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Blackjack Systems - From Basic Strategy to Expert
by D. Howard Mitchell

What began in a technical journal more than twenty years ago has come a long way since.

A new generation of more powerful blackjack systems has come on the scene, and with the wider range of choices now available, you might be wondering what kind of system is best for you. To help you make that choice, this article explains how we got where we are and divides useful blackjack systems into five distinct classes.

Blackjack In The Early Sixties

The first good blackjack system was published in the September, 1956 issue of the Journal of the American Statistical Association. Developed by Baldwin, Cantey, Maisel, and McDermott, the system was a basic strategy which gave any blackjack player using it an edge in the long run that anyone playing craps, roulette, keno, or slots did not have.

The paper received little attention at the time because it was in a technical journal. However, after reading it and trying the system, Dr. Edward 0. Thorp was inspired .to add card counting, which when used properly, actually gives the player an edge over the house. While Thorp's initial effort was a five-count system, he is best known for the ten-count system described in his Beat the Dealer, was first published in 1962. The 1966 edition has an improved version of the ten-count system, and at $2.45 per copy, it is still one of the best-selling books on blackjack.

Thorp's book contains most of what was known about blackjack in the early sixties and is still of general interest. The first edition had a discussion of the use of a side count of aces for betting purposes. It also introduced what was then thought to be the ultimate system, the use of a point count as a guide to betting while using the ten count for playing. The second edition had an improved basic strategy and a high-low count, which was introduced, by Harvey Dubner at the 1963 Fall Joint Computer Conference. With the publicity that it received and its ultimate success, Thorp's Beat the Dealer made many people aware that casino blackjack could be beaten statistically.

At that 1963 Fall Joint Computer Conference, which was held in Las Vegas, Professor Thorp was master of ceremonies for an evening panel discussion on the use of computers in studies of games of chance. The next Fall Joint Computer Conference held in Las Vegas was in 1965. While the conference was being planned, the casinos were challenged to let a computer "Play" blackjack, and one of them accepted the challenge. Although the computer's winnings were modest, they were enough to convince casino management that computers could help players and that at least one system really did work. Since the early sixties, a number of major changes have taken place:

1. The casino game of blackjack has become much more popular.
2. Casinos have taken counter measures' against card counters.
3. Easier to use card-counting systems have been developed.
4. Higher performing card-counting systems have been developed.

Popularity Increases

BeforeBeat the Dealer, the large casinos had few blackjack tables and those tables were rarely crowded. Now there are many blackjack tables at every large casino. In Professional Blackjack Stanford Wong reports that the number of blackjack tables in the large hotel casinos on the Strip in Las Vegas ranges from a high of sixty-one at the MGM Grand to a low of sixteen at the Flamingo. The casino revenue from blackjack has caught up with craps since the tables now are often filled with players. One explanation for this change has been offered by the editor of Casinos & Sports: "The craps table was the playing arena of the majority of compulsive gamblers ... They couldn't justify their losses to their peer group without a putdown because of the medium they were challenging. Many of these have switched to the more respectable counting 21 to the glee of the casinos. They keep the 'drop' somewhat respectable for the house in spite of the winnings of the good counters. Yet the compulsive can now justify his losses with honor."

Casino Countermeasures

After Thorp's Beat the Dealer the dealers no longer dealt to the last card before shuffling. At first, this change was in response to wild bet variations made by professionals who could keep track of the specific cards left in the deck. Then, the casino managers demonstrated they had more greed than common sense. They changed the playing rules to make the game less favorable for the card counters. Of course, these changes also made the game less favorable for all players, and blackjack quickly lost much of its new-found popularity. The casinos soon learned that they had to be more subtle, and they abandoned almost all of the unfavorable rule changes.

The only change left is that some of the casinos restrict doubling down to the player's first two cards so that you cannot double down on the two-card hand you get after splitting a pair. This unfavorable change was partially offset by permitting splits of pairs after the first two cards except for a pair of aces.

A more subtle change was the switch from a single deck to two or four decks under the mistaken impression that nobody could count four decks. Some casinos have since gone to five or six decks. Then along came Ken Uston's The Big Player, which shows that casinos with multiple decks are vulnerable to team play. Furthermore, team play was hurting them much more than were individual card counters.

