TL;DR:



Pictured above: a graph (and a zoomed-in view) plotting yearly volume on the x-axis, and average winrate of the players putting in that volume, on the y-axis. The standard variation of these winrates is around 1bb/100.

INTRODUCTION

In a series of threads starting with this one, I will analyze the current situation of PLO rake using statistical tools. All of this analysis is based on hand histories bought by gui166 (the Brazilian player rep in the recent Pokerstars meeting, from this thread), who was kind enough to share these HHs with me after I helped him in his analysis. These are essentially all the hands played between March 2012 and March 2012 in PLO100 on Pokerstars (a total of around ten million hands).

As we all know, the sheer variance of PLO makes it impossible to figure out the true winrate of individual players without gigantic hand samples. Therefore, we need to use statistical tools to figure out how true winrates look, to know how many players are beating the games and how significant is the rake.

In this first part I present analysis of the entire player pool, partitioned according to yearly volume. This allows me to get estimates of winrates from samples large enough to have already converged, since I'm pooling dozens or even hundreds of players with similar yearly volume together. On the flip side, this analysis tells us limited information: Suppose we know that the population of players who played between 40k and 50k hands of PLO100 have an average winnings of -0.3bb/100 pre-rakeback and 1.7bb/100 post-rakeback. What does this tell us? This population could include bad players. We don't expect all players to be winning, or even all regs to be winning: as a community, we only demand that a sizable chunk of the best players would be winning. So, keep in mind that the results here are population-wide averages, and do not tell us how many players have decent winrates.

In future installments in this series (hopefully less than a week apart from one another) I will show more sophisticated statistical analysis which will hopefully give us more idea about the actual winrate distribution of the player population. Here is a tentative plan:
Part II: cross-validation analysis of the player pool, or some other method based on partitioning the hands into chunks
Part III: Using de-convolution to get a guess for the true winrate distribution

Additional parts might come later. If this analysis proves useful, my overall future plans are to:
1. gain a good understanding of the rake paid by regs at the PLO100 level.
2. attempt to model and understand the PLO ecosystem (mathematically or otherwise) and the effect of rake on it
3. reach a consensus with the community of how much rake is "fair", both in the sense of fairly rewarding skilled players and keeping the ecosystem healthy.
4. extend this analysis to different stakes as well as different games (such as NLH or LHE), perhaps by initiating a cooperation with some of the hand-tracking websites, or by raising money from the community to buy hand histories.

I would like to solicit ideas from the 2p2 community about the content and future plans in this thread. I will be happy to do any data crunching that community members are interested in (limited by my programming time, of course), so if you think of interesting ways to analyze the data, please do tell. I have very little idea on how to approach the task of modeling the poker ecosystem (item #2 above) so I would encourage anyone who is interested in helping to start thinking about this.

I will make my source code available throughout the project (I'm a very sloppy python programmer so I can't promise the code is particularly legible).

THE ANALYSIS -- PART I

In this part, I will analyze the winrate of groups of players. So, for example, I'll look at all players who played between 10k and 20k hands, and will compute their overall winrate, as if they were one player.. They played so many hands as a group that the winrate we get is very close to the "true" winrate. Therefore, from this data you'll be able to understand what your winrate is likely to be if you are an "average" player who plans to play a particular number of hands in a single year.

To obtain these groups of players I have sorted the whole player pool according to annual volume, and started bunching together players with similar volume, until getting groups who played three million hands overall.

This analysis has various drawbacks, not least of which is self-selection: the players who played 100k PLO100 hands are the players who didn't go bust in the first 10k hands so they have some skill; on the other hand, they are the players who didn't move up to PLO200 so they are probably, on average, not the best players in this limit. There are more drawbacks, of course, which we can discuss further in the thread.

Regarding Rakeback: Since I don't have the VIP status of the players, I have computed rakeback a little arbitrarily: I assumed that play in the PLO100 level accounted for half of each player's rake for the whole year, and that the player has paid an equal amount of rake during each month. I assumed players are spending their FPP's in an efficient way. I also assumed that there are no bronzestars: I promoted all players automatically to silverstar. I assumed that each player who made supernova was also supernova in the beginning of the year. There are other drawbacks to my rakeback analysis: for example, my rake statistic is calculated on won pots (I think) while rakeback is computed based on weighted contributed; this probably won't make much of a difference, but it probably gives a little less rakeback to winning players and a bit more to losing players than in my calculation.

