Algebraic Approximation of Optimum Blackjack Strategy for Card Counters (Revised)
FROM ET FAN:
|By Arnold Snyder
(From Proceedings of the Fifth National Conference on Gambling and Risk Taking, Vol. X: The Blackjack Papers, University of Nevada, Reno, 1982)
© Arnold Snyder 1980
[Acknowledgements: This paper was originally published in 1980. Subsequent correspondence with a number of blackjack experts--notably Stanford Wong, Ph. D., Peter Griffin, Ph. D., and Bob Fisher--has led me to revise the original formula and some of the original recommendations.]
The currently employed methods of computing playing strategy indices involve high-speed computers and complex programs based on the intricate complexities of probability mathematics. Though it has been almost 20 years since the first such programs were written, there is still disagreement among experts as to the most accurate methods of approximation.
To approximate blackjack strategy tables, I will take an algebraic approach, which is far simpler than computerized methods. It has been shown that playing strategy indices cannot be accurately determined by linear methods, but it is also true that current computerized methods are imprecise approximations. I am not convinced that current computer methods are more accurate than algebraic methods, in conjunction with certain linear assumptions.
My calculations are based on Peter Griffin's Theory of Blackjack1. Anyone unfamiliar with this work will assuredly not fully comprehend my methods. I will refer often to Griffin's methods, and will neither redefine nor explain those concepts that Griffin presents so clearly in his book. I do not mean to imply that Griffin in any way suggests in his book that his calculations be used as I will use them. The theories and methods herein are my own.
For those who are unfamiliar with computer methods of obtaining strategy indices, consider the math involved in computing a single hit-stand decision. Assume a single-deck game, Vegas Strip rules, with the player holding a total of 16 versus a dealer upcard of ten (hereinafter, any 10, J, Q, K will be written "X"). The player is using the Dubner (Hi-Lo) count to keep track of the cards, and wishes to know at which "true count" or "count-per-deck" standing becomes the preferred strategy.
In this counting system, 2s, 3s, 4s, 5s, and 6s are assigned a value of +1 as they are removed from the deck. Ten-valued cards and aces are counted as -1. Count-per-deck is defined as the running count divided by the fractional proportion of one deck remaining. The first consideration in solving this problem is to realize that a player total of 16 may be composed of any number of different combinations of cards. There are, in fact, 145 different combinations of cards which would total hard 16 (See Chart #1, Appendix).
Naturally, one would be more likely to be holding a combination of X-6 than a hand of A-A-A-A-2-2-2-2-4. In fact, if dealt in that order, one would simply have split A-A. In another permutation, one would surely have stood on A-2-A-A-A-2-2, a soft 20, and the decision of how to play such an unlikely 9-card total of 16 would not have presented itself. Dealt as in the table, 4-2-2-…-A, one might conceivably face this decision.
It must be determined which of these 145 combinations are relevant to the decision, according to all the rules, procedures and options of the game. For each of these specific hands, one may determine the precise advantage of hitting or standing simply by considering the outcome of every possible series of player and dealer draws (and down-cards). By properly weighting each of these possibilities, according to how probable each hand and series of events is, one would determine basic strategy for the decision of whether to hit or stand on 16 versus X.
The amount of math involved in any single basic strategy decision is vast. That a highly accurate basic strategy was originally, and painstakingly, computed on crude adding machines is phenomenal.2 Naturally, short-cuts were taken in devising this strategy. Computers made possible precise calculations, but even after decades of mathematical research, there is still dispute over basic strategy.
Julian Braun3 says to split 2-2 vs. 3 in the single-deck game. Stanford Wong4 says to hit. Both are highly qualified mathematicians and computer programmers. Peter Griffin has computed the only 100% accurate single-deck basic strategy, but this strategy has not yet been published. Dr. Griffin informs me that on this particular strategy decision, Braun's recommendation to split 2-2 vs. 3 is the correct play.
We still have not considered the calculations involved in computing the count-per-deck at which a Dubner count system player would stand on 16 vs. X, rather than follow basic strategy. There is a precise mathematical method which will accurately determine this index number. One need only determine all of the possible deck compositions which would indicate each of the various true counts, then compute the expectation from every possible series of draws, down cards, etc., for each relevant combination of cards totaling 16, weighting each outcome to reflect its probability. The enormity of this task prohibits its being carried out, even by computer. The cost of computing such accurate indices far exceeds any card counter's, system seller's, or casino's stake in the game.