One side effect of going to multiple decks is that the game becomes less favorable for players who do not count. As a result, many non-counters have abandoned casinos with only multiple decks in favor of those which still offer single-deck play. This exodus has slowed down and probably reversed the trend toward multiple-deck games, which shows that the laws of economics apply to casinos too. Players do tend to go where they get the most for their money. Consequently, some casinos have experimented with rule changes which make the game more favorable for the knowledgeable player. One example is the introduction and slow spread of the surrender option.

However, another example of the greed of some casino managers is the barring of players suspected of being card counters in order, presumably, to increase profits. I suspect that those casinos flagrantly barring counters will actually decrease their profits. For when other players see fewer winners, they will tend to get discouraged and go to other casinos with more favorable playing conditions. Of course, the legality of barring counters is being challenged in the courts.

The only casino countermeasure which has not met serious resistance is the closing of blackjack tables when they become empty. With fewer empty tables available, the good counter finds it harder to be the only player at the table. In single-deck play, the good card counter loses a part of his advantage when forced to share the table with others if he cannot see all the cards. And, of course, he receives fewer hands per hour, having to wait while the others play.

Easier Card Counting Systems

Even though a handful of professional blackjack players were successful using the systems in Thorp's Beat the Dealer, most players find those card-counting systems too hard to use. Since Thorp's book, several card counting systems that are easier to use have been developed. We can put these systems into two classes, which we will Call intermediate and advanced. We use these terms- to emphasize that these systems are more difficult than a good basic strategy. The intermediate systems use a simpler count than the advanced systems.

The first card count which falls into the intermediateclass is Thorp's own five count. All that count involved was keeping track of the number of fives seen. When the deck has fewer fives left than normal, it is more favorable for the player, and so you bet more. When the deck is not favorable for the player, you bet less. Revere has such a five-count system in Playing Blackjack as a Business. In that system, you also have a different playing strategy than basic when the deck is favorable. Roberts' How to Win at Weekend Blackjack has both an ace count and a five-six count.

Advanced Systems


The advanced systems use a somewhat more complicated count and achieve much better performance than the intermediate systems. Table I lists the better advanced systems. Unlike the card counting systems in Thorp's Beat the Dealer, the advanced systems do not use any arithmetic beyond counting and comparing.

Roberts' Winning Blackjack uses a count of tens seen and a count of the total cards seen. Both of these counts are simpler than Thorp's count of tens left in the deck and count of non-tens left in the deck since it is easier for most people to count up than to count down. In addition, a count of all cards seen is easier than a count in which you must decide whether or not the card is a ten before you count it. Thorp next divided the number of non-tens left by the number of tens left. This tens ratio was the index used to find out how favorable the deck was. With Roberts' system, you compare the number of cards seen with the "point of favorability" for the number of tens seen. If the number of cards seen is at least as large as the point of favorability, the deck is rich and you take playing actions which you would not take if the deck were not rich. You also bet more when the deck is rich.



The other advanced systems use what is called a running count. If a card seen is one of the high cards being counted, you subtract the number shown in Table I from the running count. If the card is one of the low cards used by that system, you add one to the running count. For any other card seen, the count does not change. The running count is somewhat harder than the simple count used by the intermediate systems.

Revere's Plus Minus uses the same running count as Dubner's system used. But Dubner divided the running count by the number of cards which have not been seen to get the "true count, " and he then used the true count in much the same way as Thorp used the tens ratio. In contrast, the systems in the Advanced Class do not use a true count. They require decision making based directly on the running count. Since an ace can be valued at either one or eleven, its behavior in blackjack play is much different from that of a ten. Consequently, because the Revere system has two cards with widely differing characteristics in the same group, it is less powerful for playing decisions than the better systems which have only tens in the high group.

All of Roberts' systems offer an optional side count of aces, which is used to guide betting. When the deck remaining has more than the normal one ace per thirteen cards, the deck is ace rich and is favorable for betting purposes.

Another feature of the Roberts' systems which makes them extremely playable in a casino atmosphere is their resolution of rules so that the least amount of memory work is required. For example, the Roberts' Winning Blackjack system requires thirty seven rules, while the Revere matrix charts require some 260 rules.

Better Systems For Professionals

A number of systems for professional blackjack players have been developed since the early sixties. All of these systems use at least one true count, and that is the criterion we are using to distinguish these professional systems from advanced systems. Some of the newer systems have two or more true counts, and we will put these in the class of systems for experts. One important advantage of grouping the systems into classes is that there is very little difference in the performance between the better systems within each class except for the expert class. However, there are other differences among these systems which would tend to make some systems harder to use than others in the same class.