My (messy) code can be viewed here:
http://www.evernote.com/shard/s224/s...91d7e4a03a77d2

THE RESULTS

The results are as follows:

Annual Volume	#players	Hands	Won (bb/100)	Won+Rakeback	Rakefree Winnings
1 -- 323	38k	3.0M	-68.4	-65.5	-49.9
323 -- 779	6k	3.0M	-33.9	-31.1	-15.8
779 -- 1384	3k	3.0M	-26.4	-23.8	-9.3
1385 -- 2257	1704	3.0M	-21.5	-19.0	-5.3
2258 -- 3596	1057	3.0M	-19.5	-17.2	-4.0
3597 -- 5301	694	3.0M	-15.2	-13.0	-0.6
5304 -- 7944	470	3.0M	-11.8	-9.8	1.5
7948 -- 12k	306	3.0M	-10.7	-8.8	1.6
12k -- 17k	205	3.0M	-6.1	-4.2	5.7
18k -- 26k	140	3.0M	-4.5	-2.7	6.2
26k -- 38k	94	3.0M	-1.6	0.2	8.5
38k -- 52k	67	3.0M	-0.3	1.7	10.0
53k -- 69k	50	3.0M	0.0	2.0	10.0
69k -- 91k	38	3.0M	-0.8	1.7	9.2
92k-- 132k	27	3.0M	0.0	3.6	9.4
133k -- 212k	18	3.1M	0.2	4.2	9.6
216k -- 672k	15	5.0M	-0.9	2.9	7.8

They are pictured in the graphs in the beginning of the post.

Results for different methods of grouping the players give reasonably similar results. If anyone is interested in results from other groupings of the players, please ask.

DISCUSSION

Note that, as expected, low-volume players have abhorrent winrates. Also, maybe less expectedly, winrates actually dip for the players with close to the maximum number of hands. From working on the data I believe this is not an artifact: players who played 300k hands seem to have lower winrates than players who played only 100k hands; this might not be that surprising, since a particularly skilled player would mostly move up to 200PLO before playing 100k hands.

Very nice work.

One thing that strike me the most here is that the "whales",people with less than 323 hands aka (38k players) lost about 69bb/100 pre rakeback. I kinda often thought intuitively that due to the nature of PLO equity running closer together preflop and often on the flop that they would go bust slower than in NL. But seeing that winrate we can see that the new players are going bust superfast still.

I would really like to see the PLO "state" of the game vs the same limit at NL.

By the way did you have any zoom hands in your data or all 6 max where you can table select a bit?

Amazing data. Something I also find interesting is how much the current system of per hand rake really screws over casual players the most. Look at the lower-mid volume players. The guys losing at -21.5bb/100 are actually only losing at -5.3bb/100 pre-rake! The excessive raking of casual and typically looser player means their money disappears 400% faster than in a rake-free game. Money that would typically last them 20 days instead lasts them 5! The reason I think this is interesting is that the current trend of the the sites is all supposedly about the casual player, yet their profit system screws that particular player more than anybody else for no reason other than they like to play a lot of hands - which makes the games more fun for everybody. The sites are all currently screaming casual players at the top of their lungs yet talk about biting the hand that feeds. To say nothing of the fact that these guys are going to be getting close to 0 in rakeback given their volume.

Great work! Very interesting read. As Mig, zoom included?

This can also be a sample bias. A player that wins it's first allin ('s) is likely to play more than 323 hands. A player that looses its fist allin ('s) more likely quits.

Awsome work Eldodo. Very interesting read. What i take out of your analysis is that the part Stars takes out of the economy is sooooooooo much larger then the part of winning players.

It can be the other way round, they only played 300 hand cause they did bust superfast. Lets say they decided to give a go with a 2 buy in stop loss and after losing few flips decided to stop, the result a -69BB/100.

I mean if a whale wins for the few hundred hands he will keep on going until he is broke and will land in the 5K hands bracket with a -10BB/100 loserate.

Its hard to draw any conclusions about a sustainable whale loserate from this table.

EDIT: Ups, Joeri beat me to it.

I assume this is
"These are essentially all the hands played between March 2012 and March 2012 in PLO100 on Pokerstars "

March 2012 - march 2013?

An interesting analysis might be "are games getting thougher and at what rate"? Its fairly common to state that the games get thougher quickly lately. Maybe you have a tool to find out by deviding the data in 4 quarters and compare the 1th with the 4th? Is the rake part that stars takes out getting bigger fast relatively to the part players take out of the market?

First, a note: My description of the data was wrong: it is only 6-max non-zoom hands. It does not include zoom hands, 9-max hands or HU hands. Also, of course, I meant these are all the hands from March 2012 - March 2013.


Yes, this is very true, and this will be true of the future parts of the analysis as well. I don't know of any statistical method to get rid of this self-selection bias. If anyone has any ideas, please tell me.

(Actually, maybe something can be done by looking at all-in EV rather than bottom-line winnings and using these together in order to cancel out the self-selection bias; I wouldn't expect too much from this approach, though.)


To do this analysis I need to go on with the project for a bit, but then it'll be easy. I'll do this analysis and post it in a few days. I personally doubt that we'll see dramatic changes within the scope of one year, but it's not impossible.


