Matt Wilding (*wilding@cli.com*)

*Mon, 14 Aug 95 10:20:54 CDT*

このファイルには，1990年の夏と最近(1993年11月)にrec.gamblingで行われた カウンティングシステムに関する研究のオリジナルレポートが含まれている． rec.gambling FAQの項目(どのカウンティン グシステムが一番効果的なの？ )は，この研究により得られた物である (このファイルでは，これ以外に2つのシステムを扱っている)．

(Summer 1990)

ブラックジャックのシステムを評価するために，rec.gamblingにおいてシステムを募集した． この記事では，私の所に送られてきたシステムを紹介し，特定のルールに対す る，それぞれのシステムの力量評価の実験の結果を述べている．そして，それ ぞれのシステムを単純さと性能について評価し，一応の結論を得た．

ブラックジャックのシステムを送ってくれた次の人々に感謝します: Steve Jacobs, Jeff Jennings, Paul Kim, Ken Kubey, Steve Markowitz, and Jeff Merrick． そして，助力を申し出てくれたWayne Hathaway and Frank Irwinに感謝します offered help, but since their work would have overlapped with others they could only wait for the results. I'd also like to thank Steve Jacobs and Jeff Jennings for providing me with many tools that made this study doable in a relatively short time.

This report is organized as follows:

- Discussion of the complexity rating and power rating
- Each system with its ratings
- An interpretation of the results
- Complexity and Power Ratings

Complexity is an important aspect of any Black Jack system. 複雑性はブラックジャックのシステムにおいて重要な側面を持っている． 複雑なシステムはプレイヤーに大きな努力を要求し，実際のプレイの時にもミスが多くなる． Complex systems require greater effort by the player, and players using complex systems tend to make more mistakes while playing. A simple system with medium power will probably perform better than a complex system with slightly higher calculated effectiveness, so to compare different systems we must get a handle on complexity.

Some things to consider when analyzing complexity are:

(i) Counting system

Many systems employ a "count" where the player adds or subtracts values depending on what cards he has seen. The count is thus reflective of the composition of the deck, and can be used to better estimate the probability of certain events.

The most important difference in complexity between systems lies in the counting system. Simpler count systems have fewer non-zero values associated with cards, and also have fewer different values. The OPT-I system is an example of a simple counting system. It assigns the following values to cards:

A 2 3 4 5 6 7 8 9 10's 0 0 1 1 1 1 0 0 0 -1

Contrast this to the Uston Advanced Point Count System, which I consider a complex counting system:

A 2 3 4 5 6 7 8 9 10's 0 1 2 2 3 2 2 1 -1 -3

(ii) Side Counts

Some systems require the player to keep track of values other than the count. For example, a side count of how many aces have been dealt might be used in conjunction with the count to predict good betting situations.

(iii) Basic Strategy Deviations

The count is usually used to estimate player advantage or disadvantage for betting purposes and to estimate whether insurance is favorable to the player. Another use for the count in some systems is to modify playing decisions. This can give some additional advantage to the player (though little compared with the advantage using the count for betting and insurance) but can make things much more complicated.

Some systems have more than a hundred rules of the form "If the count is at least X then do A otherwise do B". Obviously, the more rules of this form, the more complicated things become.

I developed a very simple complexity classification system with which I rate each of the systems. Complexity points are assigned to a system in the following ways:

simple counting system: 2

complex counting system: 5

many (~50) strategy deviations: 3

side count: 2

A little fudging is necessary because some of the systems don't fit neatly into these categories. For example, the Revere plus-minus system has many deviations, but they all are depend only on whether the count is positive or negative which makes it much less complex.

A system's complexity rating shouldn't be taken all that seriously. It's only a stab at quantifying how hard it is to use - and prepare to use - a system. Even so, it conveys some idea of how difficult a system is and what the likelihood of making an error with it is.

In contrast to the complexity rating, the power rating is purely quantitative. I use a simulator to run 15,000,000 hands of black jack for each strategy using the rules described below. 20 times the system advantage - amount won or lost divided by total amount wagered

- rounded to the nearest integer is the power rating. Thus, a system with an advantage of 0.76% has a 15 power rating. I have calculated that 15,000,000 hands will provide a power rating within 1 point of a system's actual rating 99% of the time.

The rules were picked arbitrarily for the sake of concreteness. I don't make any claims that this scenario is "realistic" - though you could probably find a game not unlike it in Las Vegas. The bet spread is particularly conservative. I specify the rules only so that I can compare different systems on a level playing field.