Again, mathematicians have resorted to short-cuts. Rather than analyze every possible deck composition that would indicate each specific count-per-deck, the accepted method is to analyze carefully chosen representative deck compositions. The accuracy of indices so determined is dependent on how closely the chosen decks reflect true probability.
There exist now, and have always been, differences of opinion regarding the best method of choosing a count representative deck. Lawrence Revere,5 it has been pointed out by Julian Braun,6 erred in failing to remove neutral (0 value) cards when composing his deck subsets. The computer-derived indices, therefore, were all based on decks with an abnormally high proportion of neutral cards. A similar error had been made by Braun years earlier in composing decks for the Dubner count indices for Thorp's 1966 Beat the Dealer.7 Braun later corrected this error, and recomputed these indices, the corrected version of which appear in his How to Play Winning Blackjack.
Stanford Wong's indices for the Dubner count differ from Braun's. Wong argues that his method of choosing a representative deck will produce more accurate indices.8 Griffin points out the inherent limited accuracy of determining indices by using these artificially composed decks to represent all possible deck subsets. To quote Griffin, "…even the most carefully computerized critical indices have an element of faith in them." What Baldwin, et al., once did with adding machines to determine basic strategy is now being done with computers to determine playing strategy indices.
A simpler and, I believe, equally accurate approach, would be to precisely compute one set of strategy tables, by which any counting system could be measured, and indices calculated. On pages 74 to 85 of Theory of Blackjack, Peter Griffin provides the precise information necessary to calculate such indices by algebraic methods.
The formula is simple. Divide the favorability of no action (i.e., not hitting, not splitting, etc.) by the total effect of the count-valued cards. To obtain count-per-deck, simply multiply this by the sum of the squares of the points counted per deck. One thus obtains the critical index at which the action pivots from favorable to unfavorable, or vice-versa. (One must also account for the sum of the removed card(s)' point values, and adjust this count to reflect "count-per-deck").
The complete formula looks like this:
(mp/i) + t = count-per-deck for altering strategy
m = "mean" or "favorability", which Griffin presents in the eleventh column of his tables. It is necessary to reverse the sign (+/-) of Griffin's "mean", since he is quoting the favorability of making the action (hitting, etc.)
p = sum of the squares of the "points" counted per deck. Example: for the Dubner system, counting +1 for 2, 3, 4, 5, 6; and -1 for X and A; p=40. For Hi-Opt II counting +1 for 2, 3, 6, 7; +2 for 4, 5; and -2 for X; p=112. Simply multiply the sum of the squares of the points of the 13 different cards by 4 (for the four suits).
i = the "inner product" of the count system's point values and the effects of removal. These effects are listed in Griffin's tables. He also explains the method of calculating the inner product (p. 44).
t = the sum of the point values of the removed cards, adjusted for "true" count-per-deck.
(This formula is identical to the one in my original paper except that here I recommend p = the sum of the squares of the point values. In the original paper I recommended p = the absolute sum of the point values. For any level one count, such as the Dubner/Hi-Lo count, as will be shown in this paper, either valuation of p will produce identical indices, since 12 = 1. The few discrepancies between the charts in this paper and those of the original paper are due to slight computational and typographical errors in the original charts, discovered by Bob Fisher.
I also published a correction sheet for the first paper, which advised multiplying by "a", where a = the average point value of a counted card. With the new formula, this methodology is not advised. Both the original formula and this revised variation of it will produce identical indices for level one count systems and nearly identical indices for higher-level counts.
The new formula was developed by considering how the formula might best be applied to determining insurance indices, using Griffin's data on page 71 of Theory of Blackjack. Stanford Wong, who originally questioned the formula's validity for higher counts, pointed out to me that insurance indices were optimally calculated according to Bayesian principles, multiplying the point values of the various cards by their respective probability of being drawn. This inevitably produces a weighted count in which the ratio of the count values to one another is identical to the ratio of the respective values if all values of the count were simply squared.)
Example: 14 vs. A, single-deck, dealer stands on soft 17, using the Dubner count:
m = 18.85 (from column 11, p. 74, Theory of Blackjack)+ .44 (effect of removal of dealer's ace, p. 74, Theory of Blackjack)= 19.29 (+/-) = -19.29
Thus, a player using the Dubner count should stand with a total of 14 versus ace at a count-per-deck of +12.4. On page 137 of How to Play Winning Blackjack, Julian Braun gives this index value as +12. On page 169 of Professional Blackjack (1980), Stanford Wong gives this index as +13. To demonstrate the effectiveness of this simple formula, I will produce all 38 hit-stand indices that Braun records on page 137 of his book. Wong's indices are on page 169 of his 1980 edition. So that my work may be easily checked, I will provide the single-deck values for m and i, with the dealer upcard removed, calculated as previously explained. In all cases, p = 40 and t = the point value of the dealer's upcard. Dealer stands on soft 17. (See charts #2 and #3, Appendix.)