To me there is very little difference in difficulty between using a running count and using a true count because I have a facility for mental arithmetic. However, most people have problems with the division operation particularly, and the HI-OPT systems, for example, recommend that you divide by three-fourths when one-fourth of the deck has been played. Table I I lists and summarizes the characteristics of the more important professional blackjack systems. If you have problems with dividing by fractions, then you should not consider any of the systems in this table which call for that operation.

Table of Professional Blackjack Systems

The first four systems in Table II and Thorp's Ten Count System are of historical interest only. The early systems were quite crude when introduced because their developers were not using any of the more powerful optimization techniques now available. The Einstein system of 1968 can be regarded as a prototype for HI-OPT I. The Collver system is noted 'only because it can be regarded as a prototype for two of the later expert systems.

Although Table II contains point values for the high and for the low groups, a system is not a "point count" unless it contains more than one point value for one of the groups. Among the point count systems still of interest, the most complicated is the 1973 Revere Advanced Point Count. A point value of 1 is assigned to sevens, 2 to twos, threes, and sixes, 3 to fours, and 4 to fives. Tens have a point value of -3 and nines have a point value of -2. Aces are counted separately. This is the system which Ken Uston used, and the problems he had in mastering it are well covered in The Big Player. The simplest of the point count systems is HI-OPT II which assigns a value of +1 to twos, threes sixes, and sevens, +2 to fours and fives, and -2 to tens. Uston's point count comes in between in difficulty. The systems which use only a +1 for the low group and a -1 for the high group are obviously simpler to learn and use than the point-count systems.

Comparing Systems

It is handy to have appropriate measures of system performance. The most widely quoted measures of performance are the results of computer simulations of one million hands run by Julian Braun. They were initially published in Gambling Quarterly and were later summarized in Systems & Methods. They are now available in more complete form from the Gambler's Book Club under the title, The Development and Analysis of Winning Strategies for the Casino Game of Blackjack.

Computer simulations are useful because you don't have to worry about player errors or the ten thousand hours it would take to play the one million hands. Furthermore, all of the systems compared are exposed to the same shuffled decks. That is, the cards in the deck are in the same order. This does not mean, however, that every system will have the same sequence of player hands and dealer up cards. For if one system calls for additional cards or fewer cards than others do in the play of a hand, the subsequent player's hands will be affected in Braun's simulations. Of course, the effects of these differences cannot persist beyond the next shuffle. The overall effect is much larger with multiple-deck play because more hands can be affected by each difference.

Observing a player in action is not an adequate way to measure system performance. Even if you could achieve one hundred hours of flawless record-keeping and play, it is not enough to fix the performance of a system with an accuracy useful for comparing systems in the same class. Conversely, small differences in performance (less than 0.5 percent in expected value) should be insignificant to everyone except professionals who play for many thousands of hours during their careers. On the other hand, differences in error rates can be very important. A difference of one error per hour is normally enough to overwhelm any performance differences between systems in the same class or even differences between systems in adjacent classes.

In comparing the performance of point count systems with other systems in Table II, Braun's results show no differences between HI-OPT I and the more complicated Revere Advanced Point Count. The advertising for HI-OPT II claims a gain of 0.2, percent over HI-OPT I with a side count of aces, but the statement is not specific as to exactly what was compared. 111OPT II has a table which contains a different critical true count value for doubling down for each hard total adding up to 11. Such a table is much harder to learn and use than the HI-OPT I table with only one critical true count for the hard eleven total. Furthermore, such a small difference is meaningless unless the player's error rate is far less than one per hourand many thousands of hours are to be played using the higher performing system.

The Most Powerful Blackjack Systems Ever Devised

As noted in Table II, all of the newer professional blackjack systems offer a side count for aces, which some use for playing purposes in addition to betting. That is why that group was limited to systems with at most one true count and a side count of aces. The first system which used more complicated counts was Thorp's Ultimate System. It used a point count for betting in addition to the tens ratio for playing decisions. Characteristics of other systems which involve more than the professional systems already mentioned are summarized in Table III.

The first three systems in Table III all use the same groups of cards. Although Collver's book was published before I started my research, I did not see it until I was far beyond his system in performance. Collver did not have the benefit of any of the modern system-optimization tools; so his system was developed using tedious hand calculations. Thus, he was not able to carry a good concept very far.

Because the circumstances which led to the DHM Expert System are rather unusual, I will briefly review what happened. Although I had been to Las Vegas a number of times before I started my work in this area, I had never played the game of blackjack in a casino and believed the house could not be beaten. I expressed this opinion to friends who were amateur users of Thorp's system. The response was a challenge to read their copy of Thorp's book. Since I have a very strong background mathematical probability, I was thoroughly convinced by Thorp's exposition. I was then inspired to come up with something better.

