Expected Fluctuation for a Lifetime at the Tables

Introduction:

Zg requested that I look into the fluctuations that a counter could experience throughout their lifetime. As we all know, fluctuation in the game of blackjack is quite wild. We also know that the fluctuation tends to get smoothed out after a number of trials.

But can a lifetime provide enough trials to sufficiently smooth out the fluctuation that blackjack throws our way? I’ll examine this situation through a mathematical approach and a simulation-based approach.

Methodology:

Zg suggested that I assume a counter will get in 3-million rounds in a lifetime (100 hands per hour x 30 hours per week x 50 weeks per year x 20 years). Assuming that one plays 100 hands per hour, this represents 30,000 hours of play. He also suggested that I assume the counter uses a strong system, and that our counter has access to a pretty good double-deck game.

The Game I decided upon is of the following conditions:

2-Decks, H17, Split to 4, No RSPA, 3:2 BJ, DOA2, NS, 0.65 Pen, DAS, 1 other player, face-down (plays second)

The strong system that I decided the counter will use is HO2 (with ASC) and full-indices.

The spread Zg suggested was 1-2x4. I developed an optimal bet spread that assumed 1000 units bank and 13.5% RoR when playing with the above conditions. (Note that I had the spread set to 1-8 and just made the computer put out 2x4 instead of 1-8. 2x4 can technically be put out sooner than 1x8, but I ignored this.) Here is a summary of the spread:

https://docs.google.com/viewer?a=v&...DItNjM5Ny00NDZkLWE5NjYtOWE1MWI1MzhjMTcy&hl=fr


All simulations in this study have the following parameters:

Fixed Cut-Card, 1-Burn Card, Quarter-Deck Resolution, TC divisor = Cards in tray, Truncated TC, Rounded Deck Estimation.

The Mathematical Approach to the Problem:

I ran a simulation (400M rounds) with the above conditions and came up the following stats:

Win Rate per 100 rounds (per hour) = 2.19 units
Standard deviation per 100 rounds = 28.43 units

Some other stats that I won’t be using in calculations:

SCORE: $59.24
TBA: +1.031%
DI: 7.70

To solve this problem mathematically, I brushed up on my statistics and reread a portion of D.S.’s BJA. However, if any of my math is flagrantly incorrect, notify me immediately.

The “Expected Win” after “n” hours (100 rounds is 1 hour) = (Win Rate per hour) x (n hours)

The Standard Deviation after “n” hours = (Standard Deviation for 1 hour) x ((n hours)^½)



First, I’d like to find out one lifetime’s win rate plus or minus 1 standard deviation. After a lifetime (3 million rounds), one can expect to win (68.3% of the time):

(2.19 units per hour)(30,000 hours) +/- (28.43 units per hour)(30,000 hours)^(½) =

65,700 units +/- ~4,924 units

Expressed another way, 68.3% of the time our counter will earn between 60,776 and 70,624 units. Therefore, he has a 68.3% chance of earning between 2.03-2.35 units per 100 rounds (per hour).


Now, I’d like to find out one’s win rate plus or minus 2 standard deviations. After a lifetime (3 million rounds), one can expect to win (95.45% of the time):

(2.19 units per hour)(30,000 hours) +/- (2)(28.43 units per hour)(30,000 hours)^(½) =

65,700 units +/- ~9,848 units

Expressed another way, he has a 95.45% chance to earn between 55,852 and 75,548 units. Therefore, it follows that he has a 95.45% chance of earning between 1.86-2.52 units per 100 rounds (per hour).


Lastly, I’d like to find out one’s win rate plus or minus 3 standard deviations. After a lifetime (3 million rounds), one can expect to win (99.7% of time):

(2.19 units)(30,000 hours) +/- (3)(28.43 units per hour)(30,000 hours)^(½) =

65,700 units +/- ~14,772 units

Expressed another way, he has a 99.7% chance of earning between 50,928 and 80,472 units. Therefore, it follows that he has a 99.7% chance of earning between 1.70 and 2.68 units per 100 rounds (per hour).

The Simulation Approach to this Problem:

This portion of the study is actually what Zg had asked me to do. The mathematical approach taken above is something that I just decided to throw in. The mathematical approach can be used to reinforce the simulation approach.

Specifically, Zg asked me to run 100 3M round simulations and examine the distribution of results. This is exactly what I plan to do here. The parameters of these simulations are no different than the one described above, save for the fact that each simulation will only be 3M rounds, unlike the one above that was 400M rounds.

For each of the 100 completed simulations, I will report it’s hourly win-rate and will organize them moving from greatest to lowest (like the optimism of this organization?).