Inserting the corresponding values from Charts #2 and #3 into the formula and solving, the complete single-deck hit-stand strategy table looks like this:
Comparing these indices to Braun's, my table as a whole is quite similar. Only 4 of the 38 algebraic indices differ from Braun's by more than 1 point, when the algebraic indices are rounded to whole numbers. All of these major differences are between double-digit indices, so are not highly significant from the standpoint of player expectation. Many players do not even memorize double-digit indices. Comparing the algebraic indices to Wong's, again the table is remarkably similar. If I round all algebraic indices to the nearest whole number, only one index value differs by two points. This index value is for 16 vs. 6, for which Wong gives -12, and which the algebraic formula determines to be -13.6.
I will point out that this formula will produce index values for some decisions for which no index value actually exists (such as, with this count, 14 vs. X). Such index values will for the most part be double-digit indices that would not contribute to any notable loss of profit because of their rare application.
Consider the problem of "rounding" indices to whole numbers. Griffin has noted that this practice may introduce up to a 10% error in playing decisions. Few players can estimate a count-per-deck within fractions of a point, so indices are recorded as whole numbers. It I take liberty in rounding the algebraic indices to whole numbers in Wong's "direction," be it up or down (so that -13.6 may be rounded to -13), only 9 of the 38 algebraic indices differ from Wong's by one point, while the other 29 are the same. Note that Braun and Wong differ on 16 of these indices, four of them by 2 points.
One of Wong's points of contention with Braun's methodology is that Braun used linear methods (interpolation and extrapolation) to determine his four-deck strategy. Ironically, Braun's and Wong's four-deck strategies more closely resemble each other than do their single-deck strategies, where Braun's indices are not linear based. In his newsletter, Wong presents evidence that his methods of choosing his representative deck subsets are more accurate than Braun's.
Wong's arguments appear logical, but I have made no thorough comparative examination of their methods. Likewise, I would not recommend a player use algebraic indices instead of the computerized indices of a qualified expert like Julian Braun or Stanford Wong. I make no claim for the "superiority" of the algebraic formula. It would be of practical use to a player who desired to play a count for which reliable strategy tables were not available, or were incomplete, or were available only at a price the player did not wish to pay.
In any case, considering the extreme approximation technique of creating a "most likely" deck with a double-digit true count, I see no mathematical argument that -12, as per Wong, would be more accurate than the algebraically determined -13.6, and this is the most radical difference between any of Wong's and the algebraic single-deck hit-stand indices. What most surprises me is that a simple algebraic formula would so closely mimic the results of simulation-based data. (Readers familiar with The Blackjack Formula9 will note that I am essentially doing "more of the same" to determine indices as I did to determine profit potential. Having reduced the problem to the fewest number of variables, I make simple algebraic assumptions.)
To demonstrate the uncanny precision of this mimicry, we may use the algebraic formula to determine indices for specific player hands versus dealer upcards. Wong, on page 171 of his book, quotes the "two-card" hit-stand indices for the single-deck game, dealer stands on soft 17. For instance, both Wong's computer and the algebraic formula determine the critical index for 13 vs. 2 to be 0. However, Wong's "two-card" table shows that for the player holding X-3 or 9-4, the correct index is +2. With 8-5 the index is -2. With 7-6: -3.
Applying the formula to X-3 vs. 2:
As per Griffin (p. 86, Theory of Blackjack)
Similarly, solving each of the "two-card" player hands, and comparing the results to Wong's:
When comparing the algebraic results to hundreds of Wong's player "total" and "two-card" indices, including pair-splitting and double-down decisions, for both Wong's Hi-Lo and Halves counts (where p=44, and all values for both i and t were recalculated), where dealer both hits and stands on soft 17, the single-deck algebraic strategy is so similar to Wong's, it would take a computer simulation of many millions of hands to determine which strategy is actually superior.
The algebraic formula proves even more precise in mimicking computer-produced indices for multi-deck games, simply by weighting the removed cards according to diminishing effect. It is most convenient to simply calculate an infinite-deck strategy, and interpolate indices using the reciprocal of the number of decks (see Griffin, Theory of Blackjack, p. 115 and 127). There is very little difference between interpolated indices and indices calculated for the specific number of decks by the algebraic formula. Approximation of infinite-deck indices is quite a bit easier than calculating single-deck indices. The formula becomes simply mp/i, since t is irrelevant to an infinite number of decks.
We may further simplify by valuing p=10, and calculating i on the basis of each of the 10 different counted cards. These values are in Chart #4, Appendix.
I will point out here that an "infinite deck" is not only an impossibility, but that if one were keeping a running count of cards as they were removed from an infinite number of standard 52-card decks shuffled together, one's efforts would be pointless since "true" count would inevitably always equal 0. The term "infinite deck" is used simply to mean that the removal of any one card (dealer's upcard, in this example) will not in itself alter the ratio of the various cards to one another.
An infinite deck with a true count of +13, as per this example, means that an artificial deck would have to be created by removing "low" cards and adding "high" cards proportionately to obtain an "infinite" deck in which the ratio of the "low" cards to "high" cards would indicate that for every 52 cards in the deck, an average count of +13 would be the sum of the assigned point values. In such an artificially composed deck, one's optimum strategy would be to stand on 14 v. A. In multi-deck games, infinite deck strategies are quite accurate, since the removal of any individual card has far less effect on deck composition than in a single-deck game.
The complete infinite-deck hit-stand table looks like this:
Comparing these indices with Wong's 4-deck indices (p. 173, Professional Blackjack, 1980 ed.), rounding to the nearest whole number, no algebraic index differs by more than one point. Note that 20 of Wong's single-deck indices change when computing for 4 decks, fifteen by 1 point, three by 2 points, and two by 3 points. The algebraic formula follows Wong's changes with notable precision. When I interpolate a 4-deck strategy chart (or compute a 4-deck chart by properly weighting the effects of the cards-either method producing almost identical results), the 4-deck indices are slightly closer to Wong's 4-deck indices than are these infinite-deck indices, and no algebraic index differs from either Wong's or Braun's 4-deck indices by more than one point.
Griffin's tables may thus be used to determine a highly accurate strategy for any balanced point-count system, for any hit-stand, hard or soft double, or pair-splitting decision, for any number of decks, for both soft 17 rules. Using the surrender data Griffin provides on pages 121 and 122 (Theory of Blackjack), one may easily calculate the favorability (m) for both early and late surrender; however, Griffin does not provide the effects of removal for either surrender option. It is not correct to use the effects on pages 74-85 to calculate the value of i for surrender decisions. The effect of removing any card on one's hit-stand or pair-splitting decision, for which Griffin supplies data, will naturally differ from the effect of removing that same card on one's surrender decision.
One other common rule variation for which data is not supplied in Griffin's tables is pair splitting when doubling after splitting is allowed. Nor does Griffin provide data on splitting X-X vs. upcards of 4, 5 or 6-occasional plays for single-deck players. Dr. Griffin has informed me that the effects of removal for doubling down on A-9 vs. 4, 5 and 6, respectively, may be used to calculate these indices with notable accuracy. For the most part, Griffin's "Virtually Complete Strategy Tables" are aptly titled.
Over a weekend, using a programmable calculator, I devised relatively complete 1, 2, and 4-deck strategy tables for Hi-Opt II, Revere's Point Count, and Uston's Advanced Count. It was somewhat tedious, but consider the time and money required for computerized methods. I believe these indices to be as accurate as any devised to this point for any point count system.
I will not, by the way, make these strategy tables commercially available. In my opinion, no serious player should be without Griffin's book, which is all one needs to compute such tables. The calculations I have explained are not difficult. One need not comprehend the more advanced math of Griffin's appendices to produce strategy tables according to the algebraic method, though again, I will emphasize, one would need Griffin's book to understand and apply the methods I am proposing.
A few fine points: any time the inner product (i) is negative, as in all hit-stand decisions, the critical index for changing strategy will indicate the point at which the action (i.e., hitting) pivots from favorable to unfavorable. Any time i is a positive number, as in most double-down and pair-splitting decisions, the critical index will indicate the point at which the action pivots from unfavorable to favorable. Chart #4 (Appendix) also gives the values of i for the most common doubling and pair-splitting decisions, Dubner count, p = 10, infinite deck.
Note that of the 37 most common double-down and split decisions, only 8-8 vs. X has a negative value for i, accurately indicating that for this decision, one will be determining the critical index at which the action becomes unfavorable. This infinite-deck index value, by the way, works out to be +5.2-comparing it favorably to both Wong's and Braun's 4-deck index value of +6.
Of the 37 infinite-deck doubling and splitting indices that may be easily calculated from Chart #5, using Griffin's tables for m, 26 round off precisely to Wong's 4-deck decisions. The algebraic method consistently produces index values comparable to computer-derived values for every decision I have tested-and I have tested many examples for every type of decision for which Griffin provides data. The indices that Wong changes when switching from his Hi-Lo to Halves counts likewise change when calculated thus algebraically.
I will note that a major difference between Wong's and Braun's indices, relative to the algebraic indices, is that Wong's methodology produces indices which correspond more precisely to indices produced via algebraic and linear assumptions. An objective examination of both Wong's and Braun's methods will, no doubt, be done in time. Should Braun's methods prove superior, then certain assumptions regarding algebraic error, depending on how Wong erred, may be made. Should Wong's methods prove to be superior, the theoretical implications are interesting.
In my original version of this paper, I stressed the major value of the algebraic formula would be for players who play in single-deck games, and whose current systems do not provide "two-card" hit-stand decisions. From Griffin's table of "Average Gains for Varying Basic Strategy" (p. 30, Theory of Blackjack), note that some hit-stand decisions alone are worth more than all pair-splitting decisions combined.
Some of these most valuable hit-stand decisions, such as the 13 vs. 2 previously analyzed, can be most efficiently played in single-deck games by using two-card decisions. It occurred to me that a player might potentially realize more profit in a single-deck game from learning two-card decisions for 16 vs. X, and both 12 and 13 vs. 2 and 3, than by learning all hard and soft doubling and pair-splitting decisions combined.
From the practical point of view, the only pair-splitting indices worth learning at all are splitting X-X vs. 4, 5, and 6. Of the doubling indices, only 10 and 11 vs. X, and 11 vs. ace are worth varying basic strategy for. A sophisticated player would memorize strategy indices according to potential profitability.
Since publication of my original paper, Peter Griffin has pointed out to me that the method of computing the gain from learning two-card indices as opposed to learning a single non-composition-dependent index for any decision is to calculate the correlation of the count system for each two-card hand to obtain a weighted average correlation for the decision, and comparing this to the correlation of the non-composition-dependent decision. Learning two-card indices is not, alas, practical, as such strategies will not raise the simple correlation of the count system sufficiently to warrant the increased memory effort.
I will note, however, that the recommendations of most systems developers to learn and utilize strategy tables for pair-splitting, surrender, and most double-down decisions are ill-considered, since the potential gains from such strategies are so negligible that most players should not chance making errors by attempting to employ such indices. The information provided in Theory of Blackjack, in conjunction with the formula presented in this paper, is more than sufficient to develop a count strategy for any balanced count system, as complete as any player could practically apply at the tables.
Until system sellers analyze and incorporate into their systems the wealth of information in Griffin's Theory of Blackjack, serious players should study this book themselves.
As I pointed out in The Blackjack Formula, the financial opportunities of blackjack, as a "get-rich-quick" racket, are largely imaginary. The very effective casino countermeasures, which Thorp had predicted in his 1962 edition of Beat the Dealer were inevitable, have been largely ignored by most systems sellers since. Casinos have learned that it is in their interest to keep the blackjack profit myths alive. However, many casinos do not offer games exploitable to any profitable end by card counting. The profit potential of even the best games available in this country can only be realistically taken advantage of by highly knowledgeable players with sizeable bankrolls. A present-day professional card counter must enter the game with the same attitude, preparation, inside information, and ability to withstand fluctuations, as any investor of sizeable amounts of money on Wall Street.
In his bibliography for Theory of Blackjack, Peter Griffin states that if he were to recommend one book, and one book only, on the subject, it would be the 1966 edition of Thorp's Beat the Dealer. My recommendation for a second book on this subject would be, without question, Griffin's Theory of Blackjack. For the serious blackjack student, Griffin's work stands alone as a detailed analysis of the probabilities and possibilities of applied blackjack strategy.
Chart 1, All Combinations of Cards that Total Hard Sixteen
Chart 2, m (Favoribility) 51-card Deck, Dealer Upcard Removed (+/-)
[Note: If you are the kind of math geek, like me, who can actually make it to the end of an article like this, you may be interested in The Blackjack Shuffle Tracker's Cookbook, in which I use algebra to calculate the value of various casino-style shuffles to a shuffle-tracker, and got results that overturned the conventional wisdom of the time. — Arnold Snyder]
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