Just another std dev question

[Copernicus]

The problem is that the combined distribution under the CLT may or may not have general applicability to any one of the underlying distributions.

Take a player who faces the same players in the same home game forever, and it is a typcial loose/passive group. Change 1 player in that group to a loose/aggressive player and you can throw out any mean and standard deviation of either the first group or the combined groups.

It is interesting to play around with the statistics, but people shouldnt fool themselves into thinking they are more than they really are.

[ChipWrecked]

I'm no math whiz either, that's why I let Pokercharts figure mine.

78.25 over 218 hours, of which 86.5 are B/M 4-8. The rest are online at various limits with some low-buy NL HE, including a disasterous foray into a 3-6 game on Planet where I got my head handed to me, and my $$$ taken away... in a very short time. I figure that must have spiked the number considerably. [img]/images/graemlins/tongue.gif[/img]

[jasonHoldEm]

not a math wiz? You're the freaking Bill Gates collective.

[img]/images/graemlins/smile.gif[/img] Great pic

And I'll check out pokercharts...I never thought of the easy answer.

jHE

[BruceZ]

Looking backwards at sessions played against changing opponents in a particular casino, the data should still follow a normal distribution with some mean and standard deviation. We can then estimate what the player's true EV and standard deviation were over that period with some confidence interval. This is useful information relative to that mix of players, especially if future sessions will be played at that same casino at the same limit against approximately the same mix of player types on average. If the time intervals used are long enough, the distribution of player types should not be expected to change much. If the statistics change a great deal, then this provides an indicator that something is not the same.

For purposes of bankroll calculation, we are intrerested in the ratio sigma/EV, and this ratio is less sensitive to changes than the individual parameters alone.

You mentioned that a good player may already have a pretty good idea what his EV should be against particular opponents. In that case, this prior information about approximately where the EV should lie can be combined with actual data to provide an even more accurate and meaningful estimate using Baeysian techniques.

The main purpose is just that it lets people evaluate the likelihood that they are a winning player under some set of conditions.

[Copernicus]

The math is inarguable. My point is its barely worth the pixels its written in. Achieving additional "accuracy" in estimating a distribution that changes over time isn't going to help anyone make a better decision.

An experienced player knows when he is at the right level without a lot of math. If he moves up (or laterally to a an unknown set of players), he knows from the types of hands he is playing, winning and losing whether he is going to stick at that level a lot quicker than he will be able to improve the accuracy of his EV by a couple of tenths of a big bet.

Is someone going to modify their behavior because they realize they have an earnings rate of 1.4 BB/hour instead of 1.6 BB/hour? Right. If you believe that I have a diet plan, a quit smoking manual, and a Phil Helmuth book to sell you.

[BruceZ]

The math is inarguable. My point is its barely worth the pixels its written in. Achieving additional "accuracy" in estimating a distribution that changes over time isn't going to help anyone make a better decision.

I strongly disagree. We aren't necessarily talking about a distribution which changes over time, we are talking about sessions which change over time due to changing opponents, and we believe that the distribution which describes the aggregate of all these different sessions can be approximated over some time period as being relatively fixed. Every time you sit down, you draw a random set of opponents from the player distribution. Sometimes you draw a "loose passive game"; sometimes you draw a "tight-aggressive" game. These different games have some distribution which affectes your results just as your results are affected by the distribution of luck. Under these conditions, knowing your mean and variance over many past sessions will allow you to accurately predict your expected range of performance in the future as long as you have no reason to think that the distribution of players and game types will change significantly.

Now if you thought that the distribution of game types were going to change significantly in the future, that still doesn’t mean that your stats aren’t useful, quite the opposite. For example, if you know the games are going to get tougher, and you are barely beating them now for a few tenths of a bb/hr, then you know that you will be lucky to break even in the future, and in that case you might decide not to play. If you are beating the games now for a modest profit, and you have reason to think they are going to get looser (perhaps there is a big tournament in town) then you can expect to turn a nice profit, and you will decide to play.