The high-low count system offered several opportunities for improvement. First, tens and aces behave very differently and should not be in the same group. Second, all cards should be accounted for so that there are no assumptions for uncounted cards. Third, except for aces, the groups should be of equal size and as homogeneous as they can be. Fourth, since four-deck play was becoming more common, a count system developed for one deck play might be inaccurate for four-deck play.

My work was first published in January, 1973 in the Claremont Economic Papers as an illustration of a practical application of mathematical probability. When the Western Economics Association decided to have a session on the economics of gambling in their 1974 conference to be held in Las Vegas, mine was one of the first papers submitted. By the time that it took on the name "The First Conference on Gambling," that session had grown to two full days of meetings. Most of the papers were on economic and sociological aspects of gambling and subsequently appeared in Gambling and Society, published by Charles C. Thomas Co.

Because of my lack of playing experience, I did not know I had invented a system which I would not be able to use for several years. It didn't take too long before I was concentrating my efforts on the DHM Professional System, which I was able to master without too much effort. Then my associates contributed ideas which made the DHM Expert System manageable. It has since been refined to the point where I don't think it is any harder to learn and use than Revere's Advanced Point Count, for now even I can use it effectively in a casino.

Table of Expert Blackjack Systems

I became aware of the work of Fristedt and Heath shortly before the Second Conference on Gambling. Professor Heath had been informed about my paper by Julian Braun. Because our overall approach appeared similar, he was thinking of withdrawing the paper he and Fristedt had prepared. I encouraged him to proceed with the presentation because their paper emphasized aspects which I had covered only lightly; secondly, they presented the system while I only presented its performance. Their system has since been published in the May, 1977 issue of Winning magazine (formerly Gambling Quarterly) which is now defunct. Although I regard the manner in which they presented their system as very effective for portraying their results, it is not the least bit useful to anyone who wants to use the system in casino play. Furthermore, their presentation suggests that their optimization was less complete and less accurate than the techniques I used.

In estimating the differences in performance between the two systems, the DHM Expert System should be given credit for integrating the ace count into the playing decisions, while Fristedt and Heath used it only for betting. In addition, the better accuracy of the techniques I used could account for several tenths of a percent difference in performance, which gives the DHM Expert System a clear performance edge over the Fristedt-Heath System.

The Multiple Parameters for the HI-OPT systems represents an entirely different approach to blackjack. The mathematical foundation for the approach was laid in a paper which Professor Peter Griffin presented at the First Conference on Gambling. The paper which he presented at the Second Conference on Gambling, held in 1975, puts forth the idea itself. Since HIOPT I does not count 2's, 7's, 8's, and 9's, the multiple-parameters approach fills in the gap by maintaining side counts for each of these in addition to the side count of aces. The advertised gain over the system with only the side count of aces is only 0.2 percent. Thus, the player is asked to add four more side counts for a gain of only 0.2 percent in performance.

The multiple-parameters approach has not been compared in a simulation with either the DHM Expert or the Fristedt-Heath systems. However, in view of the rather small performance gain claimed, I believe that it is significantly inferior in performance to these other two systems.

The Sklansky Key Card concept was described in the August, 1977 issue of Gambling Times. In that article, it was recommended as an addition to a basic strategy and only a few rules for using the ten side counts were given. There are expert bridge, gin rummy, and poker player's who have demonstrated that they can and do keep track of every card in the deck. Thus, the system probably is feasible for experts. However, use of a basic strategy as the base and the limited rules presented leads me to believe that it probably is not much better in performance than the intermediate systems. I doubt that it is as good in performance as the better advanced systems in Table 1. It is certainly much harder to use.

The DHM Ultimate System was inspired by the Sklansky Key Card concept. If you are going to the trouble of counting each card separately, you should be able to do something useful with that information. At present, the best approach is to program a computer to use the ten counts and tell you what to do in each and every player hand-dealer up card situation in combination with the number andvalue of the cards remaining in the deck. Unfortunately, the use of a computer in casino play for this purpose is probably illegal-or at least would not be permitted. Thus, the DHM Ultimate System is an academic device which is being developed only to find out how close to this ultimate performance the DHM Expert System comes.