Results (this is just a list. If you'd like to see a graphical representation of the data, skip to the next link):

https://docs.google.com/viewer?a=v&...2YtZWIzOC00ZTBjLTkxNDctNTQ0ZGU1NWU2NmZi&hl=fr

Some Raw Stats:

Mean: 2.183 units per hour
Median: 2.170 units per hour
Minimum: 1.770 units per hour
Maximum: 2.530 units per hour

Here is a nice graphic summarizing the above data:

https://docs.google.com/viewer?a=v&...TMtNjY5Ny00YzYwLTg0NDctNzFmMTFiMzczY2Y1&hl=fr

Conclusions:

I’ll leave that to you all.

Disclaimers:

The fluctuations that one might experience over 3,000,000 rounds are likely larger than depicted in this study for a few reasons. First of all, the game studied was a double-deck game. It is known that the ratio of win rate : Standard Deviation is significantly lower in shoe games than it is in double-deck games such as the one studied here. It may be unrealistic to assume that one can put in 30 hours per week at the DD tables for an extended period of time, for these games tend to be more uncommon than shoe games, and they tend to be watched a bit more closely by the pit. Because one would likely have to play the occasional shoe game, one’s lifetime win rate likely will have fluctuated away from their EV more greatly than this study may indicate.
Moreover, no one plays perfect. The strategy used in this study was HO2 with ASC, full-indices and quarter-deck resolution. If you try to implement a strategy this difficult, you WILL make an occasional “mistake,” however small or insignificant it may be. Mistakes tend to increase variance and / or lower your EV, neither of which is good for the counter.
Another reason that the typical counter can’t play perfect is the occasional need for cover. It is known that betting cover is quite pricey in terms of increasing variance and lowering EV, ESPECIALLY IN CASES WHERE ONE IS NOT GETTING VERY MANY ROUNDS PER SHOE (I.e., lower number of decks or more players at the table).
I could dabble on forever as to why one’s expected win rate will likely fluctuate even further than the study indicates, but I’d probably not mention anything that you’re not aware of.

Best,

SP

your post got me to wondering about N0, and if it really meant what i thought it meant from a practical point of view.
well anyway, i plugged some numbers in Kasi's spreadsheet for a double deck game that was simmed with cvcx for a 1:6 spread.
1st image was for 261,855 hands for which at three standard deviations the worst case was lose $0 to $7
2nd image was for three million hands for which at one standard deviations the worst case was you only made $26,527. edit: it's confusing to me why for the three million hands the one standard deviation becomes the low figure of them all.:whip:

sagefr0g--

To my best understanding NO, refers to the number of rounds that one must play (given the current conditions), as to make their accumulated win equal to a 1 standard deviation swing after that many rounds. As you can see, it is related to the DI in a way; a low NO translates into a high DI, both of which are indicators of good performance. So long as one plays a strategy with +EV, there has to exist a point where there accumulated win will equal a one standard deviation swing at that point because "expected win" grows linearly with respect to the number of rounds played, whereas standard deviation grows in proportion to the square root of the number of rounds played; IOW, accumulated win starts smaller than standard deviation, but accumulated win grows quicker than standard deviation, so they must cross at some point, yes?

It seems to me that NO for the game you specified is 261,855 rounds, yes? For your 3M round case, I do not see what you're talking about when you say that for your 3M round case that the 1 S.d. swing becomes the low figure of them all. From what I'm seeing, it seems as if:

The one S.d. swing = $269,358 (EV) +/- $26,527 (1 S.d.)
The two S.d. swing = $269,358 (EV) +/- $53,054 (2 S.d.)
The three S.d. swing = $269,358 (EV) +/- $79,581 (3 S.d.)

All is as it should be, no?

Best,

SP

Southpaw, impressive work, wow. Thank you

Tallman

lol, doubtless all is as it should be, just wish i could understand it.:laugh:

but anyway far as N0, i have this definition from cvcx:
It is defined as "the number of rounds that must be played, with a fixed betting spread, such that the accumulated expectation equals the accumulated standard deviation As such, it is a measure of how many rounds must be played to overcome a negative fluctuation of one standard deviation with such a fixed spread."

errhh so it talks about accumulated expectation and accumulated standard deviation, but then it sort of concludes something about some number of rounds played to overcome a negative fluctuation of one standard deviation.

i guess i just need to try and get my mind around what that means, then re-read your posts above and see if it all starts making more sense.

Norm is right, I'm sure, when it comes to NO; I was saying that it occurs when the accumulated win is equal to a swing of 3 standard deviations, whereas your definition coming from CVCX is saying that it is the number of rounds that must be played to accumulate an expected win equal to 1 standard deviation.