This is a very good point. I personally believe that the only long-term model for raking poker is to take rake on winnings at cashout time (I have a detailed suggestion on the details of this). This will mean that losers pay no rake while winners pay the brunt of the rake, much like income tax, but it also means that rake rape is not possible. If all players are breakever, then the site makes no money.

But without making such a big reform, I think the least we should strive for is for the poker economy to be sustainable. Hopefully we can gather enough data to understand what that formally means.


Well, if I had enough NLH HHs I could do this analysis easily. When I finish the PLO rake project I intend to contact some HH-selling sites and ask them to get some hands pro bono to do this analysis (or just to give them code that they can run on their HH and send me the results). If they don't agree I'll try to get some money from the community to buy some HH to do this analysis.

small typos:

- in both graph it says "rakebacl"
- in the introduction: "These are essentially all the hands played between March 2012 and March 2012 in PLO100 on Pokerstars"

EDIT: very nice analysis. Really looking forward to the next parts.

Thanks for putting in the work, looking forward to future analysis as well.

It doesn't include deep tables as well.

Great work you're doing, eldodo!

whats the point of doing all these calculations if stars isnt going to do anything

go play on other site with better rake/rakeback

Nice analysis so far. I think another reason that the winrates of players could dip in the highest hand volume ranges is that these players are playing many tables at once to get to such a volume, and this negatively affects winrates.

You might also want to look into python-pandas, its a nice library for handling large amounts of data.

I wouldn't really assume Stars is actually aware of things as you might think. Beyond that there's two big things. One is that the online poker market is, in many ways, just starting. The future of poker in the US is starting to look quite promising and there are plenty of eyes on these forums from people with the ability to influence decisions over there. The second is that players are incredibly ignorant on average. The number of threads on here where you have ostensibly informed poker players have a discussion where somebody mentions PLO rake and another player responds something along the lines of, "Who cares. PLO players earn like 40bb/100 anyhow." is mind boggling. Any way you shake it, information is power. Please... no Cersei Lannister reenactments now.

good info, I am one of the 33 that played more than 135k hands and I am also one of the few that has quit or moved on from PLO at stars. It's pretty stressful playing full time and having stars literally take every dollar you win and then give you back a portion as you clear bonuses.

I can't imagine how the games are going to grow or continue, I would be curious how many other players have just quit PLO because of the rake (or loss attributed to 22bb/100 hands).

Really would like more details from Stars about rake and how they feel it should be measured. I think that based on hand, hour, player, or any other metric that you could come up with PLO is far ahead of other games and it just ruins it.

I was playing large volume on Full Tilt at .5/1 PLO 6-max (cap, shallow, regular, ante) for 6 or 7 months before black friday. I think I was 6-8 tabling 8-10 hours a day. I've since deleted my database in frustration, but I remember paying an extraordinary amount of rake in the plo games. I would constantly imagine what a huge winner I would be if I only paid 1/2 of the rake I was paying. Funny enough, before this I was playing a decent amount of low-stakes limit games but moved to plo after noticing that the rake was devastating me in the limit games; it was maxed a v high % of hands.

Fortunately, I managed to win at a decent clip + make, I think, something like $350/week in rakeback and bonus. My living expenses were very low, I had savings and was starting to look for soft 1/2 games right when black friday hit. Not that I have anything to show for it, ty DOJ.

Since then I am back to being a working stiff, but at the time I was completely devoted to improving my game. In addition to playing a bunch, I was receiving 1-on-1 coaching, watching videos and studying my database. I was really sure that moving up stakes and making serious cash was right around the corner. In retrospect, however, I think I was making a huge mistake trying to make it starting at .5/1 plo. I had blinders on at the time, but I should have realized that the rake is essentially unbeatable. Moving up successfully in stakes requires a parlay of positive variance with learning to be a really, really good player and to take full advantage or rakeback and bonuses.

I made a dumb decision to play full-time at .5/1 plo. Given 1.) the legislative climate, 2.) the massive rake and 3.) the relatively high level of play, it's obvious that I was a huge dog to succeed. I certainly wouldn't recommend anyone try it.

To mark my physical presence here... I'm not surprised by the figures. But don't think that the grass is greener on the other side; I expect the avg post-RB winrate on Euro networks to be similar due to poorer game selection. No money at PLO100+, 83% of stack entities are solid.

Next time when posting a table, please indicate that you measure volume in VPPs because Euro network residents like me are more used to measuring it in $, and I was first shocked to see that 18K-26K VPP players (which I thought are SNs with $18K-26K in yearly rake) lost.