The rules are:

1 deck

1 on 1 play

dealer redeals if 20 or fewer cards left
strategy must bet 4 immediately after shuffle,
and all bets must be either 2, 4, or 8
strategy can use how many cards are left in the deck
to the nearest 1/4 deck

dealer hits soft 17

double allowed on any first 2 cards

no double after split

resplitting allowed

insurance allowed

no surrender

Four systems that were sent to me are not considered in the study. (I've communicated with each of the submitters, so this listing is just for completeness sake.)

Canfield: The system from Canfield found in BlackJack_Your_Way_to_

Riches is almost exactly the Hi-Opt I system and is not handled separately. (sent by Jeff Merrick)

Modified OPT-I: This system is also so close to OPT-I that the simulator

can not differentiate between them even after 5,000,000 hands. (sent by Steve Markowitz)

QB1 and QB2: These systems have an interesting betting principle but have

poor basic strategy so little real information could be derived from their simulation. (sent by Ken Kubey)

**ANDERSON**

Complexity: 9.5

Power: 16

submitted by: Jeff Jennings

reference: "Turning the Tables on Las Vegas" by Ian Andersen
comments:

The Anderson system has a moderately complex counting system, many deviations from basic strategy, and an ace side count.

**BASIC**

Complexity: 0

Power: -5

submitted by: Matt Wilding

reference: (Taken from Steve Jacobs' basic strategy posting)
comments:

Basic strategy was added to the study for comparison purposes.

**HORSESHOE**

Complexity: 6

Power: 14

submitted by: Paul Kim

reference: simplified version based on Uston's APC (see below)
comments:

This system is an effort to simplify an already-developed system to get the important factors without losing too much power. It retains the complex counting, but reduces the number of deviations and eliminates the side count. Interestingly, HORSESHOE is to USTON APC as SUPER-SIMPLE OPT-I is to OPT-I.

**SUPER-SIMPLE OPT-I**

Complexity: 2.5

Power: 16

submitted by: Matt Wilding

reference: simplified version of OPT-I (see OPT-I below)
comments:

This is the system I use, mostly because it's simple enough that I can play it without making mistakes. It uses OPT-I's simple counting system, but eliminates almost all the deviations and the side count. (Eliminating the 6 remaining deviations to make it super-super-simple reduces the power rating by about 3.) The most unusual thing about this system is that the basic strategy is skewed so that higher-count

- and therefore higher-bet - situations are played correctly more
often.

**OPT1-6+6 **

Complexity: 5

Power: 18

submitted by: Jeff Jennings

reference: World's Greatest Black Jack Book
comments:

The OPT-I system with strategy deviations for count values between -6 and +6. The OPT-I system does pretty well despite its very simple counting system.

**OPT1-6+6 W/ ACE**

Complexity: 7

Power: 23

submitted by: Jeff Jennings and Matt Wilding
reference: World's Greatest Black Jack Book
comments:

Same as above except an ace side count is added to increase betting accuracy. The ace side count helps, though its debateable whether it's worth the additional bother and the increased chance of making a mistake.

**REVERE APC**

Complexity: 6

Power: 17

submitted by: Jeff Jennings

reference: Playing BlackJack as a Business
comments:

The Revere Advanced Point Count system is a moderately complex count system with many deviations. I believe that it is considered obsolete in favor of Revere's simpler plus minus system. This analysis sugests that the two systems have about equal power.

**REVERE PM**

Complexity: 3.5

Power: 16

submitted by: Jeff Jennings

reference: Playing BlackJack as a Business
comments:

The Revere Plus Minus system is a simple count system with many strategy deviations. One unusual thing about it is that the deviations are all dependent only on whether the count is plus or minus, thus simplifying the use of the system.

**USTON APC**

Complexity: 10

Power: 22

submitted by: Jeff Jennings

reference: "Million Dollar Blackjack" by Ken Uston
comments:

Uston's advanced point count system uses a very complex counting system, many strategy deviations, and a side count. A simpler version is the Horseshoe strategy described previously.

(According to a rec.gambler the Uston APC system now has available an improved set of strategy deviations. The system evaluated here uses the older set.)

**UTAH 10**

Complexity: 2

Power: 13

submitted by: Steve Jacobs

comments:

The Utah 10 system keeps a "perfect 10" count that allows perfect insurance betting and reasonable betting. An unusual aspect of this system is that a true count - where the count is adjusted to allow for the effect of card depletion - is not needed.

**WONG HIGH-LOW**

Complexity: 5

Power: 19

submitted by: Steve Markowitz

reference: Professional Blackjack by Stanford Wong
comments:

The Wong system uses a simple count system and many strategy deviations. It is similar to OPT-I, though its count assigns a non-zero value to aces.

