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A Review of E. Clifton Davis’ NBJ System

by Arnold Snyder (with commentary by the Boardwalker)
(From Blackjack Forum XIII #3, September 1993)
© 1993 Blackjack Forum

[Note from A.S.–WARNING! WARNING! If you are considering buying the NBJ system, you are considering investing money in a Martingale betting system! STOP! Consult with a mathematician (a real one) immediately! Emergency mathematicians are now on duty to take your call!]

1991: Enter E. Clifton Davis and NBJ (“New Blackjack”), a dressed-up martingale betting progression system for casino blackjack. The system is published by Jerry Patterson Enterprises.

Davis has been an occasional contributor to Eddie Olson’s Blackjack Confidential newsletter. Olsen is the inventor of the TARGET system, one of the first widely promoted “streak” or “trend” blackjack systems — also sold by Patterson.

I wasted a lot of ink trashing the trendies in the past. Ed Thorp, Ken Uston, Stanford Wong, Julian Braun, Peter Griffin, and Mason Malmuth have all gone on record stating their shared belief that the TARGET blackjack theories are worthless. Every independent computer simulation of non-random casino-style shuffles that I’ve seen refutes the TARGET theories about the effects of poor shuffles.

The Facts About Card Clumping

Here are the facts. When cards are dealt to players who all play perfect basic strategy, the discard stack, if dealt again without being disturbed by a shuffle, would be in an order highly favorable to players. John Gwynn and Stanford Wong independently determined that the play of the hands at blackjack puts the cards into a player-favorable order. Players can confirm this themselves with John Imming’s UBE software.

Low cards and high cards do get clumped together during the play of blackjack hands, and this clumping would be worth about .75% to the basic strategy player if the cards were not put through a shuffle before the next deal. (This is determined by running simulations with no shuffle, just picking up the discards, cutting, and resuming play.)

It’s even possible for brand new decks to favor the first base side of the table by a few tenths of a percent, and similarly hurt the third base side of the table, if you put the new decks through a weak enough shuffle (weaker than any of the moderators at BJF have seen in any casino in the world–one riffle, no cuts, no strips on one-deck).

But it does not appear possible to create a shuffle that has the huge advantages/disadvantages (+/- 5% to 20%) described by the non-random shuffle gurus. In any case, casinos do indeed put the cards through a more thorough shuffle before dealing discards or new decks, and the shuffle is sufficient in every casinos we’ve seen to eliminate this player-favorable effect.

Plus, I have personally tested Martingale systems, like those advised by Davis, vs. extremely sloppy shuffles using Imming’s UBE software. There is simply no effect, regardless of how random, non-random, or “clumpy” the cards are.

It was ten years ago — September, 1983 — when my first TARGET article appeared in Blackjack Forum. Since then, Blackjack Forum has picked up a lot of new readers, many of whom did not follow the TARGET controversy as it developed.

So, here comes E. Clifton Davis with NBJ, as well as his new improved version of NBJ, which he calls WCB (“World Class Blackjack”). I’m regularly getting news clippings of ads for “New Blackjack” introductory seminars from all over the country. Players are sending me direct mail advertisements for NBJ they’ve received from Jerry Patterson.

I’ve even got a complete copy of the 119-page “New Blackjack” Home Study Manual, written by E. Clifton Davis, provided by one of Davis’ students. I’ve got an audiotape of a 45-minute interview with Davis by Jerry Patterson, in which Davis expounds on his theories. This promotional tape is accompanied by a 9-page advertisement for Davis’ “World Class Blackjack” system.

As of the writing of this article, “New Blackjack” sells for $445 mail order. WCB is now selling for $500, though Patterson’s letter states that this is an “introductory offer” to former NBJ students, and that the price of WCB will probably be going up to $1000.

