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Old 10-02-2007, 10:47 PM
Troll_Inc Troll_Inc is offline
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Default A modest improvement in winrate estimation after Poker EV..

[I put this together as a magazine submission and it was rejected. The intro is long-winded to set it up for general reading but the first paragraph pretty much summarizes what I was trying to do. PM me with an email address if you want an easier-to-read Word document copy.

Any comments are appreciated, especially as it relates to the probability theory and statistics. There are a couple of issues that I don't think are commonplace, and could make for discussion/disagreement if you find this sort of stuff interesting.]
]


A modest improvement in winrate estimation of a big bet poker dataset after Sklansky $$$-adjustment via the new software PokerEV


What is a realistic winrate for Pot Limit Omaha?
What bankroll do I need to have to move up to the $200 buyin NL game?
Is my 17 ptbb/100winrate sustainable?
How can I tell if I’m running “hot”?
At least once a day on one of the 2+2 forums someone is asking a question related to bankroll management and winrates. This article quantitatively investigates whether a new poker analysis program is able to help derive a more accurate theoretical winrate, which would provide a better and faster estimation of a true winrate for use in more accurate bankroll calculations such as risk of ruin assessment.

INTRODUCTION
There is a good reason for wanting answers to winrate and bankroll questions, because understanding expected winrate and bankroll management ranks high on the list of fundamental concepts that many players struggle to grasp and employ. One can be a solidly winning player but will go bust if not for practicing sound bankroll (risk) management and correct game level selection. This bankroll ideal is even immortalized in Rounders by Knish’s admonishment of Mike when he breaks the Golden Rule by sitting at a game with and losing his entire bankroll: "It happens to everyone, from time to time, everyone goes bust. You'll be back in the game before you know it".
A huge problem is that bankroll advice is predicated on knowing and being “reasonably sure” about your true winrate, which ends up being difficult for several reasons, one of which is the focus of this article. Your mean (average) winrate per hand is small compared to the standard deviation of the individual hand results in big bet poker [Figure 1].
[image][/image]
Since all winrate analysis and bankroll calculations require the mean and standard deviation of your hand results, having a standard deviation much higher than the average winrate results in the need for a large (>50,000) number of hands needing to be played before practical confidence intervals can be obtained. In plain English, the confidence interval is calculated as a deviation from the mean and is proportional to the standard deviation and inversely proportional to the square root of the number of poker hands played or being examined (Figure 2 and see NOTES section).
Therefore, a faster, more accurate assessment of winrate is needed for more effective and timely BR management. I hypothesize that soon there may be a much more satisfying way to answer the bankroll and winrate questions asked above so that all rounders can “follow the Golden [censored] Rule and leave themselves some outs (bankroll)” with which to stay in the game. This article tests the hypothesis that an objective adjustment to your winrate delivered by the application of an old poker concept by a new poker software tool will tighten the confidence interval and make winrate/bankroll analysis more viable. By reducing the variance caused by the results of big all in hands, I hypothesize that your true winrate will emerge faster due to this objective reduction of the standard deviation with which everyone is more familiar.

The invention I speak of is a new poker software tool.

A couple of months ago in the deep, dark recesses of 2+2 Software forum, some posters started talking about and programming an application that would calculate Sklansky $$$ for large numbers of hands stored in a user’s Pokertracker database. For those not in the know, Sklansky $$$ is an important concept that assesses whether or not you are getting your money in “good”. In other words in big bet poker when large sums of money are shoved in the pot can you expect a positive return (expectation)? I discuss this concept in greater detail as it relates to variance calculations (see SKLANSKY $$$ EXPLAINED).

SUMMARY OF CURRENT BANKROLL THINKING
While bankroll management has been discussed ad naseum, I need to break a couple of points down to help explain how my hypothesized improvement would work mathematically. All accepted bankroll theory revolves around the calculation of winrate and standard deviation calculations. Pure statistical analysis of winrate data from either sessions or hands shows wide variance, and therefore wider confidence intervals. Meaning if you have played 10,000 hands at NL200 (Blinds $1/2) and have a solidly positive winrate, then it is quite possible you are not guaranteed of being a winning poker player. The fact that standard deviations are many times larger than average winrates (Figure 1), makes poker hand result data quite unusual.
To illustrate my point, let us pretend that net yards per rush in the NFL were subject to variance as high as is seen at the poker table. You would laugh if someone told you that there was a decent chance that LaDainian Tomlinson was really a runner that does not have a positive yardage expectation running the ball eventhough over his 7 year career he has averaged 4.4 yds/rush over a total of 2107 carries. While that sounds ludicrous, it is exactly the nature of poker results. We poker players deal with an odd beast, but indeed it is often a beast.
I think that in practical terms, the first and largest problem with getting to know your true winrate, is that you have to play so many hands to offset the high standard deviation to get a mean winrate with a reasonable confidence interval. Wizened poker players admonish, “Go play 100,000 hands if you want to know something about your winrate”. Such advice is not terribly useful and can be downright misleading since many important factors that affect winrate will change over the time that it takes to play the prescribed all those hands, which means your hand samples are being taken from multiple populations and not useful for predicting future results. By the time it takes to play even 20,000 hands your ability level will hopefully change and hopefully you will have moved to play higher stakes. The quality of your opponents fluctuates and so does the pokernomics as evidenced by the recent governmental actions and poker industry reactions and player population changes. Today’s fast-paced internet poker world is in constant flux.