Finally, even if it were true that these stats don’t help you to make a better decision, so what? Most people want to know how they are doing, and whether their results are due to skill or merely to luck. People constantly post their data and ask questions of the form “am I good or just lucky”, or “am I bad or just unlucky”. For most of these players, stats provide the only objective way these questions can be answered. They provide a figure of merit, a way of keeping score which is more meaningful than just the winnings, and for that they are invaluable.

What I said about improving the “accuracy” of the estimate through Bayesian methods is a separate issue, and a rather esoteric one in the context of poker. I just brought that up to show that even if you already have knowledge about what your EV is based on experience, that knowledge can be rolled into the stats to make a more meaningful estimate. If you are reasonably certain that you can beat the games you play in for at least 0.5 bb/hr, then there are ways to bias your estimate to reject anything below that value. This is an extremely important (and somewhat controversial) topic in statistical estimation theory used widely across diverse fields. It is used for the very reason you mention, that you have experience which supercedes what the numbers themselves can tell you in the short term, so you perform an analysis that takes advantage of that experience. If you don't have this experience, then you just do a maximum likelihood estimate. So whether you have experience or not, statistics can help you make a more accurate estimate, and more informed decisions.


An experienced player knows when he is at the right level without a lot of math.

That's nonsense, there are tons of experienced players, and the vast majority of them haven't a clue as to what level they should be playing at for their bankroll and ability, and they don’t have any feel whatsoever as to how sensitive their risk of ruin is to very small changes in their bankroll. This is a big reason why even most professional players eventually go broke. One very good book I know gives bankroll requirements for a 5% risk of ruin which actually would produce a 26% risk of ruin. These numbers look perfectly reasonable, and over many years if they were questioned at all, most people thought the bankrolls were too large!

We all know "experienced" players who judge themselves better than their competition “based on the way hands play out”, and it’s some of these very players who keep the games good when they are oblivious to the fact that their judgment is completely wrong. It takes an "expert" player to make this type of evaluation correctly, not just an experienced player, and experts are few and far between.

Many players judge their ability based merely on their winnings over a short period of time. These people are truly fooling themselves. If there were any way to estimate EV based on results any faster than the statistical methods, those methods would BECOME the statistical methods. The various statistical estimates are proven to be optimal in ways that have been well-defined.


Is someone going to modify their behavior because they realize they have an earnings rate of 1.4 BB/hour instead of 1.6 BB/hour? Right. If you believe that I have a diet plan, a quit smoking manual, and a Phil Helmuth book to sell you.

Would you care if you found out that your salary was to be cut by 12.5% this year, or wouldn't that make any difference to you or your lifestyle? A better question is would it make a difference if your standard deviation were 12 instead of 10? I'll answer. Yes, that is a gigantic difference which changes your bankroll requirements, risk of going broke, and swings so dramatically that it could easily determine what level you play at or if you play at all. Now if you think that you can distinguish a sigma of 10 from one of 12 without any math based only on how your opponents play, then perhaps I can interest you in a full collection of John Patrick videos.

In summary, far from being a waste of pixels, understanding these simple stats represents one of the single most important tools that is sadly lacking from the toolbox of most poker players, and most gamblers in general. GTAOT states that computing one's standard deviation is very important for most players, even though it is seldom ever computed. Many of the topics in GTAOT assume that you know your EV and sigma, and I can assure you that this book is certainly not a waste of ink (though you should substitute the formulas I have given for bankroll and risk of ruin). Finally, given the ease with which these stats can be automatically computed with modern spreadsheets, there is really no reason for a player to not regularly generate them other simple ignorance or laziness.

[Mason Malmuth]

Hi Copernicus:

Several comments. First the main driver of your standard deviation is how you play, not how they play even though how they play will influence your result.