Reports on NBJ from Players

This is one letter I received from an NBJ player:

[Regarding] E. Clifton Davis’ NBJ course… I asked for (and got) my money back because I didn’t play it to the 5%-15% advantage that the system touts. I did, however, win with it. My records reveal that I played it to a 1.96% advantage during three weeks last spring — certainly not a large enough trial to compel anyone to change religions (so to speak), but I wasn’t exactly disgusted with the outcome either, since seldom do I perform that well using a conventional counting system.

If I may be allowed to free-associate (don’t a lot of Californians do that???), my thoughts on NBJ are as follows:

I went to the seminar as advertised, and the following assertions were really made: That currently, the typical house advantage against a basic-strategy player is about 20%. No, not a hold percentage of 20% (which is believable), but a house advantage of 20%. In addition, they claim that because of like-card clumping, the current dealer bust percentage is about 12%…

Furthermore, Davis pledges that to keep the casinos from making any wholesale changes in their procedures resulting from any of the material in this course, under no circumstances will he sell his course to more than two percent of the blackjack-playing public.

Let’s do the math, shall we? 30 million blackjack players times 2% times $445 each (the cost of the course) equals $267 million. You gotta admire the man’s character for placing those kinds of economic restrictions on himself.

Finally, Davis asserts that 92% of his students win, compared to 1% for card counters, and that one fellow in Minnesota won 41 straight sessions with this system…

As for the course itself: I think that Davis’ advice on using betting progressions is ludicrous. To me, it’s like tracking roulette results, determining a pattern, structuring a progressive betting system to match that pattern, and hoping that you’re not too wrong if that pattern collapses. In his course, Davis has the student ascertain a “game type,” which he says sustains itself from one shoe to the next, and then use a betting progression to exploit it. Maybe it was just me, but for me the game types just didn’t remain the same from one shoe to the next.

I also think his strategy charts need a little refinement in places. The basis of NBJ, as Davis says, is non-random cards (due to insufficient shuffling, particularly in shoe games). As a result, sometimes you’re instructed to violate basic strategy plays, depending on whether high or low cards “are running.”

One such play is to split 6’s when the dealer has an ace up and “tens are running.” When I called Davis myself to ask him about this, his response was that the player’s objective in this situation is to go for an overall push by winning one of the two hands. Whaaaat?

But I also believe that there are some viable concepts in his system, particularly with respect to the notion that cards are not sufficiently shuffled so as to be randomly distributed. Although there was one occasion where I didn’t split a pair of aces that I was dealt (because “low cards were running”), then got two tens, there seemed to be quite a few occasions where his methods could be used to determine whether the dealer had a stiff hand with that ten up, and/or what my next hit card would be based on the values of the previous four or five cards.

As is the case with first-basing or playing tells (in blackjack or poker), you don’t have to be right 100% of the time to make your system work. And if I can consistently use this non-random-card business to change an average of two losing hands into two winning hands each hour (over and above the number of winning hands turned into losers by using this method), I’ll be changing careers real soon.

Conclusion: I’m still on the fence. I’ll be experimenting heavily later this year to see if non-random card analyses can be used in conjunction with conventional counting to eke out another percent or so advantage. I’ll let you know the results, both in a simulation environment, and in the casinos.

I think that NBJ might warrant a comment from you and/or other blackjack authorities since, according to Davis, his graduates now number over 1,000. Since one could argue that there is some subjectivity involved when making NBJ playing decisions, do you think that it’s conceivably possible for a computer jock somewhere to simulate this method?

Possible, yes. Probable, no. Many reputable programmers have lost interest in testing betting systems because it has been proven over and over again that such systems do nothing.

Again, I have personally tested Martingale systems, like those advised by Davis, vs. extremely sloppy shuffles using Imming’s UBE software. To repeat: there is simply no effect, regardless of how random, non-random, or “clumpy” the cards are.

The author of the above letter seems to be a thoughtful and intelligent person. His questions about NBJ reflect a genuine interest in whether or not some aspect of Davis’ approach might be valid, even if there are many aspects of the system which he finds hard to swallow.