Conclusions and Recommendations

There noware five distinct classes of useful blackjack systems:

Basic - no counting;

Intermediate - a simple side count;

Advanced
- a running count;

Professional - a true count plus an optional side count of aces;

Expert - everything more complicated than professional systems including the most powerful blackjack systems ever devised.

People who play blackjack should be able to learn and use a good basic strategy such as the one contained on page 13 of this issue of Gambling Times. Lessons are available for those who find it hard to learn from a book or article but who can easily learn from a teacher. If you are not able or willing to learn a good basic strategy, then you will just have to put up with long-run losses, which will generally be higher than they would be in games like baccarat or craps. For in those games, the luck factor is dominant, but the casino has only a small edge against you.

With a good basic strategy properly used, you would in the long run break even in single-deck play at a typical large casino on the Strip in Las Vegas. A few of these casinos offer slightly more favorable rules than others. With four-deck play, the basic systems drop off about 0.5 percent in performance as they do with the typical Reno-Tahoe rules. Two deck play comes in between as does play at the typical downtown Las Vegas casinos in which the dealer hits a soft 17.

If you are going to play casino blackjack at least sixteen hours per year, it will probably be worth your while to move up to a higher performing level once you have mastered a good basic strategy. If used correctly, the intermediate systems are sufficient to give you a clear edge for single-deck play in Las Vegas and to allow you at least to break even elsewhere. There is no way to have a long-run edge in your favor without accurate card counting. The intermediate systems have the simplest counting systems, but the price of this simplicity is lesser performance potential than the higher level systems. However, an edge in your favor no matter how small is much better than playing against a house edge as you would do in any other casino game.

The better advanced systems listed in Table I give you a bigger edge if you can use them correctly in casino play. They are good enough to provide an edge in your favor even for four-deck play. Furthermore, they do not use any mental arithmetic beyond counting. The developers of these systems have tried to make them as easy as possible to use with reasonable performance and to keep the rules as simple as possible, If tables are used at all, they are much less complicated than those used in the professional-level systems.

At present, few players are successfully using the professional and expert systems. While reliable estimates are not available, I doubt that there are more than one hundred players in the world today who regularly use a professional system in casino play with an error rate much less than one per hour. I doubtthat there are ten players successfully using an expert system. Simply buying one of these systems doesn't make one a professional or an expert. It took me a fair amount of effort to learn to use the DHM Professional System, which I regard as the simplest of the better professional systems. In addition, I had to brush up and practice before each trip to Las Vegas. I never did bother with the side count of aces because it wasn't worth the extra effort to me. The ace count is easier and more accurate in the DHM Expert System; so I am now using that one. However, it does take more effort to learn the expert system and more practice to get up to the necessary speed before using it in a casino.

Except for the various; expert systems, there is little difference in performance between the better systems within any one class. On the other hand, there are differences in what you have to learn and what you have to do to use the various systems in the same class. Once you choose the level which is best for you, you should consider the characteristics of the better systems in that class and choose the one with which you are most comfortable.

Don't forget, you can't be sure the level is right until you have mastered the system and actually used it successfully in a casino. Since no system can guarantee that you will win in the short run, your error rate and how you feel using the system are more important in deciding whether the level is right for you than is how much you won. The safest approach is to move up only one level at a time, and you don't move up until you are absolutely perfect in the level you are using. Because the differences between a basic strategy and the intermediate systems are small compared to the differences between the intermediate systems and the higher levels, you may be able to skip one of those two levels.

If you have sufficient motivation and the appropriate skills, you may be able to make it all the way up to the expert level. If you succeed, it will take a smaller bankroll to reach the same income level and you can get by with a smaller average bet. In addition, the probability of an unfavorable session is sharply reduced. Among the expert systems listed in Table III, there are only four which I am sure outperform the better professional systems. On that basis these four can be called the most powerful blackjack systems ever devised. One of them was given that title in the article describing it. However, that system (Fristedt-Heath) was described in a way which I don't believe any professional blackjack player can use. Another of these systems, the DHM Ultimate, requires use of a computer during play; so it is not usable in a casino. The other two-HI-OPT with Multiple Parameters and the DHM Expert System-are usable in casino play.


D. Howard Mitchell is president of DHM Associates, a computer systems consulting firm. He was educated in New York City and completed his graduate work [MA in mathematics and PhD in economics) in Southern California. Besides developing better blackjack systems, Dr. Mitchell has used computers to solve a variety of problems, ranging from how to get more natural gas through existing pipelines to designing a system to process efficiently a flood of data from a Defense department satellite.

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