Second, in my book Gambling Theory and Other Topics I address the complaint you are making. I call it a non self weighting effect.

Something else to keep in mind is that your win rate and standard deviation do not remain constant. For example, when you are running bad in a game, it usually encourages opponents to take shots at you, and these shots generally cost you money. That is, your win rate temporarily goes down, and your standard deviation temporarily goes up. The opposite also can happen. Notice that what I have just described is a non-self-weighting effect, and as already pointed out, non-self-weighting effects are statistically equivalent to reducing the sample size. In plain English, the numbers given in the tables may actually be too small. By how much I don’t know, but the serious player may want to increase his bankroll allocations by 20 percent to 30 percent.

Finally, I have been using these numbers for many years and have found them to be accurate and consistent with my overall results. If you're a serious player, they are well worth tracking.

Best wishes,
Mason

[ChipWrecked]

[ QUOTE ]
For purposes of bankroll calculation, we are intrerested in the ratio sigma/EV, and this ratio is less sensitive to changes than the individual parameters alone.


[/ QUOTE ]

Is this the Kelly formula, or similar?

Also, I'm confused as to how one can figure EV in poker. I understand in blackjack, that it's decided by table rules, decks etc. But poker is so fluid, it seems difficult to pin down. Bets won per hour maybe?

[BruceZ]

Is this the Kelly formula, or similar?

Kelly depends on sigma/EV. I had in mind the bankroll formula B = [-sigma^2/(2*EV)]*ln(risk of ruin) which actually depends on variance/EV, and this is even more stable than sigma/EV. (sigma/EV)^2 determines how long it takes to reach the long run, that is, how long it takes to be ahead for any given confidence level.

Also, I'm confused as to how one can figure EV in poker. I understand in blackjack, that it's decided by table rules, decks etc. But poker is so fluid, it seems difficult to pin down. Bets won per hour maybe?

Your actual bets won per hour is the maximum likelihood estimate of your EV for poker, that is, the EV that makes your results most likely. Note that this is different from saying it is your "most likely EV". In blackjack, EV and standard deviation are computed accurately by software simulation if you know the specific rules, decks, penetration, counting method, bet size, bet variation, and playing strategy.

Although it is difficult to know your EV and sigma accurately for poker, for a significant winner sigma/EV is generally better than it is for card counting. sigma/EV for card counting typically ranges around 20-40+, with the low 20s being for rare single deck games with good penetration and bet spread which are hard to find and tend to not last very long. In poker, one shoots for sigma/EV to be around 10. Compared to 20, this means we reach the long run 4 times as fast and for EV=1 we require 1/4 the bankroll. Compared to 40 we reach the long run 16 times as fast and for EV=1 we require 1/16 the bankroll for EV=1.

As your standard deviation gets larger and/or your EV gets smaller, poker can start to degenerate towards a good BJ game, e.g., 0.5 bb/hr with a sigma = 12 gives sigma/EV = 24. This is why it is important when the games become more aggressive that you're sure your win rate will increase enough to track the increase to your standard deviation. Of course if you are barely a break-even player, your win rate can be as close to 0 as you please, and sigma/EV can become arbitrarly large which demands much larger bankrolls to keep the risk of going broke to modest levels. In a shorthanded game, your standard deviation may be around 20 due to the speed of the game, and also due to playing a wider range of hands playing them more aggressively. It will be even higher online. If you are not a good shorhanded player, your win rate may actually be lower in these games, and this can greatly increase your risk of going broke and bankroll requirements. This is why it is so important to track your win rate and standard deviation and know how it compares to these benchmarks.

[BruceZ]

This is why it is important when the games become more aggressive that you're sure your win rate will increase enough to track the increase to your standard deviation.

One time you might make an exception to this is when you are moving to a higher limit. In that case sigma/EV may go up, and you will require a larger bankroll for the same risk of ruin, and it will take longer to reach the long run. You may choose to accept these things because you will be making more dollars.