This article quantitates the effect that calculating Sklansky $$$, instead of your your actual results, has on winrate assessment in the form of confidence intervals. I hypothesize this Sklansky $$$-adjustment will lead to great reduction, of at least several fold, of winrate standard deviation and therefore confidence intervals by eliminating the normally distributed randomness of how the cards fall once you have shoveled large amounts of money into the pot. Meaningful theoretical winrate analysis could then occur with the playing of fewer hands.


SKLANSKY $$$ EXPLAINED
You are playing a heads up No Limit Hold’em cash game and are dealt AA three times in a row against a maniac LAG who has been abusing you all night. With little effort, you get your LAG villain to push all in preflop all three times. The villain ends up having 77, QQ, and KK and miraculously hits a set in the first two hands but your hand holds up the last time for net results of -$100, -$100 and +$100 for an actual overall net of -$100. You got unlucky for sure, but just how unlucky?

Well on my own, I came up with the idea to just calculate the EV on hands that go to showdown to make the EV calculations on each street when all the money went in. But apparently, someone older, and maybe just as wise, beat me to the punch. The origin of Sklansky $$$ remains a mystery to this author as it is probably hidden in the dark recesses of the 2+2 archive server or some book no one ever references, but the concept is quite intuitive. Sklanksy $$$ are simply a calculation of what you expect to win on average if the hand was run hundreds of thousands of times so that the expected values of the final net outcome, of playing your poker hand the way you did, is reached. High stakes poker players love to “run it twice” especially in common all in situations in Pot Limit Omaha to reduce the variance on their bankroll; a Sklansky $$$-adjustment is just “running it hundred thousand times”. In mathematical terms this is simply referred to as the estimated value (EV) of the hand or a decision. To drill this important point into your skull, let us examine the actual hands.

1st hand AA vs 77
The pot is $200, which you have a 80.5% chance of winning [see NOTES]. Therefore, your total EV of that hand is calculated as: ($200 * 0.805) = $161. Your net EV is calculated as total EV minus your cost to play the hand ($100), simply calculated as $161-$100 = $61 because eventually if this situation happened over and over again you would win $200 enough times and lose $100 enough times so that it averaged out to the expected value of $61. Another way to look at it is that if this hand were run many, many times you can expect on average to win $61. Unfortunately for you, your opponent spiked a set and won the pot, so you actually lost $100, a $161 deviation from your EV.

2nd hand AA vs QQ
81.6% to win the hand, but in Sklansky $$$ you won $163.20 total, or a net $63.20

3rd hand AA vs KK
81.95% to win the hand, so your Sklansky $$$ for this hand is $163.90, or a net $63.90.

Now of course these examples are quite extreme but if we graph the actual results in red and the Sklanksy $$$-adjusted EV values in blue (Figure 3), we start to see what Sklansky $$$ is measuring and why just looking at a small number of results is not necessarily going to give you a lot of information on how well you are playing or what your longterm winrate. Calculating the winrate from the actual raw hand results (squares, Figure 3) and average ($62.70) and the Sklansky $$$-adjusted winrate (triangles, Figure 3) and average ($-33.33).

[image][/image]

Visually one can see why the standard deviation is reduced since standard deviation is correlated to the difference between the average winrate and the individual hands results (Figure 2) and clearly the individual raw data hand results are further from the raw data winrate than the Sklansky $$$-adjusted grouping. This graph visually frames the hypothesis that I am testing in this article: Namely, that converting the results of your hands into net EV (Sklansky $) will facilitate statistical bankroll calculations on a lower number of hands because of this improvement in a reduction of the standard deviation.