Blackjack Forum’s “Atlantic City Update” columnist—who writes under the nom de plume The Boardwalker—was less kind in his remarks about NBJ. The Boardwalker never bought the system; he just attended one of the $10 NBJ introductory seminars that appeared in his area. This is his report:

NBJ = (FOH + BBS)^CST

The newspaper ad read like a carnival pitchman’s bark: E. Clifton Davis’ New BlackJack is sweeping the country! Why counting no longer works! Triple your bankroll! Win more hands than the dealer! How casinos make you lose! Win 75% of your double downs! Why basic strategy makes you lose!

Being a doubting Thomas for the ’90s, you couldn’t have kept me out of this seminar for an RFB comp. So, I plunked down a sawbuck and grabbed a chair right up front.

Arriving early, I scanned the four page NBJ sales leaflet, which had 10 playing tips from Davis on the front. Tip #7 said to always insure a natural against a dealer’s ace. Knowing this is a 4% basic strategy error, my antenna went into high gain. Tip #9 said to treat a dealer’s deuce like a ten. Not even if you held a gun to my head. Well, maybe for that.

Inside were more gems such as: win 80 units per hour; learn a winning system in 20 minutes; win 54% of your 15’s and 16’s; win 67% of your insurance bets. And all this without having to count. Could I plan to retire on just that last one alone?

A few of the 100+ blackjack curious in attendance and I had a chance for a friendly chat before the main event. I played the part of blackjack moron and let them spill their guts. If their knowledge was any kind of a representative sample, then in my opinion, many couldn’t recite proper basic strategy if asked and those who said they knew how to count cards wouldn’t know a ruin formula if it jumped off the page and bit them on the nose! Cherries, ripe for picking, if you ask me.

Enter the pitchman, Michael Simpson, a full time investment banker and part-time NBJ player/salesman. He began with the words of a confidence man, saying, “You won’t hear the truth anywhere else.” Opening remarks included highly questionable statements like, “Basic strategy used to work and so did card counting.”

Simpson claimed that basic strategy players win 40% of the hands and the dealer wins 60%, therefore, the casino enjoys a 20% advantage. When asked about the effect of doubles and naturals, he muttered some mumbo jumbo that made no relative sense and changed the subject.

Simpson said that NBJ has nothing to do with counting cards, yet in the same breath noted that while a 9 is of little value to card counters, it is a valuable card in the NBJ strategy. Go figure.

In a continuing ramble, he submitted that NBJ is based on card clumping and that after cards are recognized as being properly clumped, they can be predicted, giving the NBJ player a significant advantage over the house.

Simpson went so far as to say dealers intentionally perform high-low stacks while picking up completed hands, then shuffle in such a manner as to complete the stack. (Kudos to Steve Forte for educating me on these techniques in his Gamblers’ Protection videos.)

Besides the fact that I have never, ever, witnessed this in my 1000-plus hours of casino time over the years [note from A.S.–now 15,000 hours and still playing], there is absolutely no legal way the casino could use this to their advantage. Yet Simpson claims NBJ players can recognize this happening and can use it to predict hit and hole cards with 50-75% accuracy. Not likely, in my opinion.

I asked Simpson if NBJ has an insurance strategy and he replied, “Yes, we do insure and we do it very well.”

I asked Simpson how NBJ players assess whether or not they are playing properly and he replied, “If we won, then we made the correct play.”

Finally, I asked how many hours he had used NBJ in actual casino play and how much he was ahead. Simpson said he would rather not say in a public place, but added that he has played recreationally for 2 years and was ahead multiple thousands of dollars. Modest, but not too much so and also still very much in the short run, wouldn’t you say?

On my way out the door, I was given an NBJ newsletter for prospective pigeons. It says 40 units per hour is often earned in a good “type 1” game, whatever the heck that is. NBJ recommends players start small and gradually build up a bankroll, and gives these players specific low risk procedures to follow for a while. I guess once you’ve started making your fortune, you can advance to higher risk play.

