Just another std dev question

[jasonHoldEm]

Hi guys,

Been going through my first cold streak recently. As a new player I'm still developing an appreciation for the natural swings in the game (and sadly not having this appreciation has caused me to lose some of my edge and make things worse).

I walked away from the tables for a few days after losing my entire buy-in Sunday night in about a half hour. I had three big hands that went sour (and I probably could have gotten away from two of them had I been thinking more clearly), the other hands I played I just threw my money away, so I think a break is the best thing for me at this point.

I've been trying to gain a better understanding for the swings I am to expect playing poker and since I'm not a math wiz, I'm turning to you guys for help.

Soooo here goes:

I have 36 sessions logged so far. Since I'm still new to the game I keep my sessions fairly short (they range from 12 mins to 126 mins). I converted the minutes to hours so that I could use Mason's formula (from his essay on the main page)...will using decimals (ex: 0.73 hr) throw off this calculation somehow?

I followed the example he gives in the essay (my N=36 instead of 10) and I came out with a variance of 620.5 and a sd of 24.91. I should point out that I play 1/2 at this point.

I'm pretty sure I calculated these correctly (if they seem unrealistic please let me know), but now I get to show my ignorance by asking what exactly do these numbers mean? I squeaked through stats 101 with a D, so please use small words. /forums/images/icons/grin.gif

Thanks in advance,
Jason

[BruceZ]

will using decimals (ex: 0.73 hr) throw off this calculation somehow?

No, that's how it should be done.

I followed the example he gives in the essay (my N=36 instead of 10) and I came out with a variance of 620.5 and a sd of 24.91. I should point out that I play 1/2 at this point.

I'm pretty sure I calculated these correctly (if they seem unrealistic please let me know), but now I get to show my ignorance by asking what exactly do these numbers mean?

That is a realistic standard deviation. It should be around 10-12 big bets, and it is 12.5. Lower is better, and it will become more accurate with more sessions. It will be higher if you play a lot of shorthanded games or aggressive games. If you win more than 1 bb/hr your standard deviation will usually be higher, but it should still be around 10-12 times your average hourly win. Ways to reduce your standard deviation include improving your hand reading skills, tightening up before the flop if necessary, and avoiding chasing in marginal situations.

Divide this number by the square root of the number of hours played, in this case 24.91/6 = 4.15. Now compute the average amount you have won or lost per hour played. This should lie within $4.15 of your true hourly rate 68% of the time (1 standard deviation), and within 2*$4.15 = $8.30 95% of the time (2 standard deviations). This will become more accurate as you play more hours. This gives you an estimate of your true hourly rate or EV.

You can also use these numbers to figure out your chance of going broke and how much bankroll you need by plugging into these formulas:

ror = exp(-2uB/sigma^2)
B = -[ sigma^2/(2u)]*ln(ror)

ror = risk of ruin (going broke)
B = bankroll
u = hourly rate
sigma = hourly standard deviation

The problem is that you have to play enough hours to get accurate estimates of u and sigma, but if your bankroll too small you may go broke several times before you find that out.

[irchans]

Two thoughts about your post:

1) Typically you need at least a hundred hours to estimate your win rate to within 1 BB / hr or 400 hour for an accuracy of 1 SB/ hr.

2) I think it is very hard to win at 1/2 in a B&M casino because the rake is typically a dollar per hand. On line the rake is often half that, but the games are much tougher.

[jasonHoldEm]

Bruce / irchans,

Thanks for the advice and explanations. Actually irchans answered what was going to be my next question (how long before you can "accurately" estimate your win rate). I plan to keep my eye on this now that I understand it better and as my hours pile up I know the calculations will be more valuable.

Thanks,
Jason

[BruceZ]

Actually irchans answered what was going to be my next question (how long before you can "accurately" estimate your win rate).

Notice that you get irchans' numbers from what I said above. If your hourly standard deviation is 10 times your hourly rate, say hourly rate = 1 bb/hr and hourly standard deviation = 10 bb, then if you play 100 hours you divide 10 by sqrt(100) = 10 to get 1. This means that the standard deviation of your hourly rate will be 1 big bet after this amount of time, or that your true hourly rate will be +/- 1 big bet from your actual hourly rate 68% of the time as I explained. After 400 hours it would be within 10/sqrt(400) = 10/20 = 1/2 big bet or 1 small bet. Notice that it will take even longer than this if your standard deviation is greater than 10 times your hourly rate. This standard deviation of your hourly rate is actually called the "standard error" of the hourly rate. It is a measure of how accurate your estimate of your hourly rate is. Note that it takes a very long time to have a very accurate estimate of your hourly rate. If you want the standard error to be 10%, you would have to play 10,000 hours, since 10/sqrt(10,000) = 10/100 = 10%. Note that this doesn't mean you really know what it is to within 10%, just that it will lie within 10% 68% of the time (68% confidence). 95% of the time it will lie within twice this or 20% etc. Note that it takes much less time for your standard deviation to become accurate than for your hourly rate to become accurate.

[Copernicus]

How "accurate" is any of it. Unless you play the same stakes against the same players your "true" (as well as your "sample") mean and SD change. It is somewhat illusory to think that more data means more accuracy when the conditions arent the same.

[jasonHoldEm]

ooooh that's good.

Making much more sense now, thank you. I actually remember the words standard error, but that was like five years ago so things are pretty rusty...if only I hadn't sold my stats book on ebay. /images/graemlins/frown.gif

Thanks,
jHE

[redsamurai]

This weekend a college kid came into the cardroom I play $4-$8 at with a stats book in hand. The table gave him a $160 lesson in 2 hours with Pedro joking about taking the white boy's tuition money the whole time. If he comes back I can see if he'll sell the book for a few chips.

[BruceZ]

There is a version of the central limit theorem (Liapounov version) that applies even to random variables that are not from the same distribution (changing). Under these conditions, the overall distribution will still be normal with a mean that is the sum of the means, and a variance that is a sum of the variances of the individual random variables. The changes due to the individual distributions average out in time. This explains why so many distribution in nature that are comprised of many different distrubutions are in fact normal. The convergence can take longer than if they are from the same distibution. See DeGroot p. 276.

[BruceZ]

Sounds like an excellent tactic to make good players play incorrectly.

[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. /images/graemlins/tongue.gif

[jasonHoldEm]

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

/images/graemlins/smile.gif 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.