SOFTWARE ANALYSIS OF SKLANSKY $$$ WITH PokerEV SOFTWARE
As I was industriously reinventing the wheel with the already-discovered Sklansky $$$ concept, and doing these calculations by hand using Pokertracker, along came 2+2 user “Phil153” and his beta-version of the program PokerEV. PokerEV’s major innovation is calculating Sklansky $$$ for all the hands in your database that go to showdown and then presenting your Sklansky$$$-adjusted results as easy-to-read graphs. It also allowed me to more quickly test my hypothesis that EV adjustment of my big hands would greatly reduce the variance of my dataset and tighten the confidence interval.

In Excel, I placed a selection of all 8,557 $1/$2 PLO hands that I had played in 2007. Then I used PokerEV to adjust the all in hands that fell outside 3 standard deviations from my mean winrate. A total of 151 positive and 71 negative hand results exceeded this arbitrary benchmark. Simple analysis showed me that I was running “hot” as my actual winrate was $ 0.38 per hand and after a Sklansky $$$-adjustment, my EV-adjusted winrate was just $0.28/hand, or 74% of my actual winrate (Figure 1). The standard deviation was tightened and reduced to 66.5%, or from 29.60 to 19.69. All further analysis in this article was performed on this Sklansky $$$-adjusted dataset. As an aside, the full PokerEV analysis of my 8,557 hands and my corrected data set have a very similar, which proves that the majority of Sklansky $$$-adjustment improvement lies in this small minority of hands from big pots.

This improvement may sound nice, but left me wholly unsatisified when we recall the original goal of narrowing down a right and practical confidence interval. If we look at 95% confidence intervals (p-value of 0.05), then the raw data leaves me with a winrate range of -$0.23 to $ 0.99/hand. The Sklansky adjusted winrate goes from -$0.12 to $0.68/hand. Sadly one could still spend a whole month playing these 8,000+ hands, and still not be very confident about being even a winning player, let alone figuring out if they will make enough money in the future to rely on poker as a source of income.

CONCLUSION
The analysis in this article shows that on one set of PLO data, Sklansky $$$ adjustment of poker results provides an approximate 25% correction of a true EV winrate, 45% improvement on standard deviation, and therefore a partial narrowing of the winrate confidence interval. Frankly, this was a disappointment to me as I had guarded optimism that the standard deviation would drop several fold to a manageable point. Of course more analysis is needed beyond one dataset, including the more popular No Limit Hold’em game. I guess once multiple datasets are examined we will see that Sklansky $$$-adjustment allows for no more than a 20-30% long term adjustment, which leaves open the door for further improvement; the development of new theories, tools, and analysis.

Providing quick access to your Sklansky $$$-adjusted winrate is very valuable though because there are definitely times where your results do not match the value of your decisions and it would be good to be sobered from the drunk enthusiasm that often stems from running hot or conversely brighten your pessimism about your results if you are running abnormally cold. Also, Poker EV provides other analysis tools that will allow the astute poker player to reap the benefits of advanced play analysis.

WHERE DO WE GO FROM HERE?
These results do lead us further down the road of thinking about the difference between what happened previously in hands that you have played (sample), and what is likely to happen (population). Accounting for the variance created between actual reslts and expected Sklansky $$$-adjusted results must necessarily be cleared first before these other topics/conceptrs/issues are tackled. Perhaps searching for ways of estimating your winrate from the population of poker hands you have already played, and those that you have yet to play, is indeed a modern day search for the Northwest Passage. Amazingly a game that seems so simple on the surface may indeed take “a lifetime to learn”.

NOTES
1. All hand analysis is done on propokertools.com.

2. Confidence interval explanation. The confidence interval estimation of a population parameter is associated with an associated probability (p) that is generated from a random sample of an underlying unknown probability. In probability-speak, a 90% confidence interval means that based on a specific set (sample) of data one can be 90% certain the true winrate is in between x and y, where x and y are determined by standard deviation and the number of hands sample (see Figure 2 for formula). An increase in standard deviation causes a proportional broadening of this confidence interval, while an increase in the number of hands sampled decreases the confidence interval inversely to the square root of the number of hands sample. There are several assumptions that must be true for the assessment of a confidence interval a mean winrate calculated from a sample of hands to be within the confidence interval of the true population mean winrate. The discussion of these assumptions is fairly complicated and best left for a future discussion. In Figure 2, x bar is your mean winrate, s is your standard deviation, n the number of hands played, and z stands for z-score which is a modifier adjusted for the percent confidence interval you want to analyze, i.e. 68, 90, 95, 99%, etc.

[image][/image]
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  #2  
Old 10-03-2007, 12:21 AM
pzhon pzhon is offline
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Default Re: A modest improvement in winrate estimation after Poker EV..