 One comment that got my water boiling was that NBJ players don’t stand out like card counters because they don’t cheat. That’s right. NBJ players stand out at the casino cash advance machines. Sorry, I added that. Couldn’t resist.

Also included in the newsletter is a 2 1/2 page testimonial to E. Clifton Davis by Jerry Patterson, which attempts to put Davis on a higher pedestal than Thorp, Griffin, Wong, Uston, et al. But let’s face it. While I’ve heard Patterson is a pretty nice guy, his credibility in the blackjack world has been questionable since the early eighties.

The remaining pages of the newsletter contain a table of contents from the NBJ manual, which includes such amusing topics as “Telltale Signs of Clumping,” “Testing the Water,” “Negative Betting Progressions,” and “Insuring for Less.” Tell me something, if you have a known positive insurance expectation, why would you do it for less? Better yet, would you spend $445 on the NBJ system to find out? I think not. The friendly folks at NBJ will also sell you a different “World Class” system for another $500.

The NBJ people did seem to be genuinely friendly because I called and talked to a couple of them. I spoke with Marv, who considers himself a professional blackjack player, and Suzanne, who is a part time NBJ player and has contracted with Davis to sell the NBJ systems in my area.

Marv sounded like a really nice chap over the phone and was more than willing to tell me about how important 1st and 3rd bases are to NBJ players and that he can control the table from 3rd base, presumably by taking the dealer’s bust card or sticking him with one.

He felt positive that he could guess the dealer’s hole card at the rate of about 80% the other night using NBJ techniques. But Marv was quick to add that you don’t want a whole table full of NBJ players because they take all the good cards from each other! I’m sure the casinos would say the more the merrier.

Likewise, Suzanne raved about how well the system has worked for her. I asked her if Davis supplied mathematical proof for his theories and for some reason she began telling me about his credentials.

She did contradict Marv, however, by telling me the dealer will break more often with more NBJ players at the table and that with NBJ, one can predict hole cards up to 90% of the time.

Marv and Suzanne did agree on two things, though. One of the keys to playing NBJ is using the “tens ratio,” which you derive by observing the number of tens on the table. But hey, that’s not counting because NBJ players don’t cheat.

Also, I just had to ask them both if NBJ included anything on ruin probabilities or anything like that. Both were quick to respond that NBJ uses a 12-unit stop-loss money management strategy. Marv even goes one better. If he loses 3 units, he walks.

I could go on for at least another few pages, but I’m sure the readers of Blackjack Forum have the general idea. However, in case you’re curious and haven’t tried to figure it out yet, the title of my report means New BlackJack is Full Of Holes and Basically BullSh*t to the Casinos Say Thanks power!

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What’s Taught at the NBJ Seminar

Okay, so the Boardwalker wasn’t exactly enthralled by the NBJ seminar he attended. I have since spent a considerable amount of time reading and analyzing the NBJ Home Study Manual because NBJ is not a system which will only appeal to dummies (as one might assume from the Boardwalker’s report).

NBJ is a complex system, which requires that the player first ascertain which one of six “game types” he is playing in. The patterns of wins and losses determine the game type. The player’s betting strategy will then be one of many recomended Martingale, or sometimes reverse-Martingale, betting progressions, depending on the game type. The hands are played according to a variable strategy, depending on how the player reads the clumping effects of the high and low cards. I couldn’t even begin to explain the details of the system in a short review, but that’s not necessary. What is most intriguing about Davis’ NBJ system are the theories behind it.

Regarding Davis’ Tip #9, from the hand-out which the Boardwalker received—“Treat a dealer two up like a ten”—I would assume Davis is advising less splitting and doubling when the dealer shows a two up, and more hitting on stiffs.

Most of Davis’ deviations from basic strategy, which seem weird at first perusal, would fall into the category of bankroll conservative. The other two deviations on this tip sheet—not splitting 8s vs. 9 or 10, and always insuring blackjacks—would both tend to reduce fluctuations even if they are technically incorrect plays. The fact that Davis includes advice to never split tens (Tip #8) indicates that he may be assuming a low level of expertise in many of the players who take his course.