3. Conclusions

Remember that 1 power unit = 0.05% of advantage. The system ratings, sorted by simplicity, are:

name complexity power BASIC 0 -5 UTAH 10 2 13 SUPER-SIMPLE OPT-I 2.5 16 REVERE PM 3.5 16 OPT1-6+6 5 18 WONG HIGH-LOW 5 19 HORSESHOE 6 14 REVERE APC 6 17 OPT1-6+6 W/ ACE 7 23 ANDERSON 9.5 16 USTON APC 10 22

Super-simple OPT-I appears best on the low end of complexity. Revere Plus-Minus has similar power, though has many (simple) deviations.

OPT-1 (with most of its deviations and no side count) and Wong High-Low are the best moderate-complexity systems, with similar complexity and power.

At the high end, full-blown OPT-I dominates. Uston's system has similar power but significantly greater complexity.

Personally, after all this I think I'll stick with my super-simple system. I had hoped to find a system of comparable simplicity with greater performance, but the only systems significantly better were much more complicated. I could gain as much as 0.35% theoretical advantage by going to a full blown OPT-1 system, but it would be a big pain and I wouldn't get the full increase anyway because I'd likely make more errors.

A final note. Some blackjack players - especially beginners - don't
appreciate the high level of variance in blackjack. I've noticed that
many rec.gambling craps players think blackjack card counting allows a
player to "grind out" profits, making a few more dollars each hour.
Nothing could be further from the truth. It truly is gambling, it's
just that in the long run you'll win more than you lose. Perfect play
with a solid system will bring you as much as a 1% advantage with the
standard deviation of 1000 hands of blackjack at around 3.5%. This
means that about 1 time in 3 you will LOSE while playing 1000 hands.
**1000 HANDS IS MORE HANDS THAN YOU CAN PLAY IN A FULL DAY, SO MORE THAN**
**1 DAY IN 3 YOU WILL LOSE WHILE PLAYING BLACKJACK WITH A SYSTEM THAT**
**HAS A SIGNIFICANT ADVANTAGE!**

If you've a system that has not been analyzed in this study, feel free to send it to me. When I get more time I'll analyze it and send you the results. If enough people do this, I'll write up another report and post it.

Good Luck!

matt wilding

university of texas at austin

(December 1993)

Q:B19 What counting system is easiest to use? A:B19 (Matt Wilding)

Background: Lots of systems are available. There is an important tradeoff between complexity and theoretical power, as more complex systems are harder to use and more error-prone.

Answer: You pick'em. A rec.gambling study was accomplished that compared different systems, and here a summary of what came out:

Complexity is a subjective measure with guidelines described in the results paper. Power is the integer closest to p/0.05%, where p is the % advantage of the strategy one-on-one in a single deck, dealer hits on soft 17, no DDAS, resplitting-allowed game that's dealt down to 20 cards and using a 1-4 betting spread. 15,000,000 hands guarantee correctness to within 1 point 99% of the time.

name complex power card weights reference A 2 3 4 5 6 7 8 9 X -------------------------------------------------------------------------- BASIC 0 -5 Steve Jacobs UNBALANCED 10 2 13 1 1 1 1 1 1 1 1 1 -2 Steve Jacobs SUPER-SIMPLE OPT-I 2.5 16 1 1 1 1 -1 WGBJB (1) REVERE PM 3.5 16 -1 1 1 1 1 1 -1 PBaaB RED SEVEN 3.5 19 -1 1 1 1 1 1 R:1 -1 BiB OPT1-6+6 5 18 1 1 1 1 -1 WGBJB WONG HIGH-LOW 5 19 -1 1 1 1 1 1 -1 PB ZEN 5 19 -1 1 1 2 2 2 1 -2 BiB HORSESHOE 6 14 1 2 2 3 2 2 1 -1 -3 MDB (2) REVERE APC 6 17 -2 1 2 2 2 2 1 -2 PBaaB OPT1-6+6 W/ ACE 7 23 1 1 1 1 -1 WGBJB ANDERSON 9.5 16 -2 1 1 1 2 1 1 -1 -1 TtToLV USTON APC 10 22 1 2 2 3 2 2 1 -1 -3 MDB

- WGBJB: "World's Greatest BlackJack Book" by Humble and Cooper
- PBaaB: "Playing Blackjack as a Business" by Lawrence Revere
- BiB: "Blackbelt in Blackjack" by Arnold Snyder
- PB: "Professional Blackjack" by Stanford Wong
- TtToLV: "Turning the Tables on Las Vegas" by Ian Anderson
- MDB: "Million Dollar Blackjack" by Ken Uston
- (1) with modifications by Matthew Wilding
- (2) with modifications by Paul C. Kim

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