[ QUOTE ]
I used PokerEV to adjust the all in hands that fell outside 3 standard deviations from my mean winrate.

[/ QUOTE ]
Why didn't you adjust all of the all-in hands? Setting a threshold like that can bias the results, just like adjusting only the hands you won would bias the results.

Since the folded hands are not random, the Sklansky dollar calculation is not exact. This can introduce a bias into the results.

[ QUOTE ]
The analysis in this article shows that on one set of PLO data, Sklansky $$$ adjustment of poker results provides an approximate 25% correction of a true EV winrate, 45% improvement on standard deviation, and therefore a partial narrowing of the winrate confidence interval.

[/ QUOTE ]
The 25% change in win rate is not particularly meaningful. You could easily get a radically different figure, e.g., if you came out slightly behind, you could end up adjusting your win rate by -12000%.

The 35% (not 45%) reduction in the standard deviation is meaningful.
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Old 10-03-2007, 09:29 PM
Troll_Inc Troll_Inc is offline
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Default Re: A modest improvement in winrate estimation after Poker EV..

[ QUOTE ]
[ QUOTE ]
I used PokerEV to adjust the all in hands that fell outside 3 standard deviations from my mean winrate.

[/ QUOTE ]
Why didn't you adjust all of the all-in hands?


[/ QUOTE ]

I did this because these were the number of hands I checked and verified and I didn't want to do all of them.

Also, as I said in the article, the allin hands for smaller amounts didn't affect the calculated winrate or standard deviation much.

But as far as actual % changes in winrate and standard deviation for this particular dataset isn't as much the point as seeing what other's have.

My analysis was more of a case report which I had hoped would spark enough interest for a 2+2 epidemiological study. If you look through the Software thread on Poker EV (and other forums), you get an idea of the average winrates people see. It might be interesting to see what others report for standard deviations calculated from PokerEV-adjusted datasets.

****************

Other thoughts in general

What I'm am hypothesizing is that the average and standard deviation from the Poker EV are better predictors of the mean of the population and confidence intervals than the same from the raw data. I putting this as a hypothesis, not this is absolutely wrong or right.

Here are the choices for assessing the mean and standard deviation of the population. Numbers which are necessary for assessing the true mean winrates of a player, confidence intervals of that mean, bankroll considerations, etc...

1. Standard way. Average and standard deviation from raw data.
2. What I suggested in my article. Average and standard deviations from Poker EV-adjusted data.
3. Average from Poker EV and standard deviation from raw data.
4. Bayes choice. I suppose you could take the average from Poker EV and a Bayes standard deviation based on the standard deviations from
2+2 posters, those with similar winrates, similar playing styles or some combination.
5. I suppose there are other ways.
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Old 10-04-2007, 08:40 PM
DrVanNostrin DrVanNostrin is offline
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Default Re: A modest improvement in winrate estimation after Poker EV..

Neat idea. But there is a problem. Sorry in advance if I'm missing something.

The play that maximizes one's EV does not necessarily maximize one's EV in Sklansky bucks.

Let's say you know two poker players. Player A plays extremely well. Player B plays just like player A until the river, where he frequently makes very loose calls. Player A has a higher true win rate, but due to the Sklansky bucks player B gets back at the showdown he'll have a higher expectation in Sklansky bucks. (I realize that in PL and NL games player A may do better in Sklansky bucks, but the difference in their true win rates will still be underestimated using Sklansky bucks.)

Even with a large sample size there still will be residuals that result from the differences in style of play. So if you're going to draw a conclusion about a player's true win rate using their Sklansky bucks you need to throw in this uncertainty.

I'm not sure of this, but intuitively when you throw this uncertainty into the equation you'll get exactly what you would if you just used win rate and standard deviation. Perhaps someone more knowledgeable than me can confirm or deny this.
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Old 10-05-2007, 10:06 PM
Troll_Inc Troll_Inc is offline
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Default Re: A modest improvement in winrate estimation after Poker EV..

[ QUOTE ]

So if you're going to draw a conclusion about a player's true win rate using their Sklansky bucks you need to throw in this uncertainty.

I'm not sure of this, but intuitively when you throw this uncertainty into the equation you'll get exactly what you would if you just used win rate and standard deviation. Perhaps someone more knowledgeable than me can confirm or deny this.

[/ QUOTE ]

What you say is true in that Sklansky/EV adjustments do have errors. However, what I would counter with is that just because there is flaw with EV adjustments, it doesn't mean that it isn't better than the current winrate/bankroll analysis.