But putting it straight, those who can't rake $1500 a month just can't play enough hands at stakes high enough to make poker a full-time job [take a look in the mirror, cåån, your lazy bones have had only one $1500+ month ]. I view SN as a distinction of true grinders from casual players. The latter just don't rely on cashback that much as a source of income, that's why Stars ripped them off it, they're a for-profit business and have the right to screw their customers as they like within the law if they're sure they'll have enough customers. It's indeed just price differentiation as OP stated in a meeting report thread, nothing special.

Hey c.o.o.n,

I guess I made the rookie mistake of not marking my axis properly. The "yearly volume" is in hands, not in VPPs or dollars raked.

Also, my goal for the project isn't to compare Plo winrates in pokerstars to PLO winrates in euro networks. Rather, it is to compare PLO winrates to NLH winrates (or just to the winrate of someone not playing -- 0bb/100) in order to figure out if PLO rake needs to change in order to keep PLO a viable game in the long run. If Pokerstars changes the rake, I think the other networks will follow.

I'm far from an expert but regarding the guys who play longer or shorter based on whether they win or lose, I believe that doesn't matter as on balance there will be the same number of each on either side pro-rated to the number of hands either side that they would have played on average. Or something. I'm sure there's a better way to express that.

Hey joeri,

I finally finished parsing the hand histories on my own (via FPDB) and I now have the data needed to start answering your question:

First Quarter (March 2012 - June 2012)

Annual Volume	#players	Hands	Won (bb/100)	Won+Rakeback	Rakefree Winnings
1 -- 865	21825	3.0M	-46.1	-43.3	-28.2
866 -- 3136	1890	3.0M	-17.2	-14.8	-1.6
3144 -- 9922	558	3.0M	-8.4	-6.5	4.0
9976 -- 27942	183	3.0M	-1.7	0.0	8.9
28368 -- 249676	87	5.0M	0.7	3.1	10.1

Fourth Quarter (December 2012 - March 2013)

Annual Volume	#players	Hands	Won (bb/100)	Won+Rakeback	Rakefree Winnings
1 -- 1018	20254	3.0M	-43.6	-40.8	-25.8
1020 -- 4898	1433	3.0M	-13.1	-10.9	1.2
4901 -- 18672	326	3.0M	-3.7	-2.0	7.4
18835 -- 301315	138	6.0M	-0.6	1.6	8.6

I was hesitant about checking smaller Hands parameters because they'd make the standard deviation even bigger. Even as is, the standard deviation of each winrate (separately) is about 1bb/100, so while the results for the fourth quarter seem worse (for the regs) than the results for the first quarter, the results are not statistically significant enough to say that the level of play increased for sure.

If you want more detailed analysis done on this question, please ask and try to think of ideas about what should be measured.

I'd like to take this chance to ask the community again for ideas on methods of analysis. I now have the data in an easy-to-work-with format, and can do most forms of analysis relatively easily.

I've been making some progress on the next parts in the series, and I hope to publish a new part in a few days.

Also, if there is anyone in the audience with in-depth training in statistics (a Ph.D. student or higher should have the required background, but knowledgeable amateurs would also be appreciated) who is interested in helping me do some theory-work for the analysis, please PM me. I have various ideas on statistical methods to analyze the data, but they seem to touch the cutting edge, and I'm only an amateur so some guidance would help.

Wow, I think that is a huge difference. From plus 0,7bb to minus 0,6bb for the regs... I get that it is not a statistically significant difference, but i wasn't expecting that it would be (where you?). But regs complain all the time that the games are getting thougher quickly, if you then find these results i think that it is a pretty good confirmation.

Also, people may think its "only" a 1.3bb drop. But if you look at the won+rakeback line, you see that the net return halves from 3.1 to 1,6bb/100. If we expect this trend to continue, plo volume will decrease drastically. A semi-professional player who plays 3k hands a day with 1.6bb/100 net result, expects to make 48$ a day... I think we will loose a lot of the high volume regs this way over time and this will reduce Pokerstars rake.

This is mere conjecture. You can't read into the results like that if eldodo42 says they are not statistically significant.

^^This. It's a huge drop, pretty good confirmation that games have been much tougher lately.

I disagree. Its not like where testing some random datasets. We had a hypothesis that games are getting thougher at a fast pace lately. The results eldodo shows are in line with this and thus support this view.

"the results are not statistically significant enough to say that the level of play increased for sure."

The way Eldodo wrote this, makes me feel that it was close.

Jeah offcourse we cant make any significant claims. But it does support the view and it shows more analysis on more data might be good.

Eldodo, If it doesnt take too much time; can you do the same for quarters 2 and 3? If 1>2>3>4 holds true that might also be a strong indication.

This is a form of confirmation bias though. If the variation in the dataset allows for the results eldodo obtained to not be statistically significant, we have to be very careful with drawing conclusions.

I agree with you on this. It might also be that a different methodology on the same dataset would lead to a more significant correlation (assuming the chosen methodology is fitting).