completely fails for a 0 value.

Anyway, my findings have way higher probability than 95%. You're looking at the error bars on the values in the control group, taking the lowest player out of 48 in the control group, and then saying "what would happen if this lowest player actually had a much lower value but just happened to hit a 5% shot to come out as high as it did?" Hopefully when I phrase it that way it's clear why this isn't the right way to look at things.

The right thing to do is to look at the error bars on the suspect accounts values and see if the fact that they're such huge outlier can be explained statistically. In the LittleZen and Kinetica case, the data was clear enough that I was able to just directly compare their results to the lowest value in the control group. For example:

So as far as I'm concerned that's totally conclusive that LittleZen played differently against knockstiff than anyone else. Note that this is actually a pretty generous way to look at things. In fact, LittleZen's data shows that he barely varied his 3-bet % against different opponents (if he did at all), so comparing to the average 3-bet might be more reasonable:

Kinetica data is basically the same with different numbers. In the knockstiff and 40putts examples, stuff was much murkier, but I think that what I just said addresses your concerns pretty well.

Edit: I wrote that between your second and third posts. I think most of the stuff you asked in the third post is answered above, but I'll answer a few things that I think aren't:

Yeah.. oops. Looking back through this now is really annoying for me because there are all these stupid typos I can't correct .

Yeah.. weighted means that I took the total number of 3-bets in the control group and divided out by the total number of opportunities to 3-bets, so players with higher samples counted more.

Neither method is perfect, but in the case where you have no extreme outliers with large samples, taking the weighted average is better. I don't think the numbers will be much different if I don't weight the average, but if you'd like me to do that I will.

Please let me know if you feel I haven't addressed anythign sufficiently.

BTW, I think I forgot to say that I believe that the cheating happened in non-CAP games on FTP as well (it actually looks like it was more extreme there, but obviously stack sizes make this more complicated). I also believe that the cheating occurred both before and after they were accused of cheating in March '09.

Noah, I'm still not getting it 100%.

So two things:
1) I see you got my point (for the 4% guy) - i.e. summarized it exactly what I meant, but why are you saying this isn't a correct way to look into that? What if I take take this lowest guy (the 4% one) and similarly conclude (looking at #std from mean) that he colluded with kinetika? That I can approach in two ways:
a) Without taking the 95% std, just comparing with the mean (using the 4% guy std)
b) If you take the 95% interval for the 4% guy (in which case the comparison to the average would have much higher deviation from the mean)
Would be good to give some numbers here (or maybe, you can send me the excel that produced this graph and I can try it) to at least get to a number X meaning "If LittleZen colluded based on comparison with the mean and his std, then the 4% guy colluded as well with probability X%" - or something in that direction). I just want to look into how much X is.

2) I don't get why due to the small std of LittleZen against kinetika that this implies that he played differently. I agree with your paragraph - it proves without doubt that LittleZen 3bet range is very close to 2%. It is a very small std - based both on the low mean, and on the big number of hands you have for these two guys, so indeed, the probability that his 3bet range is higher, just 2% higher, is ridicolously low. But what evidence is here to prove that he played differently, which means he didn't have the same range against other players too?

1)
a) Fair question. I didn't realize you were asking this earlier. The odds that the 4.4% guy came from the average 3-bet % of 8% are about 7/1000. So that's the sort of outlier that isn't too surprising in a sample of 48, but still you only get an outlier like that about 28% of the time (1- (1-7/1000)^48). If I allow that he was someone who LittleZen 3-bet a little less, say like 7% of the time, then the odds go up to about 3.5%.

b) This is what I was trying to explain in my previous post. Let's say you're looking at my first 3-bet % graph except without the knockstiff point present and you want to conclude how much LittleZen varies his 3-bet % by player against regulars. It would not be reasonable to say that there's a 5% chance that LittleZen's lowest 3-bet % vs. a regular is under the 95% confidence interval of the bottom point. This is because by singling out the bottom point in a sample size of 48, you are almost definitely choosing a data point that was skewed low by random chance. Indeed, as I showed above, if LittleZen's lowest 3-bet % against any player was 7%, it wouldn't be surprising at all to see the 4.4% guy.

(However, if LittleZen's lowest 3-bet % against any player was even 4.4%, which as I think I've explained above is almost certainly a low estimate, LittleZen's 3-bet % against knockstiff would be a one in 3 million shot.)

Does that make sense now?

2) I showed data on 48 other players that he 3-bet much more than this.