Qualitatively, I will take the mistakes that EV adjustments make on smaller hands vs the large mistakes made when you just use raw data from big bet poker hands.
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Old 10-05-2007, 10:46 PM
DrVanNostrin DrVanNostrin is offline
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Default Re: A modest improvement in winrate estimation after Poker EV..

After thinking about it more, you could use other statistics to estimate the flaw in Sklansky bucks. For example you could estimate one's true win rate as:

EV = sEV + kW$SD

-k is an unknown constant

If you did more research you could probably better determine what factors relate EV to sEV. A multiple regression might be a good idea.
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Old 10-05-2007, 11:03 PM
Troll_Inc Troll_Inc is offline
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Default Re: A modest improvement in winrate estimation after Poker EV..

[ QUOTE ]
After thinking about it more, you could use other statistics to estimate the flaw in Sklansky bucks. For example you could estimate one's true win rate as:

EV = sEV + kW$SD

-k is an unknown constant

If you did more research you could probably better determine what factors relate EV to sEV. A multiple regression might be a good idea.

[/ QUOTE ]

The k and WSD is an interesting idea.

Do you mean multiple regression of sEV for each hand within my dataset? or for sEV's calculated from different datasets?
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Old 10-05-2007, 11:15 PM
Phil153 Phil153 is offline
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Default Re: A modest improvement in winrate estimation after Poker EV..

[ QUOTE ]
Neat idea. But there is a problem. Sorry in advance if I'm missing something.

The play that maximizes one's EV does not necessarily maximize one's EV in Sklansky bucks.

Let's say you know two poker players. Player A plays extremely well. Player B plays just like player A until the river, where he frequently makes very loose calls. Player A has a higher true win rate, but due to the Sklansky bucks player B gets back at the showdown he'll have a higher expectation in Sklansky bucks. (I realize that in PL and NL games player A may do better in Sklansky bucks, but the difference in their true win rates will still be underestimated using Sklansky bucks.)

Even with a large sample size there still will be residuals that result from the differences in style of play. So if you're going to draw a conclusion about a player's true win rate using their Sklansky bucks you need to throw in this uncertainty.

I'm not sure of this, but intuitively when you throw this uncertainty into the equation you'll get exactly what you would if you just used win rate and standard deviation. Perhaps someone more knowledgeable than me can confirm or deny this.

[/ QUOTE ]
I believe (I may be mistaken) that the Sklansky bucks calculations used in Troll's analysis was simply the expectation of hands that were all-in before the river vs the actual result.
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Old 10-06-2007, 01:47 PM
DrVanNostrin DrVanNostrin is offline
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Default Re: A modest improvement in winrate estimation after Poker EV..

[ QUOTE ]
I believe (I may be mistaken) that the Sklansky bucks calculations used in Troll's analysis was simply the expectation of hands that were all-in before the river vs the actual result.

[/ QUOTE ]

Troll, is this correct? If so I missed something important.

[ QUOTE ]
[ QUOTE ]
After thinking about it more, you could use other statistics to estimate the flaw in Sklansky bucks. For example you could estimate one's true win rate as:

EV = sEV + kW$SD

-k is an unknown constant

If you did more research you could probably better determine what factors relate EV to sEV. A multiple regression might be a good idea.

[/ QUOTE ]

The k and WSD is an interesting idea.

Do you mean multiple regression of sEV for each hand within my dataset? or for sEV's calculated from different datasets?

[/ QUOTE ]
There should also be an unknown constant in front of sEV.

I was suggesting a multiple regression to determine EV. sEV would only be one of those factors.

So your final estimate of EV would be:

EV = k_1*sEV + k_2*factor_2 + ... k_n*factor_n

You could rewrite this equation to estimate the flaw in sEV:

EV - k_1*sEV = k_2*factor + ... k_n*factor_n
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Old 10-06-2007, 03:46 PM
Troll_Inc Troll_Inc is offline
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Default Re: A modest improvement in winrate estimation after Poker EV..

[ QUOTE ]

I believe (I may be mistaken) that the Sklansky bucks calculations used in Troll's analysis was simply the expectation of hands that were all-in before the river vs the actual result.

[/ QUOTE ]

I used your Poker EV calculations for the adjustment for all the largest hands that went to showdown (don't necessarily have to be allin).

I thought the Doctor's point was that the EV will miss the true EV calculations of hands where someone calls big bets but where the hands do not see showdown, as discussed throughout the threads for PokerEV:

http://forumserver.twoplustwo.com/showfl...p;vc=1&nt=6

http://forumserver.twoplustwo.com/showfl...1&fpart=all
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