Think of it this way then, it is the number of rounds that must be played under the specified conditions such that you have a 84.15% chance of being even or positive at that point.

Along with others you may wonder where I got the 84.15% figure from. It is the percent chance that you find yourself NOT away from EV (in the negative direction) by more than 1 S.d. (I.e., only 15.85% of the area under a normal distribution curve is found to the left of negative 1 S.d.)

SP

Impressive

And to think that I and the Great Paddy had the honor of breaking in this young Phenom at his first 2 times on the real tables!

I think SP could become the greatest AP player to come along since JG arrived on the scene.

We can all someday say we knew him when!

CP

Are you saying that after 3m hands (not rounds) I have a 32% chance of being behind? zg

No, 68.3% of the time you will be within one standard deviation from your E.V.

In this case, you'll be in the following range (as I mentioned in my original post):

65,700 units +/- ~4,924 units

Therefore, in 68.3% of cases he will have earned between 2.03-2.35 units per 100 rounds throughout his lifetime.

Note that this study indicates that it is virtually impossible to be behind after 3M hands, assuming you follow the strategy to a tee and are playing in the designated game. In fact, in 99.7% of cases, you land in the range of being up

65,700 units +/- ~14,772 units

Therefore, in 99.7% of cases the counter will have earned between 1.70 and 2.68 units per 100 rounds throughout their life.

Hope this helps,

SP

I just gonna work on growing my business, easier and better roi.:laugh:

To each his own.

SP

well actually around 29,090 rounds, but we cleared that up with the definition of N0.

right, right, i was interpreting the three million spreadsheet data incorrectly. so just as you say above the three sd. figure would be the low figure of $189,701 as the second image in this link: http://www.blackjackinfo.com/bb/showpost.php?p=209668&postcount=2 shows. my bad, no wonder i was confused.
so anyway, i guess that sort of data can give an idea of how things would work out with respect to ZG's inquiry.

I'd really recommend that people seek to understand the concepts in this thread. This will help them to put their actual playing results into context - have they been lucky, have they been unlucky, are their calculated EV levels correct?

It helped me a lot when I learned this things to keep an objective view of my playing, and to understand how much fluctuation i was exposing myself too. And whenever I am evaluating new strategies, I ALWAYS calculate the EV, the variance/SD and the resulting N0. And N0 is one of my key indicators for selecting what games to play. It will also help to compare different businesses, for example stock/FX trading vs card-counting vs poker.

Regularly people post threads asking how likely it is that they have had x amounts of consecutive winning/losing sessions, that they are up/down $z after w hours. If people understand the concepts of EV, standard deviation and know how to calculate SD for a given amount of rounds/hours/number of hands, then they will be able to answer their own questions.

Great research Southpaw!!

CP, you give me too much credit

Thanks, Matt!

I was wondering how the original stated ROR of 13.5% may or may not fit into all this.

On the one hand, you seem to say it's virtually impossible to be behind after 3MM hands but, otoh, the original ROR would suggest he might experience so much fluctuation he will go bust once every 7-8 bankrolls and could never make it to 3MM hands?

It's always sunshine and lollipops when only EV and variance are considered millions of hands later ignoring risk.

These sound like valid points. zg

Going bust prematurely does not make it impossible to be ahead after 3M hands. The simulator may or may not go bust while performing the simulation, but it will continue playing, knowing that it will be ahead once again. The win rates only indicate how much the simulator ended with, but does not at all guarantee that it did not go bust early on in the simulation. In fact, when planning a simulation, you do not tell it "this amount is your BR." IOW, simulations only look at the end result of how many units they are up.

Therefore, what I said still stands. Assuming you follow the strategy to a tee and are playing in the designated game, it is virtually impossible to be behind after 3M rounds. But this does not mean that one did not go bust somewhere along the way.

For a discussion of RoR and hitting the barrier prematurely vs. being behind after the said number of rounds, see D.S.'s chapter on RoR in BJA.

Hope this helps,

SP

No & nooooooooooooooooooooooooooo

for fixed bets
NO=1sd
4NO=2sd
9NO=3sd

for Kelly resizing
4NO=1sd
16NO=2sd
36NO=3sd

Probably no one continuously resizes their bank based on wins and losses. However, if you resize at all your NO will be much closer to the Kelly resizing NO.:joker::whip:

Book Smart?

Book smart does not always convert to street smart. I would say the 3 biggest factors after skill is:

Does he have money
Does he play
Is he daring

A good math background can be very nice.

good cards
:joker::whip: