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Basic strategy and expectation in casino Blackjack
ORGA1NIZATI0iXTAL BEI-IAVIOR AND I=[U1VIAi'~ PERFORMANCE 19'~ 413--428
(1974)
Basic Strategy and Expectation in Casino Blackjack NICHOLAS A. BOND, JR. California State University at Sacramento Fifty-three Blackjack gamblers in four Nevada casinos were unobtrusively observed, and scored according to whether their playing decisions corresponded to the optimal "Zero-Memory" ]Basic decision strategy. Of the 940 hands recorded, at least 16% were played differently from the Basic prescription; presumably as a result of this "nonoptimal" play, average losses at Blackjack were quite high, on the order of ten times the theoretical loss rate for t~asic play. Also, by wagering small bets in a subfair game, Blackjack gamblers practically guaranteed loss of their betting capital to the casinos.
Blackjack, or "21," is the only casino game which offers the public an expected return greater than zero. I t is also the only game t h a t allows the player to m a k e tactical decisions during play: Unlike the Craps and Roulette bettor, the Blackiack gambler can significantly influence the likelihood of his winning by his choice behavior after play begins. The decisions required of the player are varied, and in some cases they defy common sense. Furthermore, the game is fast, and there are several possibilities for distributing money over the hands and the side bets. These properties m a k e the game an intellectual challenge, and they also lead to gambling behavior t h a t is complex. This study sought to record significant aspects of t h a t behavior in a commercial casino setting. I n N e v a d a Blackjack, a dealer represents the "house," and there are one to six players. Each card from 2-10 in the standard 52-card deck receives a value according to its number; picture cards have a value of 10; Aces can have a value of either 1 or 11, at the player's option. A hand is "soft" if it contains an Ace and t h a t Ace can be counted as 11 without the total exceeding 21. Thus, an A-6 is a "soft 17." All aceless hands are " h a r d " ; 9-8 is a "hard 17." After bets are placed, each player and the dealer receive two cards. One of the dealer's cards is exposed, the "up" card; the other card is "down" or concealed. The player's object is to beat the dealer's numerical total, without exceeding 21. The player can request additional cards or "hits" as long as his total is less t h a n 21 ; when he is satisfied, he "stands." 413 Copyright © 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.
4]4-
NICHOLAS A. BOND; J R .
If the player's cards total more than 21, he "busts" and loses his bet immediately; if his cards total less than 21, and the dealer busts, then the player wins the amount of his original bet. An original two-card combination of an Ace and a 10-valued card is a Blackjack, and wins over all other hands; if a player gets a Blackjack, he is paid 1.5 times the amount of the bet. In a tie hand between dealer and player, no money changes hands. Under some circumstances, there are special choices available to the player: He can split pairs of the same numerical value, such as two 6% or two 8's, and place the value of his original bet on each of the two "new" hands, or he can "double down," in which case he doubles the initial bet and draws just one additional card. There is also an "Insurance" option when the dealer's up card is an Ace; the player can bet that the dealer's down card is a 10 card, and if a 10 is concealed, then the player receives double the amount of the bet. Figure 1 gives a flow chart of the binary decisions that the player must make on every deal. For the majority of hands, the stand-or-draw choice is the most significant decision for the player. The dealer has to play according to a mechanized strategy: He must stand on dealer totals of 17 or better, and must hit until then. The best strategies for playing Blackjack involve detailed memory and require the counting and weighting of each card played from the deck.
I
Do you have a pair? I
)o
the pair?
Double down? Yes
Draw?
;I
Yes~
Is it a pair of Aces?
1
Stand (End)
FIO. i. Flowchart of player decmions in B1aokjaok. Modified ~om Beat the Deale~ by EdwaM
Thorp, 1966. Copyrighted by Random
House, New York, 1966.
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When the depleted deck exhibits a composition which is favorable to the player (say when all the 5's are gone, or when the deck has a relatively high proportion of 10's), then bets may be increased according to the favorability of the player's expectation, with minimum bets being placed at other times. Machines and computer programs have been designed to evaluate the effects of cards as they are played, and positive expectations on the order of +.05 can be achieved with their aid (Epstein, 1967). Casinos do not, of course, permit such devices to be used in actual play. Human memory cannot store all the data or perform all the calculations necessary for truly optimal play, but approximate systems have been devised which approach the optimum and which are feasible for a determined player to use (Thorp, 1966; Epstein, 1967; Revere, 1973). In these systems, tile player does not have, to compute and store the precise effects of every card that is exposed; for example, he may need to count and remember only the "highs" (10's and Aces) and "lows" (2's through 6's) that remain in the deck, and bet an amount roughly proportional to the high-low difference. Even when simplified, however, these card-counting systems are rather difficult to practice, and require attention to every card dealt, during the game. "Zero-memory" or "Basic" strategies have also been derived (Baldwin, 1956; Epstein, 1967; Revere, 1973; Thorp, 1966). These strategies are much simpler: They assume that a single player competes against the dealer, and that only three cards are known (the player's two cards and the dealer's up card); no count is kept of previous cards played in the deck, hence the term "zero memory." Gains or losses from the hit, stand, split, or double-down decisions have been estimated from computer simulation of many Blackjack hands in the remaining 49-card deck, and the player is advised to take the decision with the highest simulated expectation. Since the player's two cards can comprise 55 different two-card value combinations, and the dealer's up card can take on any one of ten values, there are 550 possible card arrangements facing the player. For each one of these he must be able to decide quickly whether to draw, stand, split, or double down. A player using the optimalzero-memory strategy does not really have to memorize a 55 × 10 matrix; a 28 X 10 table is sufficient, as shown in Table 1, taken from Epstein (1967). This reduction is possible because some of the different hands are treated the same way ; a player holding of a 9 and a 6 is treated exactly like an 8 and a 7. Remembering the 28 × 10 decision table can be facilitated by rules which specify several table entries: "Always stand on hard 17 or better," and "Always double down on 11." Since some casinos do not permit certain bets, such as soft doubles, the decision table
416
NICHOLAS A. BOND, JR. TABLE 1 SUMMARY OF OPTIlYIAL ZERo-MEMoRY STRATEGYg
Dealer's up card 2
3
4
5
6
7
8
9
10
A
X X X X X D D X X X
X X X X X D D X X X
X X XI X X D X X X X
X X X X X D X X X X
X X X X X
X X X X X X
X X X X X X
X X X X X
S S S
S S
S
S
S
S
S
S
Player's hard total 17 16 15 14 13 12 11 10 9 8 7
a X D D D X X
Xb D D D X X
~ D D D X X
D D D Xd X
D D D D~ X
X X X X X D D X X X
Player's soft total 19 18 17 16 15 14 13
D X X X X
D D X X X X
D D D D D D
D D D D D D
D D D D D D
X X X X X
S S
S S
S S
S S S
S S
S S
S S S S
S S S S
S S S S
S S S S
S S S S
S S S
S
S
S
S
S
S
Player's pair 2's 3's 4's 5's 6's 7's 8's 9's 10's A's
a Draw with hand (3,10) vs a 2. b Stand with hands (3,9), (4,8), and (5,7) vs a 3. c Draw with hand (2,10) vs a 4. d Double down with hand (3,5) vs a 5. Draw with hand (2,6) vs a 6. ] Stand with hand (7,7) vs a 10. • g Symbolism: X = draw; D = double down; S = split. Note. Reprinted from Theory of Gambling and Statistical Logic, by R. A. Epstein, 1967. Copyrighted by Academic Press, 1966.
STRATEGY AND EXPECTATION IN CASINO BLACKJACK
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is reduced even further. The zero-memory Basic strategy can be learned in a few hours and played at casino speed after a little practice and observation. The Basic strategy is strictly optimal only for the first play of the game, because it assumes a 49-card deck with equal probabilities of drawing each remaining card, and this assumption is palpably false with a depleted deck. Nevertheless, Basic gives a surprisingly good return. It yields a theoretical expectation of about +0.0015 under the best Las Vegas rules, and an expectation of about -0.0055 for South Lake Tahoe and Reno casinos (Thorp, 1966). All four casinos at South Tahoe forbid doubling down on soft totals, and on all totals after pair splitting; also, the dealer draws to soft 17 instead of standing on any 17; and three casinos forbid doubling on hard 9. These rules work slightly against the player, and together they reduce his expectation by about 0.006, from +0.0015 at Las Vegas casinos to -0.0055 at South Tahoe. In the only empirical test of Basic that has been reported, Baldwin et al. (1957) sustained losses of $26.00 over 3530 hands, for an expectation of about -0.007. Thus, a player who follows the optimal zero-memory strategy over a long series has about an even-money bet. On the Las Vegas Strip, he could anticipate winning an average of $1.50 per 1000 plays at $1 each; in Northern Nevada, because of less favorable rules, he should anticipate losing $5-6 in 1000 plays, a rather attractive return for a strategy that does not require counting cards or calculating ratios, and that works iust as well when the cards are continually reshuffled. And, it does offer the best possible play, as long as information is confined to the dealer's up card and player's hand. Given that the Basic strategy offers a relatively good casino gamble, is widely known, and is easy to learn, how many Blackjack players follow it? To explore this question, the present study recorded actual Blackjack gambling decisions. As casinos do not ordinarily permit observations of their customers' behavior by outsiders, our data-collection plan had to be completely unobtrusive, with no clipboards, recording devices, or questionnaires in sight. It was also decided to concentrate on player strategy and win-loss return or expectation, and to ignore other features such as the timing of gambler decisions, superstitions about sitting to the far left of the dealer, size of the gambling bankroll, and so forth. METHOD Research Site
Observations were carried out in four commercial casinos at South Lake Tahoe, on the California-Nevada border. This is one of the prin-
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NICHOLAS A. BOND~ JR.
eipal Blackjack gambling areas of the world; the game is played around the clock, with hundreds of players at the tables in afternoon and evening hours. Fortunately for observational purposes, the four casinos are on adjoining lots, just a few yards from each other.
Sampling Plan The population of interest was the complete set of Blackjack hands played during 5 days in the summer of 1973, at the four South Lake Tahoe casinos during the hours of 10-2 AM. The characteristics of interest were (a) the proportion of hands played according to Basic, and (b) the net return in bets won or lost. There is no list or count of all these Blackjack hands; the total count of dealer hands is probably around 600,000, with a total number of gambler hands on the order of 1.75 million (70 hands per hr × 3.1 gamblers per table ;4 100 tables × 16 hr × 5 days = 1,736,000). With no master list, it was impossible to take a simple randora sample of these hands, and facilities were not available for monitoring simultaneously a large number of tables in the casinos. Therefore, a two-stage cluster sampling procedure was set up which designated the active Blackjack tables as the primary sampling units (psu's), and chose a random sample of n tables from the total of N psu's. After that, a small number m of hands were selected from the total number M at each table in a second-stage sampling (Raj, 1968). This procedure proved to be feasible, since a complete list of active psu's (tables) could be prepared, checked, and sampled just before observing began; the actual observations were then concentrated in only a few tables. In midsummer 1973 there were altogether 206 Blackjack tables in the four casinos, but for much of the time only about half of them were operating. A map layout of these tables was made and each table was assigned a three-digit number from 001-206. There were 5 observation days, with each one running from 10-2 AM. Two on-the-hour starting times were randomly chosen each day from the 16 hr. On a typical day the starting times were 10 AM and 6 PM. Just before 10 Am, a random drawing of several three-digit numbers between 001206 was made, with the results listed in order of drawing. Then a quick count of active Blackjack tables operating in all four casinos was taken. The number of tables actually observed was the integer nearest to ½o of the number of tables operating; thus the sampling fraction fl of psu's (tables) was .05. If 55 tables were running at 10, AM, then the first three table numbers from the three-digit series were designated for that period. During the early evening hours, say at 6 ~M, 132 tables were operating, and so a randomly selected set of seven tables was chosen for study starting at that time. This procedure assured that active tables were
STRATEGY AND EXPECTATION IN CASINO BLACKJACK
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represented according to their numerical prevalence in the four casinos. It turned out that about 40% of the tables selected were in the largest casino, and about 4% of the tables were in the smallest casino. After the sampling decisions were made on how many and which tables to study, the observer proceeded immediately to the first table in the list. He then randonlly chose one of the six seats, by rolling a die, and watched exactly 20 hands which were played by the gambler(s) sitting in that seat. If a seat was vacated during the 20-hand series, then the observer shifted his attention to the next seat to the left if the first random number was odd, and to the right if it was even. The 20-hand segment, which is presumed to be a random sample of the hands played at that table during the sampling period, represents a very small sampling fraction indeed; the number of hands played at a continuously operating table over a 16-hr period is variable, but will probably average above 3000 (3.1 gamblers × 16 hr × 70 hands per hr = 3472). Thus, the secondary sampling fraction f2 might be estimated as (20 × 2)/3472 or about .011, with the estimated ]lf~ product, or total sampling fraction, as .05 × .011 = .00055 (Raj, 1968).
Recording Procedure UsualIy, the observer could see the player's "down" cards just by standing behind him; there are many people who watch the games and look at the cards, and there is no reason for the player to conceal them anyway. If the player being observed did temporarily conceal his cards, the observer had to remember the dealer's first up card, wait for all cards to be exposed at the payoff, and then infer whether the decisions made by the player were Basic plays. Counting conventions were as follows. If either the dealer or the player received an immediate Blackjack, the player was considered to have played according to the Basic model even though he didn't "play" at all; when a pair that should have been split was not split, that hand was counted a non-Basic play, but when an opening pair was (properly) split, the two resulting hands were analyzed and counted separately. There were no "excluded" hands; all were counted. A three-number mental coding scheme was used to keep track of the decisions and the returns for the 20-hand series. The first number was the number of the hand being playedl and it varied from 1-20; the second number was the number of hands out of this block of 20 which conformed to the Basic strategy; and the third number was the return enjoyed by the player. This last figure was simply the aggregate number of bets won or lost, not the actual amount of money bet; the procedure estimates the single,trial probability of winning. Supposing for illustra-
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NICHOLAS A. BOND, JR.
tion that the sixteenth hand had just been played, then the observer might have been counting as follows: 16-13-0ff 2. At this point, 16 hands have been played, 13 of them were played according to the Basic (3 were not), and the player had a net loss of 2 bets. No recording reliability cheeks were made in the casinos; however, the observer's accuracy was tested in a noneasino setting. Two-hundred hands were dealt at about a hand a minute, and the observer (mentally) recorded the results in his three-number code. After each block of 20 hands, he wrote down the three numbers (gambling hands, Basic plays, net returns). Meanwhile, two assistants independently made objective records of each card dealt, and then reconstructed the whole series of 600 numbers. When the observer's reports were compared with a true record of these 200 hands, three errors were found; all were only one digit off. This suggests a recording error on the order of 1-2% for noneasino conditions. After watching 20 hands at one table, the observer wrote down the results of the 20 preceding observations, reset his mental counters to 0-0-0, went on to the next designated by the random series, and continued observing. The whole procedure must have been quite unobtrusive, as the observer was never questioned by casino employees or by any of the players he watched. In two instances, the table randomizing scheme called for observing isolated tables where there was only one player. Steady peeking over a single player's shoulder for 15 min might have been noticeable. So for those two occasions only, the observer sat at the table himself and played the Basic strategy with minimum bets, while mentally recording the other player's results for 20 hands: Interviews
Under casino conditions, it was impossible to obtain a probability sample of Blackiaek players for interviews. But a few specific questions were addressed to a grab sample of the gamblers. Whenever the schedule permitted, the observer would wait by the table after finishing a block of 20 hands. If the player left the table within 20 rain, the observer followed the player, and made an opening remark to the effect that he had seen the player refuse an Insurance bet at the table. This refusal was so common that it was safe to assume the player really had turned down such a bet. The interviewer, who had a Blackjack instruction booklet in his hand, then explained that he was trying to learn the game, and was: a little uncertain about some of the plays. If the player seemed cooperative and willing to talk, then the following six questions were addressed to him.
STRATEGY AND EXPECTATION IN CASINO BLACKJACK
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1. When do you take the Insurance? (Correct answer: Never take Insurance unless the ratio of non-10's to 10% in the deck is less than 2.00). 2. W h a t do you do when you have 7-7 and the dealer has a 6 up? (Correct answer: Split the pair). 3. Do you hit (the player's) 12 when the dealer has a 6 up? (Correct answer: No). 4. W h a t is the minimum hard standing number you ought to have if the dealer has 8 up? (Correct answer: 17). 5. Over at Casino X, they let you double down on hard 9; when do you do that? Do you always double on 10 and 11? (Correct answer: When dealer has 2-6 up; always on 11; always on 10 except for 10-A up). 6. When do you split a pair of 9's? (Correct answer: Always except for 7, 10, or A up). Fourteen gamblers were interviewed; answers to each question were scored as right or wrong. RESULTS A total of 940 Blackjack hands (4:7 sets of 20 hands each) was recorded for the 5 days; the slowest day had 140 observations, the busiest day had 240. These hands were played by 53 people, 29 males and 24 females, all presumably above 21 y r of age. The minimum bet placed on a single hand was $1 and the maximum bet was $100; a large fraction of the bets, probably a majority, were either $2 or $1.
Basic Play Of the 940 hands observed, 793, or about 84%, were played according to the Basic decision table shown in Table 1. For a 20-hand sequence, the mean number of Basic plays was 16.8 with a SD of 1.75, n =- 47. There were three "perfect" series of 20 hands each, with all decisions in accord with the Basic strategy; the "worst" player made Basic decisions on only 13 of his 20 hands. The 84% figure was inflated by scoring all "natural" Blackjacks (those occurring on the first two cards) as Basic plays. In a head-to-head game with one player and the dealer, naturals occur about 10% of the time. When the natural hands were eliminated, only about 75% of those played were zero-memory optimal. In two-stage cluster sampling, the best estimate for the population mean is the sample mean. Hence for the total population of more than 1.7 million hands, we anticipate that about 84% were Basic plays on the liberal scoring rule, with about 75% Basic plays when naturals are ex-
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NICHOLAS A. BOND~ JR.
eluded. Population variance estimates can be made by combining the variabilities of second-stage units (hands) inside psu's (tables) with the variance of psu totals around the grand sample mean (Raj, 1968). Under reasonable assumptions about sampling fractions, the standard error estimate of the population proportion was less than 2%. W h y were there so many non-Basic (and presumably nonoptimal) decisions? Table 2 indicates that many of the gamblers simply did not know the correct plays for some of the nonobvious situations. Interview responses to items 2-6 were about equally split between rights and wrongs. Item 4 was the only one for which everybody knew the right answer; and this item was designed to be easy, since it can be handled by a simple rule: "Always stand on hard 17 or better." Some of the gamblers did not appear to know or to use the Basic strategy in a strict way. T h e y probably memorized a few rules that reproduced most, of the Basic table, and then played these pretty consistently. But when borderline decisions came up, the player often wavered and changed his tactics from one occasion to another. The mental recording system did not permit fine-grMned error tabulations, but m a n y stand-hit mistakes were noted on player totals of 12 against dealer up cards of 4-5-6; the player with 12 should stand, except when he has 2-10 against the dealer's 4. The A-6 "soft 17" hand was also something of a puzzle and was often wrongly accepted as a suitable soft standing total; double-down and pair-splitting opportunities were sometimes ignored. Item 1, on the Insurance option, was perhaps the most revealing one of the six interview questions. When the dealer has an Ace showing, the gambler can enter a side wager that the dealer has a 10 as hole card. If the dealer in fact has a 10 in the hole, then double the amount of the side bet is paid to the gambler. Now in evaluating the Insurance bet, an analytical gambler might reason as follows: The dealer has an Ace up; I have two cards in my TABLE 2 RESPONSES TO INTERVIEW QUESTIONS Item
Correct
Incorrect
n
1
1
2 3 4 5 6
2 7 14 9 3
13 12 7 0 5 ll
14 14 14 14 14 14
36
48
Total
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hand (assume these are non-10's). There are 49 cards left in the deck, of which 16 are 10's, so tl~e chance of the dealer's hole card being a 10 is 16/49, and the chance of it being a non-10 is 33/49. If it is a 10, I get twice my bet back; but 2 X 16/49 = 32/49, which is still less than 33/49. Therefore, I won't take the bet. The Insurance bet is even worse for the gambler if he has one or two 10's in his own hand, since that reduces the dealer's likelihood of getting a 10 to 15/49 or 14/49. And thus a general answer to Item 1: Never take the Insurance bet, unless you are counting cards and are reasonably sure that the proportion of non-10's/10's is ~2.00. Only one respondent gave the right answer to this question. A frequent response was t h a t the bet was usually refused, but was accepted sometimes "when the cards are running right." Expectation
The 53 gamblers had net losses of 55.5 bets during their 940 plays, for an expectation of about - . 0 6 . Over a 20-hand series, the expected loss was 1.18 bets (SD = 1.7, n = 47). Seventeen of the 20-hand blocks won and four broke even, so the overall probability of break-even-or-better was 21/47 or 0.45, with a likelihood of 17/47 or 0.36 of making an overall profit. Conditional probabilities of break-even-or-better improved to 0.56 if 18 or more of the 20 plays were Basic. As far as payoff is concerned, Blackjack can be considered to be a simplified Red-and-Black game (Dubins & Savage, 1965), where the gambler can stake any amount in his possession on the outcome. He wins back his stake and as much again with probability p, and loses with probability q. Let p = 0.47 and q = 0.53, so t h a t the expectation corresponds with our results. Then for unit bets in n plays of the game, the mean expectation is n ( p - q ) , or - 0 . 0 6 n ; the variance is n [ p + q (p _ q)2] = 0.996n. For 100, plays at $1 per hand, a gambler could expect to lose $5 or $6. With a standard deviation of a little less than ten, he also can expect to come out ahead only about 20% of the time he plays out the 100-hand series. The expectation for Basic play at South Tahoe was about -0.006, yet the gamblers in this sample were losing nearly ten times t h a t much, on the average deal. These losses were surprisingly high. If such losses are indeed characteristic of the population of Blackjack hands, then Craps and Roulette, with best-bet expectations of - 0 . 0 1 4 and -0.053, respectively, m a y yield better money returns to the casual gambler, even though these games are relatively trivial and are completely impervious to player strategy. When the losses of the most unfortunate 7 sequences of the 47 are removed, then average losses
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NICHOLAS A. BOND, JR.
were reduced to about 3%. Thus less than a quarter of the players lost half of the bets. As played by these gamblers, Blackjack was a subfair game, since p < q (Dubins & Savage, 1965). Anybody who plays a subfair game can expect to be ruined if the game goes on long enough. The controlling ruin parameters are the probability of winning p, the resources available to the player and the house, and the betting policy, that is, whether wagers are "bold" or conservative. As one illustration, if an infinitely rich house is assumed and the estimated number of triMs-to-ruin is sought, the calculation is as follows:
E(n), = z/(p -- q)(1 -- r), where E(n)~ z p q r
= = = = =
(1)
Expected number of trials before ruin, Gambler's initial fortune, expressed in numbers of bets, Single-trial probability of winning, Single-trial probability of losing, Single-trial probability of tying.
Solving this expression for p = 0.47, q = 0.53, r = .00, and a gambler with an initial fortune of $100 played in $1 bets, yields E(n)loo = 1667. At 60 or 70 hands per hour, the gambler should last for 25 hr or so before going broke. Since ties occur in Blackjack quite frequently, the actual number of plays prior to ruin should be even larger (for p = 0.43, q = 0.49, r = 0 . 0 8 , E ( n ) l o 0 --~ 1812.
Optimum Wagers in a Subfair Game Suppose that a Blackjack player knows he is in a subfair game, but still desires to play. How should he allocate his money, in a few large bets or in a succession of small bets? Mathematicians have proven that if the gambler is playing a subfair game and has the fixed objective of increasing his initial fortune z to an amount A - z, then it is best for him to play boldly, that is, always to bet his entire fortune z, or just enough of it to reach the objective A - z (Epstein, 1967; Dubins & Savage, 1965; Marina & Deutsch, 1973). The remarkable effects of bold versus cautious betting can be demonstrated in two calculations. Assume a cautious gambler with an initial fortune of $100, a desired fortune of $200, bets of $1 per hand, p = 0.47, and q = 0.53, and infinite credit. Then, from Dubins and Savage (1965), (p)lOo p($100, 8200, cautious) =
[1 __ (p/q)100] X [1 (p/q)~Oo] ~ (47"~~°° =\~/ ~ 6 . 0 6 X 10-6.
(2)
STRATEGY AND EXPECTATION IN CASINO BLACKJACK
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Thus, there is virtually no chance (six times in a million) of doubling the original fortune via small bets. Now assume the same initial fortune of 8100, the same values of p and q, but an objective of $1000 and bold wagering. For this case, Dubins and Savage (1965) prove that p($100, 81000, bold) = p4(1 + q)(1 - p2q2)-I~ .08. Then, p($100-200) cautious p($100-1000) bold
.000006 6 X 10-5 .08 = 8 × 10-2 - 7.5 × 10.3 .
(3)
Thus a most surprising conclusion: Gambler A has $100 and plays subfair Blackjack (p = 0.47) with the intent of doubling his $100; he bets $1 per hand. Gambler B has $100 and plays the same game with the goal of increasing his fortune tenfold to $1000. He plays boldly. It is 7500 times more likely that Gambler B will reach his $1000 goal, than it is that Gambler A will double his money to $200. Very few of the Blackjack gamblers seemed to appreciate the statistical approach to playing a subfair game. As far as we could tell, gamblers in our sample usually did not set a fixed fortune objective, and then stop gambling when that obieetive was reached. And they certainly tended to avoid bold play. THE MOTIVATION TO GAMBLE The statistician views Blackjack play as a technical problem in maximizing the likelihood of achieving some desirable, well-definedobiective. But our Blackjack gamblers did not always perceive the game as a strict adversary encounter, to be played in a specific optimal way. Many other aspects of the gambling situation were relevant besides the technical decision choices; we note two of them here. A first aspect was the escape or diversion motive for gambling. Da Silva and Dorcus (1963) believe that the need for escape is important in explaining the behavior of horse race fans: " . . . The bettor is preoccupied in selecting winners, making bets, calculating the odds on the totalisator board, observing performance, etc. In these things, he has a moment's respite from his daily problems, and his nervous tensions are released." The Blackjack table presents some of these same diversionary features, and offers much faster action. Besides the activities of the game itself, the atmosphere of a crowded casino is exciting to many people. And casino package deals serve to facilitate the escape: transportation, hotel rooms, food, drink, and entertainment are provided at reasonable cost in the casino area. The losses at the table, then, might be considered
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NICHOLAS A. BOND, JR.
as part of the entertainment expense. Furthermore, if the game is entertMning, then the tendency toward smMl bets is rational, because it "makes the game last as long as possible." Thus a gambler who made the right decisions three-fourths of the time, and who lost a few dollars per hour in $i bets, could feel he was playing a pretty good game, and getting his money's worth. Indeed, he might be surprised to find his behavior regarded as nonoptimM. A second factor was the "feel" of the game, the "way the cards were running." M a n y of the gamblers we watched and interviewed remarked that, at certain times, things suddenly "felt right." When this happened, the player "just knew it," and he might raise his stakes, or move to a more expensive table. And occasionally the feeling did coincide with a winning streak, say over a short series of 20-40 hands. This coincidence appeared to be extraordinarily pleasurable and memorable to the gambler, and it seemed to provide him with very satisfying feelings of competence and mastery. In our view, such feelings are illusions of the mind, but they are vividly documented in such literary sources as Dostoyevski's novels (Bolen & Boyd, 1968). Perhaps there are individual differences in the extent to which people seek meaning in a series of intrinsically random events, and in the satisfaction they get from the illusory meaning that they perceive. This recognition of meaningful trends in a random series extends also to casino personnel. At coffee and shift breaks, dealers often observed that the "cards are running strong (against the house) tonight." Actually, with a fixed advantage, hundreds of thousands of small bets per day, and a relatively stationary population of players, it necessarily follows that day-to-day variance in the house "take" will be small, and will be highly correlated with the total number of bets placed. Even experienced praetitioners in the gambling industry may have erroneous notions about the laws of probability I about the significance of occasional runs in a series, and about optimal play. It might be particularly instructive to observe experienced Blackjack dealers when they go to the other side of the table, and play the game as gamblers. Many other hypotheses have been offered to explain the utility of gambling. A popular psychoanalytic view is that the compulsive gambler is an orally regressed neurotic who feels guilty; punishment via the gambling losses will help to remove the guilt, so gambling persists until losses are catastrophic. In this special sense, the gambler really wants to lose (Bergler, 1970; Bolen & Boyd, 1968). Bergler elaimed to have cured 30 compulsive gamblers by tracing the self-punishment need back to some forbidden ehildhood emotions. The psychoanalysts admit, though, that the majority of gamblers are recreational gamblers, and want to win.
STRATEGY AND EXPECTATION IN CASINO BLACKJACK
427
Most studies find that gambling tendencies are not simple traits. Slovic (1962) correlated several measures of risk-taking propensity, and found negligible correlations between them. Burisch (1970') and Hochauer (1970) found wide individual differences in risky choice behavior, and inconsistencies in wager variances by the same people over time. At least one unpublished survey of "gambling bus" passengers found a relation between age and preferred gambling game: Women past 55 were more likely to play slot machines than were younger women. Speculations about individual motivation to gamble are difficult to validate, until extensive biographical, observational, and interview data can be obtained on each member of a suitable sample. There are, however, many intriguing behavioral questions that could be answered if a researcher could obtain permission to film Blackjack gambling decisions and bets from behind the one-way security mirrors in the casinos. For instance, do the players betting high stakes play better Blackiaek than those who bet $1 or $2 per hand? We know, from computer simulation, which playing decisions are "closest," or most marginal, in their returns: Are these the decisions which produce the most hesitation and unreliability in the average gambler? How many players seem to be using one of the "counting" systems, and what kinds of mistakes do they make? How do successive losses, or gains, affect the amounts wagered? Do the empirical likelihoods of winning over a series of 100 hands, or over a period of several hours, match the theoretical probability distributions? A few days of film records could yield satisfactory answers to such questions, and provide a highly detailed data base for simulation models of gambling behavior. REFERENCES BALDWIN, R., CANTEY, W., MAISEL, H., & McDERmOTT, J. Playing blackjaclc to win: A new strategy ]or the game o] 21. New York: Barrows, 1957. BALDWIN, R., CANTEY, W., MAISEL, H., & McDER~OWT, J. The optimum strategy in blackjack. Journal o] the American statistical association, 1956, 51, 429-439. BERGLER, E. The psychology o] gambling. New York: International University Press, 1970 (first published, 1958). BOL~N, D. W., & BOVD, W. H. Gambling and the gambler. Archives o] general psychiatry, 1968, 18, 617-630. BURISeH, M. An attempt to demonstrate utility of gambling. Acta psychologica, 1970, 34, 367-372. DA SILVA, E. R., & DORCUS, R. M. Science in betting. Garden City, New York: Doubleday, 1963. DUBINS, L. E., & SAVAGE,L. J. How to gamble i] you must. New York: McGrawHill, 1965. EPsTeIN, R. A. The theory o] gambling and statistical!logic. New York: Academic Press, 1967. HOCI~AVER, B. Decision-making in roulette. Acta psychologica, 1970, 34, 357-366.
42~
NICHOLAS A. BOND~ JR.
MARMA, V. J., & DEUTSeI-I, I~. W. Survival in unfair conflict. Behavioral science, 1973, 18, 313-334. RAJ, D. Sampling theory. New York: McGraw-Hill, 1968. REVERE, L. Playing blackjack as a business. Los Angeles: Paul Mann, 1973. SLowc, P. Convergent validation of risk taking measures. Journal o] abnormal and social psychology, 1962, 65, 68-71. THoRP, E. O. Beat the dealer: A winning strategy ]or the game o] twenty-one. New York: Random House, 1966.
RECEIVED:
JANUARY
2, 1974
(1974)
Basic Strategy and Expectation in Casino Blackjack NICHOLAS A. BOND, JR. California State University at Sacramento Fifty-three Blackjack gamblers in four Nevada casinos were unobtrusively observed, and scored according to whether their playing decisions corresponded to the optimal "Zero-Memory" ]Basic decision strategy. Of the 940 hands recorded, at least 16% were played differently from the Basic prescription; presumably as a result of this "nonoptimal" play, average losses at Blackjack were quite high, on the order of ten times the theoretical loss rate for t~asic play. Also, by wagering small bets in a subfair game, Blackjack gamblers practically guaranteed loss of their betting capital to the casinos.
Blackjack, or "21," is the only casino game which offers the public an expected return greater than zero. I t is also the only game t h a t allows the player to m a k e tactical decisions during play: Unlike the Craps and Roulette bettor, the Blackiack gambler can significantly influence the likelihood of his winning by his choice behavior after play begins. The decisions required of the player are varied, and in some cases they defy common sense. Furthermore, the game is fast, and there are several possibilities for distributing money over the hands and the side bets. These properties m a k e the game an intellectual challenge, and they also lead to gambling behavior t h a t is complex. This study sought to record significant aspects of t h a t behavior in a commercial casino setting. I n N e v a d a Blackjack, a dealer represents the "house," and there are one to six players. Each card from 2-10 in the standard 52-card deck receives a value according to its number; picture cards have a value of 10; Aces can have a value of either 1 or 11, at the player's option. A hand is "soft" if it contains an Ace and t h a t Ace can be counted as 11 without the total exceeding 21. Thus, an A-6 is a "soft 17." All aceless hands are " h a r d " ; 9-8 is a "hard 17." After bets are placed, each player and the dealer receive two cards. One of the dealer's cards is exposed, the "up" card; the other card is "down" or concealed. The player's object is to beat the dealer's numerical total, without exceeding 21. The player can request additional cards or "hits" as long as his total is less t h a n 21 ; when he is satisfied, he "stands." 413 Copyright © 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.
4]4-
NICHOLAS A. BOND; J R .
If the player's cards total more than 21, he "busts" and loses his bet immediately; if his cards total less than 21, and the dealer busts, then the player wins the amount of his original bet. An original two-card combination of an Ace and a 10-valued card is a Blackjack, and wins over all other hands; if a player gets a Blackjack, he is paid 1.5 times the amount of the bet. In a tie hand between dealer and player, no money changes hands. Under some circumstances, there are special choices available to the player: He can split pairs of the same numerical value, such as two 6% or two 8's, and place the value of his original bet on each of the two "new" hands, or he can "double down," in which case he doubles the initial bet and draws just one additional card. There is also an "Insurance" option when the dealer's up card is an Ace; the player can bet that the dealer's down card is a 10 card, and if a 10 is concealed, then the player receives double the amount of the bet. Figure 1 gives a flow chart of the binary decisions that the player must make on every deal. For the majority of hands, the stand-or-draw choice is the most significant decision for the player. The dealer has to play according to a mechanized strategy: He must stand on dealer totals of 17 or better, and must hit until then. The best strategies for playing Blackjack involve detailed memory and require the counting and weighting of each card played from the deck.
I
Do you have a pair? I
)o
the pair?
Double down? Yes
Draw?
;I
Yes~
Is it a pair of Aces?
1
Stand (End)
FIO. i. Flowchart of player decmions in B1aokjaok. Modified ~om Beat the Deale~ by EdwaM
Thorp, 1966. Copyrighted by Random
House, New York, 1966.
STRATEGY AND E X P E C T A T I O N
IN
CASINO B L A C K J A C K
415
When the depleted deck exhibits a composition which is favorable to the player (say when all the 5's are gone, or when the deck has a relatively high proportion of 10's), then bets may be increased according to the favorability of the player's expectation, with minimum bets being placed at other times. Machines and computer programs have been designed to evaluate the effects of cards as they are played, and positive expectations on the order of +.05 can be achieved with their aid (Epstein, 1967). Casinos do not, of course, permit such devices to be used in actual play. Human memory cannot store all the data or perform all the calculations necessary for truly optimal play, but approximate systems have been devised which approach the optimum and which are feasible for a determined player to use (Thorp, 1966; Epstein, 1967; Revere, 1973). In these systems, tile player does not have, to compute and store the precise effects of every card that is exposed; for example, he may need to count and remember only the "highs" (10's and Aces) and "lows" (2's through 6's) that remain in the deck, and bet an amount roughly proportional to the high-low difference. Even when simplified, however, these card-counting systems are rather difficult to practice, and require attention to every card dealt, during the game. "Zero-memory" or "Basic" strategies have also been derived (Baldwin, 1956; Epstein, 1967; Revere, 1973; Thorp, 1966). These strategies are much simpler: They assume that a single player competes against the dealer, and that only three cards are known (the player's two cards and the dealer's up card); no count is kept of previous cards played in the deck, hence the term "zero memory." Gains or losses from the hit, stand, split, or double-down decisions have been estimated from computer simulation of many Blackjack hands in the remaining 49-card deck, and the player is advised to take the decision with the highest simulated expectation. Since the player's two cards can comprise 55 different two-card value combinations, and the dealer's up card can take on any one of ten values, there are 550 possible card arrangements facing the player. For each one of these he must be able to decide quickly whether to draw, stand, split, or double down. A player using the optimalzero-memory strategy does not really have to memorize a 55 × 10 matrix; a 28 X 10 table is sufficient, as shown in Table 1, taken from Epstein (1967). This reduction is possible because some of the different hands are treated the same way ; a player holding of a 9 and a 6 is treated exactly like an 8 and a 7. Remembering the 28 × 10 decision table can be facilitated by rules which specify several table entries: "Always stand on hard 17 or better," and "Always double down on 11." Since some casinos do not permit certain bets, such as soft doubles, the decision table
416
NICHOLAS A. BOND, JR. TABLE 1 SUMMARY OF OPTIlYIAL ZERo-MEMoRY STRATEGYg
Dealer's up card 2
3
4
5
6
7
8
9
10
A
X X X X X D D X X X
X X X X X D D X X X
X X XI X X D X X X X
X X X X X D X X X X
X X X X X
X X X X X X
X X X X X X
X X X X X
S S S
S S
S
S
S
S
S
S
Player's hard total 17 16 15 14 13 12 11 10 9 8 7
a X D D D X X
Xb D D D X X
~ D D D X X
D D D Xd X
D D D D~ X
X X X X X D D X X X
Player's soft total 19 18 17 16 15 14 13
D X X X X
D D X X X X
D D D D D D
D D D D D D
D D D D D D
X X X X X
S S
S S
S S
S S S
S S
S S
S S S S
S S S S
S S S S
S S S S
S S S S
S S S
S
S
S
S
S
S
Player's pair 2's 3's 4's 5's 6's 7's 8's 9's 10's A's
a Draw with hand (3,10) vs a 2. b Stand with hands (3,9), (4,8), and (5,7) vs a 3. c Draw with hand (2,10) vs a 4. d Double down with hand (3,5) vs a 5. Draw with hand (2,6) vs a 6. ] Stand with hand (7,7) vs a 10. • g Symbolism: X = draw; D = double down; S = split. Note. Reprinted from Theory of Gambling and Statistical Logic, by R. A. Epstein, 1967. Copyrighted by Academic Press, 1966.
STRATEGY AND EXPECTATION IN CASINO BLACKJACK
417
is reduced even further. The zero-memory Basic strategy can be learned in a few hours and played at casino speed after a little practice and observation. The Basic strategy is strictly optimal only for the first play of the game, because it assumes a 49-card deck with equal probabilities of drawing each remaining card, and this assumption is palpably false with a depleted deck. Nevertheless, Basic gives a surprisingly good return. It yields a theoretical expectation of about +0.0015 under the best Las Vegas rules, and an expectation of about -0.0055 for South Lake Tahoe and Reno casinos (Thorp, 1966). All four casinos at South Tahoe forbid doubling down on soft totals, and on all totals after pair splitting; also, the dealer draws to soft 17 instead of standing on any 17; and three casinos forbid doubling on hard 9. These rules work slightly against the player, and together they reduce his expectation by about 0.006, from +0.0015 at Las Vegas casinos to -0.0055 at South Tahoe. In the only empirical test of Basic that has been reported, Baldwin et al. (1957) sustained losses of $26.00 over 3530 hands, for an expectation of about -0.007. Thus, a player who follows the optimal zero-memory strategy over a long series has about an even-money bet. On the Las Vegas Strip, he could anticipate winning an average of $1.50 per 1000 plays at $1 each; in Northern Nevada, because of less favorable rules, he should anticipate losing $5-6 in 1000 plays, a rather attractive return for a strategy that does not require counting cards or calculating ratios, and that works iust as well when the cards are continually reshuffled. And, it does offer the best possible play, as long as information is confined to the dealer's up card and player's hand. Given that the Basic strategy offers a relatively good casino gamble, is widely known, and is easy to learn, how many Blackjack players follow it? To explore this question, the present study recorded actual Blackjack gambling decisions. As casinos do not ordinarily permit observations of their customers' behavior by outsiders, our data-collection plan had to be completely unobtrusive, with no clipboards, recording devices, or questionnaires in sight. It was also decided to concentrate on player strategy and win-loss return or expectation, and to ignore other features such as the timing of gambler decisions, superstitions about sitting to the far left of the dealer, size of the gambling bankroll, and so forth. METHOD Research Site
Observations were carried out in four commercial casinos at South Lake Tahoe, on the California-Nevada border. This is one of the prin-
418
NICHOLAS A. BOND~ JR.
eipal Blackjack gambling areas of the world; the game is played around the clock, with hundreds of players at the tables in afternoon and evening hours. Fortunately for observational purposes, the four casinos are on adjoining lots, just a few yards from each other.
Sampling Plan The population of interest was the complete set of Blackjack hands played during 5 days in the summer of 1973, at the four South Lake Tahoe casinos during the hours of 10-2 AM. The characteristics of interest were (a) the proportion of hands played according to Basic, and (b) the net return in bets won or lost. There is no list or count of all these Blackjack hands; the total count of dealer hands is probably around 600,000, with a total number of gambler hands on the order of 1.75 million (70 hands per hr × 3.1 gamblers per table ;4 100 tables × 16 hr × 5 days = 1,736,000). With no master list, it was impossible to take a simple randora sample of these hands, and facilities were not available for monitoring simultaneously a large number of tables in the casinos. Therefore, a two-stage cluster sampling procedure was set up which designated the active Blackjack tables as the primary sampling units (psu's), and chose a random sample of n tables from the total of N psu's. After that, a small number m of hands were selected from the total number M at each table in a second-stage sampling (Raj, 1968). This procedure proved to be feasible, since a complete list of active psu's (tables) could be prepared, checked, and sampled just before observing began; the actual observations were then concentrated in only a few tables. In midsummer 1973 there were altogether 206 Blackjack tables in the four casinos, but for much of the time only about half of them were operating. A map layout of these tables was made and each table was assigned a three-digit number from 001-206. There were 5 observation days, with each one running from 10-2 AM. Two on-the-hour starting times were randomly chosen each day from the 16 hr. On a typical day the starting times were 10 AM and 6 PM. Just before 10 Am, a random drawing of several three-digit numbers between 001206 was made, with the results listed in order of drawing. Then a quick count of active Blackjack tables operating in all four casinos was taken. The number of tables actually observed was the integer nearest to ½o of the number of tables operating; thus the sampling fraction fl of psu's (tables) was .05. If 55 tables were running at 10, AM, then the first three table numbers from the three-digit series were designated for that period. During the early evening hours, say at 6 ~M, 132 tables were operating, and so a randomly selected set of seven tables was chosen for study starting at that time. This procedure assured that active tables were
STRATEGY AND EXPECTATION IN CASINO BLACKJACK
419
represented according to their numerical prevalence in the four casinos. It turned out that about 40% of the tables selected were in the largest casino, and about 4% of the tables were in the smallest casino. After the sampling decisions were made on how many and which tables to study, the observer proceeded immediately to the first table in the list. He then randonlly chose one of the six seats, by rolling a die, and watched exactly 20 hands which were played by the gambler(s) sitting in that seat. If a seat was vacated during the 20-hand series, then the observer shifted his attention to the next seat to the left if the first random number was odd, and to the right if it was even. The 20-hand segment, which is presumed to be a random sample of the hands played at that table during the sampling period, represents a very small sampling fraction indeed; the number of hands played at a continuously operating table over a 16-hr period is variable, but will probably average above 3000 (3.1 gamblers × 16 hr × 70 hands per hr = 3472). Thus, the secondary sampling fraction f2 might be estimated as (20 × 2)/3472 or about .011, with the estimated ]lf~ product, or total sampling fraction, as .05 × .011 = .00055 (Raj, 1968).
Recording Procedure UsualIy, the observer could see the player's "down" cards just by standing behind him; there are many people who watch the games and look at the cards, and there is no reason for the player to conceal them anyway. If the player being observed did temporarily conceal his cards, the observer had to remember the dealer's first up card, wait for all cards to be exposed at the payoff, and then infer whether the decisions made by the player were Basic plays. Counting conventions were as follows. If either the dealer or the player received an immediate Blackjack, the player was considered to have played according to the Basic model even though he didn't "play" at all; when a pair that should have been split was not split, that hand was counted a non-Basic play, but when an opening pair was (properly) split, the two resulting hands were analyzed and counted separately. There were no "excluded" hands; all were counted. A three-number mental coding scheme was used to keep track of the decisions and the returns for the 20-hand series. The first number was the number of the hand being playedl and it varied from 1-20; the second number was the number of hands out of this block of 20 which conformed to the Basic strategy; and the third number was the return enjoyed by the player. This last figure was simply the aggregate number of bets won or lost, not the actual amount of money bet; the procedure estimates the single,trial probability of winning. Supposing for illustra-
420
NICHOLAS A. BOND, JR.
tion that the sixteenth hand had just been played, then the observer might have been counting as follows: 16-13-0ff 2. At this point, 16 hands have been played, 13 of them were played according to the Basic (3 were not), and the player had a net loss of 2 bets. No recording reliability cheeks were made in the casinos; however, the observer's accuracy was tested in a noneasino setting. Two-hundred hands were dealt at about a hand a minute, and the observer (mentally) recorded the results in his three-number code. After each block of 20 hands, he wrote down the three numbers (gambling hands, Basic plays, net returns). Meanwhile, two assistants independently made objective records of each card dealt, and then reconstructed the whole series of 600 numbers. When the observer's reports were compared with a true record of these 200 hands, three errors were found; all were only one digit off. This suggests a recording error on the order of 1-2% for noneasino conditions. After watching 20 hands at one table, the observer wrote down the results of the 20 preceding observations, reset his mental counters to 0-0-0, went on to the next designated by the random series, and continued observing. The whole procedure must have been quite unobtrusive, as the observer was never questioned by casino employees or by any of the players he watched. In two instances, the table randomizing scheme called for observing isolated tables where there was only one player. Steady peeking over a single player's shoulder for 15 min might have been noticeable. So for those two occasions only, the observer sat at the table himself and played the Basic strategy with minimum bets, while mentally recording the other player's results for 20 hands: Interviews
Under casino conditions, it was impossible to obtain a probability sample of Blackiaek players for interviews. But a few specific questions were addressed to a grab sample of the gamblers. Whenever the schedule permitted, the observer would wait by the table after finishing a block of 20 hands. If the player left the table within 20 rain, the observer followed the player, and made an opening remark to the effect that he had seen the player refuse an Insurance bet at the table. This refusal was so common that it was safe to assume the player really had turned down such a bet. The interviewer, who had a Blackjack instruction booklet in his hand, then explained that he was trying to learn the game, and was: a little uncertain about some of the plays. If the player seemed cooperative and willing to talk, then the following six questions were addressed to him.
STRATEGY AND EXPECTATION IN CASINO BLACKJACK
421
1. When do you take the Insurance? (Correct answer: Never take Insurance unless the ratio of non-10's to 10% in the deck is less than 2.00). 2. W h a t do you do when you have 7-7 and the dealer has a 6 up? (Correct answer: Split the pair). 3. Do you hit (the player's) 12 when the dealer has a 6 up? (Correct answer: No). 4. W h a t is the minimum hard standing number you ought to have if the dealer has 8 up? (Correct answer: 17). 5. Over at Casino X, they let you double down on hard 9; when do you do that? Do you always double on 10 and 11? (Correct answer: When dealer has 2-6 up; always on 11; always on 10 except for 10-A up). 6. When do you split a pair of 9's? (Correct answer: Always except for 7, 10, or A up). Fourteen gamblers were interviewed; answers to each question were scored as right or wrong. RESULTS A total of 940 Blackjack hands (4:7 sets of 20 hands each) was recorded for the 5 days; the slowest day had 140 observations, the busiest day had 240. These hands were played by 53 people, 29 males and 24 females, all presumably above 21 y r of age. The minimum bet placed on a single hand was $1 and the maximum bet was $100; a large fraction of the bets, probably a majority, were either $2 or $1.
Basic Play Of the 940 hands observed, 793, or about 84%, were played according to the Basic decision table shown in Table 1. For a 20-hand sequence, the mean number of Basic plays was 16.8 with a SD of 1.75, n =- 47. There were three "perfect" series of 20 hands each, with all decisions in accord with the Basic strategy; the "worst" player made Basic decisions on only 13 of his 20 hands. The 84% figure was inflated by scoring all "natural" Blackjacks (those occurring on the first two cards) as Basic plays. In a head-to-head game with one player and the dealer, naturals occur about 10% of the time. When the natural hands were eliminated, only about 75% of those played were zero-memory optimal. In two-stage cluster sampling, the best estimate for the population mean is the sample mean. Hence for the total population of more than 1.7 million hands, we anticipate that about 84% were Basic plays on the liberal scoring rule, with about 75% Basic plays when naturals are ex-
422
NICHOLAS A. BOND~ JR.
eluded. Population variance estimates can be made by combining the variabilities of second-stage units (hands) inside psu's (tables) with the variance of psu totals around the grand sample mean (Raj, 1968). Under reasonable assumptions about sampling fractions, the standard error estimate of the population proportion was less than 2%. W h y were there so many non-Basic (and presumably nonoptimal) decisions? Table 2 indicates that many of the gamblers simply did not know the correct plays for some of the nonobvious situations. Interview responses to items 2-6 were about equally split between rights and wrongs. Item 4 was the only one for which everybody knew the right answer; and this item was designed to be easy, since it can be handled by a simple rule: "Always stand on hard 17 or better." Some of the gamblers did not appear to know or to use the Basic strategy in a strict way. T h e y probably memorized a few rules that reproduced most, of the Basic table, and then played these pretty consistently. But when borderline decisions came up, the player often wavered and changed his tactics from one occasion to another. The mental recording system did not permit fine-grMned error tabulations, but m a n y stand-hit mistakes were noted on player totals of 12 against dealer up cards of 4-5-6; the player with 12 should stand, except when he has 2-10 against the dealer's 4. The A-6 "soft 17" hand was also something of a puzzle and was often wrongly accepted as a suitable soft standing total; double-down and pair-splitting opportunities were sometimes ignored. Item 1, on the Insurance option, was perhaps the most revealing one of the six interview questions. When the dealer has an Ace showing, the gambler can enter a side wager that the dealer has a 10 as hole card. If the dealer in fact has a 10 in the hole, then double the amount of the side bet is paid to the gambler. Now in evaluating the Insurance bet, an analytical gambler might reason as follows: The dealer has an Ace up; I have two cards in my TABLE 2 RESPONSES TO INTERVIEW QUESTIONS Item
Correct
Incorrect
n
1
1
2 3 4 5 6
2 7 14 9 3
13 12 7 0 5 ll
14 14 14 14 14 14
36
48
Total
STRATEGY AND EX:PECTATION
IN
CASINO B L A C K J A C K
423
hand (assume these are non-10's). There are 49 cards left in the deck, of which 16 are 10's, so tl~e chance of the dealer's hole card being a 10 is 16/49, and the chance of it being a non-10 is 33/49. If it is a 10, I get twice my bet back; but 2 X 16/49 = 32/49, which is still less than 33/49. Therefore, I won't take the bet. The Insurance bet is even worse for the gambler if he has one or two 10's in his own hand, since that reduces the dealer's likelihood of getting a 10 to 15/49 or 14/49. And thus a general answer to Item 1: Never take the Insurance bet, unless you are counting cards and are reasonably sure that the proportion of non-10's/10's is ~2.00. Only one respondent gave the right answer to this question. A frequent response was t h a t the bet was usually refused, but was accepted sometimes "when the cards are running right." Expectation
The 53 gamblers had net losses of 55.5 bets during their 940 plays, for an expectation of about - . 0 6 . Over a 20-hand series, the expected loss was 1.18 bets (SD = 1.7, n = 47). Seventeen of the 20-hand blocks won and four broke even, so the overall probability of break-even-or-better was 21/47 or 0.45, with a likelihood of 17/47 or 0.36 of making an overall profit. Conditional probabilities of break-even-or-better improved to 0.56 if 18 or more of the 20 plays were Basic. As far as payoff is concerned, Blackjack can be considered to be a simplified Red-and-Black game (Dubins & Savage, 1965), where the gambler can stake any amount in his possession on the outcome. He wins back his stake and as much again with probability p, and loses with probability q. Let p = 0.47 and q = 0.53, so t h a t the expectation corresponds with our results. Then for unit bets in n plays of the game, the mean expectation is n ( p - q ) , or - 0 . 0 6 n ; the variance is n [ p + q (p _ q)2] = 0.996n. For 100, plays at $1 per hand, a gambler could expect to lose $5 or $6. With a standard deviation of a little less than ten, he also can expect to come out ahead only about 20% of the time he plays out the 100-hand series. The expectation for Basic play at South Tahoe was about -0.006, yet the gamblers in this sample were losing nearly ten times t h a t much, on the average deal. These losses were surprisingly high. If such losses are indeed characteristic of the population of Blackjack hands, then Craps and Roulette, with best-bet expectations of - 0 . 0 1 4 and -0.053, respectively, m a y yield better money returns to the casual gambler, even though these games are relatively trivial and are completely impervious to player strategy. When the losses of the most unfortunate 7 sequences of the 47 are removed, then average losses
424
NICHOLAS A. BOND, JR.
were reduced to about 3%. Thus less than a quarter of the players lost half of the bets. As played by these gamblers, Blackjack was a subfair game, since p < q (Dubins & Savage, 1965). Anybody who plays a subfair game can expect to be ruined if the game goes on long enough. The controlling ruin parameters are the probability of winning p, the resources available to the player and the house, and the betting policy, that is, whether wagers are "bold" or conservative. As one illustration, if an infinitely rich house is assumed and the estimated number of triMs-to-ruin is sought, the calculation is as follows:
E(n), = z/(p -- q)(1 -- r), where E(n)~ z p q r
= = = = =
(1)
Expected number of trials before ruin, Gambler's initial fortune, expressed in numbers of bets, Single-trial probability of winning, Single-trial probability of losing, Single-trial probability of tying.
Solving this expression for p = 0.47, q = 0.53, r = .00, and a gambler with an initial fortune of $100 played in $1 bets, yields E(n)loo = 1667. At 60 or 70 hands per hour, the gambler should last for 25 hr or so before going broke. Since ties occur in Blackjack quite frequently, the actual number of plays prior to ruin should be even larger (for p = 0.43, q = 0.49, r = 0 . 0 8 , E ( n ) l o 0 --~ 1812.
Optimum Wagers in a Subfair Game Suppose that a Blackjack player knows he is in a subfair game, but still desires to play. How should he allocate his money, in a few large bets or in a succession of small bets? Mathematicians have proven that if the gambler is playing a subfair game and has the fixed objective of increasing his initial fortune z to an amount A - z, then it is best for him to play boldly, that is, always to bet his entire fortune z, or just enough of it to reach the objective A - z (Epstein, 1967; Dubins & Savage, 1965; Marina & Deutsch, 1973). The remarkable effects of bold versus cautious betting can be demonstrated in two calculations. Assume a cautious gambler with an initial fortune of $100, a desired fortune of $200, bets of $1 per hand, p = 0.47, and q = 0.53, and infinite credit. Then, from Dubins and Savage (1965), (p)lOo p($100, 8200, cautious) =
[1 __ (p/q)100] X [1 (p/q)~Oo] ~ (47"~~°° =\~/ ~ 6 . 0 6 X 10-6.
(2)
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425
Thus, there is virtually no chance (six times in a million) of doubling the original fortune via small bets. Now assume the same initial fortune of 8100, the same values of p and q, but an objective of $1000 and bold wagering. For this case, Dubins and Savage (1965) prove that p($100, 81000, bold) = p4(1 + q)(1 - p2q2)-I~ .08. Then, p($100-200) cautious p($100-1000) bold
.000006 6 X 10-5 .08 = 8 × 10-2 - 7.5 × 10.3 .
(3)
Thus a most surprising conclusion: Gambler A has $100 and plays subfair Blackjack (p = 0.47) with the intent of doubling his $100; he bets $1 per hand. Gambler B has $100 and plays the same game with the goal of increasing his fortune tenfold to $1000. He plays boldly. It is 7500 times more likely that Gambler B will reach his $1000 goal, than it is that Gambler A will double his money to $200. Very few of the Blackjack gamblers seemed to appreciate the statistical approach to playing a subfair game. As far as we could tell, gamblers in our sample usually did not set a fixed fortune objective, and then stop gambling when that obieetive was reached. And they certainly tended to avoid bold play. THE MOTIVATION TO GAMBLE The statistician views Blackjack play as a technical problem in maximizing the likelihood of achieving some desirable, well-definedobiective. But our Blackjack gamblers did not always perceive the game as a strict adversary encounter, to be played in a specific optimal way. Many other aspects of the gambling situation were relevant besides the technical decision choices; we note two of them here. A first aspect was the escape or diversion motive for gambling. Da Silva and Dorcus (1963) believe that the need for escape is important in explaining the behavior of horse race fans: " . . . The bettor is preoccupied in selecting winners, making bets, calculating the odds on the totalisator board, observing performance, etc. In these things, he has a moment's respite from his daily problems, and his nervous tensions are released." The Blackjack table presents some of these same diversionary features, and offers much faster action. Besides the activities of the game itself, the atmosphere of a crowded casino is exciting to many people. And casino package deals serve to facilitate the escape: transportation, hotel rooms, food, drink, and entertainment are provided at reasonable cost in the casino area. The losses at the table, then, might be considered
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as part of the entertainment expense. Furthermore, if the game is entertMning, then the tendency toward smMl bets is rational, because it "makes the game last as long as possible." Thus a gambler who made the right decisions three-fourths of the time, and who lost a few dollars per hour in $i bets, could feel he was playing a pretty good game, and getting his money's worth. Indeed, he might be surprised to find his behavior regarded as nonoptimM. A second factor was the "feel" of the game, the "way the cards were running." M a n y of the gamblers we watched and interviewed remarked that, at certain times, things suddenly "felt right." When this happened, the player "just knew it," and he might raise his stakes, or move to a more expensive table. And occasionally the feeling did coincide with a winning streak, say over a short series of 20-40 hands. This coincidence appeared to be extraordinarily pleasurable and memorable to the gambler, and it seemed to provide him with very satisfying feelings of competence and mastery. In our view, such feelings are illusions of the mind, but they are vividly documented in such literary sources as Dostoyevski's novels (Bolen & Boyd, 1968). Perhaps there are individual differences in the extent to which people seek meaning in a series of intrinsically random events, and in the satisfaction they get from the illusory meaning that they perceive. This recognition of meaningful trends in a random series extends also to casino personnel. At coffee and shift breaks, dealers often observed that the "cards are running strong (against the house) tonight." Actually, with a fixed advantage, hundreds of thousands of small bets per day, and a relatively stationary population of players, it necessarily follows that day-to-day variance in the house "take" will be small, and will be highly correlated with the total number of bets placed. Even experienced praetitioners in the gambling industry may have erroneous notions about the laws of probability I about the significance of occasional runs in a series, and about optimal play. It might be particularly instructive to observe experienced Blackjack dealers when they go to the other side of the table, and play the game as gamblers. Many other hypotheses have been offered to explain the utility of gambling. A popular psychoanalytic view is that the compulsive gambler is an orally regressed neurotic who feels guilty; punishment via the gambling losses will help to remove the guilt, so gambling persists until losses are catastrophic. In this special sense, the gambler really wants to lose (Bergler, 1970; Bolen & Boyd, 1968). Bergler elaimed to have cured 30 compulsive gamblers by tracing the self-punishment need back to some forbidden ehildhood emotions. The psychoanalysts admit, though, that the majority of gamblers are recreational gamblers, and want to win.
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Most studies find that gambling tendencies are not simple traits. Slovic (1962) correlated several measures of risk-taking propensity, and found negligible correlations between them. Burisch (1970') and Hochauer (1970) found wide individual differences in risky choice behavior, and inconsistencies in wager variances by the same people over time. At least one unpublished survey of "gambling bus" passengers found a relation between age and preferred gambling game: Women past 55 were more likely to play slot machines than were younger women. Speculations about individual motivation to gamble are difficult to validate, until extensive biographical, observational, and interview data can be obtained on each member of a suitable sample. There are, however, many intriguing behavioral questions that could be answered if a researcher could obtain permission to film Blackjack gambling decisions and bets from behind the one-way security mirrors in the casinos. For instance, do the players betting high stakes play better Blackiaek than those who bet $1 or $2 per hand? We know, from computer simulation, which playing decisions are "closest," or most marginal, in their returns: Are these the decisions which produce the most hesitation and unreliability in the average gambler? How many players seem to be using one of the "counting" systems, and what kinds of mistakes do they make? How do successive losses, or gains, affect the amounts wagered? Do the empirical likelihoods of winning over a series of 100 hands, or over a period of several hours, match the theoretical probability distributions? A few days of film records could yield satisfactory answers to such questions, and provide a highly detailed data base for simulation models of gambling behavior. REFERENCES BALDWIN, R., CANTEY, W., MAISEL, H., & McDERmOTT, J. Playing blackjaclc to win: A new strategy ]or the game o] 21. New York: Barrows, 1957. BALDWIN, R., CANTEY, W., MAISEL, H., & McDER~OWT, J. The optimum strategy in blackjack. Journal o] the American statistical association, 1956, 51, 429-439. BERGLER, E. The psychology o] gambling. New York: International University Press, 1970 (first published, 1958). BOL~N, D. W., & BOVD, W. H. Gambling and the gambler. Archives o] general psychiatry, 1968, 18, 617-630. BURISeH, M. An attempt to demonstrate utility of gambling. Acta psychologica, 1970, 34, 367-372. DA SILVA, E. R., & DORCUS, R. M. Science in betting. Garden City, New York: Doubleday, 1963. DUBINS, L. E., & SAVAGE,L. J. How to gamble i] you must. New York: McGrawHill, 1965. EPsTeIN, R. A. The theory o] gambling and statistical!logic. New York: Academic Press, 1967. HOCI~AVER, B. Decision-making in roulette. Acta psychologica, 1970, 34, 357-366.
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MARMA, V. J., & DEUTSeI-I, I~. W. Survival in unfair conflict. Behavioral science, 1973, 18, 313-334. RAJ, D. Sampling theory. New York: McGraw-Hill, 1968. REVERE, L. Playing blackjack as a business. Los Angeles: Paul Mann, 1973. SLowc, P. Convergent validation of risk taking measures. Journal o] abnormal and social psychology, 1962, 65, 68-71. THoRP, E. O. Beat the dealer: A winning strategy ]or the game o] twenty-one. New York: Random House, 1966.
RECEIVED:
JANUARY
2, 1974