Indy,

I could be wrong, but I don't think that there is any way a professional poker player could ever be profitable with these ranges against all players.

indiana,

I'm not sure why you keep saying that it doesn't prove that they played differently against each other. The 2% is the percentage of the time that Littlezen 3 bet's Kinetica when he has the opportunity. The other numbers are the percentages of time that LittleZen 3 bet's other players when he has the opportunity. So he 3-bets 2% against Kinetica but much more against other players.... his 3-bet range is only 2% against Kinetica but much higher against all other players. This is him playing against Kinetica differently

good work, itt. Big thanks OP

Report the post with description of typos so they can fix it.

just post the data anyway to get things into perspective.

wanted to post to say that if a) he was in fact cheating (seems so) and b) he was smart about it (he is certainly a good player so i assume he was) that he had a HUGE edge.

I am a regular in the cap games and have been for years. I am not great but am a marginal winner at 5/10-25/50. I have logged thousands of hands with him. The edge in cap is so small and the game is so systematic that small edges are SO important. I can think of 5-10 different plays that would make a significant difference long term in WR.

To cheat in a cash game by multi accounting does not happen very often mostly because it is difficult imo. most stories we all hear about multi accounting is about our friends doing it in $3 SnGs and failing horribly. This leads to ppl thinking multi accounting in cash games barely works. I dont think this is true. if a person has a very solid foundation in poker, which there is no doubting stox did, think about the damage someone can do. If i were given two accounts today, i think i could immediately increase my winrate. and that is without getting the hang of it. think of the things you would learn after 1 million hands too. stox was also always a better player than me.

regardless of how much it increased, any amount is huge when someone is grinding 10/20-25/50 for hundreds of thousands of hands.

what im trying to say is I would not be surprised if he gained 100k-500k (probably towards the lower end) from cheating. it could very well be one of the bigger cheating scandals and should not be over looked

Noad, jchristo,

I'll try to explain with an example. It's visible he played differently, but you need to go into some sort of analysis like in point 1) above to prove it (or come to some cheating number).

Let me put an example, take a random player A and draw the same graph for him. If you draw such 3bet ranges graph there will always be a player B against which the 3bet range of A is the lowest. So in general, you cant just draw such graph, take the lowest point, and say - see, it's very different of how he played against the rest.

Honestly in this case for these both players [kinetika] isn't just the lowest point that happened to be there - they know each other and so on (which is yet another indication on top of the already very very strong evidence against stox). I just wanted to find a number that represents this probability.

I agree, you're right. 2% 3bet range is completely unusual and I also think it can't be profitable.

Last point, I think stox said in his last statement that he never colluded with the accounts that he created and used, but he never addressed issues outside of this. I think at this point if stox doesn't say anything about this, and come with some explanation then everyone should accept that he colluded. If he just briefly tells his strategy and tries to explains what is seen, it would be easier to match the expected with the observed numbers, and in case he tries to come with a fake explanation/strategy it would be easy to bust it.

At this point it's pretty clear that there was a collusion, and my feeling is that in the best case (for stox) it could have been some sort of tacid collusion, that is:
a) Both players knew each other strategy very well - for whatever reason - were close friends, roomates, studied it via HH, etc.
b) They never made an an agreement to play like this against each other, but somehow indepenantly came to the conclusion that this is the best way to play knowing how the other guy plays (rather "expecting" as if I use "knowing" that would imply reliable thing which is then a collusion).
c) Even in such case there would be quite some doubt, as these two guys played more hands with each other than normally, which (again in the best case) would be a 2nd level of tacid collusion.

actually fees comment was about as ignorant as it gets.

indiana,

I think you need to reread carefully. Noah addressed these points. He even took Kinetica out of the equation and analyzed the 2nd lowest 3-bet%. I am not going to go back and find the numbers for you but the conclusion was that the 2.2% 3-bet% was a one in 3.5 millionish statistical anomaly compared to his mean 3-bet%... implying that he very certainly was playing considerably differently against Kinetica. Conversely the 2nd lowest 3-bet % was 4.4%. With the sample size for that player, it was a 1 in 40 ish (or something like that) statistical probability that he would get that 3 bet % despite his mean.

So in a 36 person sample, the lowest 3-bet % (kinetica) is a one in 3 million + anomaly and the 2nd lowest is a 1 in 40 ish anomaly. Obviously if we removed kinetica from the equation, then we wouldn't think the player with a 4.4% 3-bet percentage cheated because he still falls well within a normal distribution. It's not just because kinetica is the lowest #, its because he is so extremely far from the mean, and over a large sample of hands, that he is almost DEFINITELY softplaying/colluding with him.

Noah thanks for all the good work.

Out of curiosity what were their winrates in games they played together vs games they were in alone.