Tip #10—When in doubt — hit—would also strike most knowledgeable players as very strange advice. If we consider, however, that the most consistent error of poor players is failing to hit stiffs vs. dealer high cards, Tip #10 might not be such bad advice for a lot of neophytes. Nobody is going to lose his shirt by always insuring his blackjacks. Some pros do this religiously as a form of low cost camouflage.

The Boardwalker is correct that this is an expensive error—but it does not come up frequently enough to hurt anyone significantly in the long run. It is also a fact that always insuring your blackjacks will reduce your bankroll fluctuations. Again, not significantly because of the infrequency of the hand, but combined with all of the other strategy deviations Davis recommends, an NBJ player would experience significantly less severe fluctuation than a basic strategy player.

The most effective technique Davis uses to reduce fluctuations is his conservative double down strategy. NBJ never advises doubling down on any player hand vs. any dealer upcard as basic strategy. Doubling down is only advised after the player predicts both the dealer’s likely hole card and the likely hit card the player would receive, depending on the game type, whether or not high cards or low cards are running, etc.

Davis really attacks the double down basic strategy, and all of the so called “basic strategy experts,”—and he likes to put “experts” in quotes. He says basic strategy and card counting fail because the cards aren’t random. NBJ players, au contraire, play on a “higher level.” They exploit the win/loss trends by card predicting. According to Davis, this is why NBJ players win such a high percentage of their double downs.

Actually, NBJ players should win a greater percentage of their double downs—but not for the reasons Davis states.

Any blackjack “expert” knows that a player who never doubles down (assuming he follows the optimal hit/stand strategy) will win more hands than a player who follows correct double down basic strategy. This is elementary. Any time you double down vs. a dealer high card (7-A), you relinquish the right to take another hit should your double down card make you stiff. Doubling down is essentially agreeing to win a smaller number of hands in order to win more money in the long run due to more action on hands that are worth the risk.

Doubling down less often, as Davis advises, would not only result in NBJ players winning a greater proportion of their hands, as he says they do, but they would also experience less volatile fluctuations to their bankrolls than if they followed double down basic strategy. (Hey, if you’re going to sell a Martingale betting progression to the general public as a wise investment, you’ve got to take what steps you can to reduce fluctuations!)

Basic strategy was not devised to reduce fluctuations, but to optimize the player’s expectation in the long run. Doubling down less often is not really a wise long run strategy for a player who wants to beat the house. If you do not take every opportunity to risk more money on favorable hands, you will not beat this game. If your bankroll cannot afford the risks associated with doubling down, you probably shouldn’t be playing blackjack. Watch out! Wake up!

The NBJ variable strategy—as opposed to basic strategy—is really not that bad. True, there is some weird advice that I’ll be damned if I can figure out. (There is a distinct possibility that E. Clifton Davis may actually be from another planet…)

But basic strategy is advised for most hands, with variable strategies allowed for the more borderline hands (depending on how the cards are “running”). The variable (and therefore less frequent) pair splits and double downs are the major differences from traditional basic strategy.

I tried playing the NBJ strategy against one of my computer practice programs (Blackjack—Your Key to Winning Play)—which, of course, must be a sacrilege of some sort. I’m not claiming I actually learned this system… I get the feeling nobody could ever quite “learn” it, since decisions are “intuition” based.

Attempting to employ the system, with the book in my lap, I discovered that—intuitively—I almost never doubled down vs. dealer high cards. Consider: if highs are “running,” there is too much risk that the dealer’s hole card will be high, giving him a pat hand. If lows are “running,” there is too much risk that I’ll make myself stiff. About the only time it ever seemed safe to double down was when the dealer showed a low card, and highs were “running.”

Double downs vs. dealer low cards are, in fact, the most profitable double down plays. Double downs vs. dealer high cards (even when correct) are the riskiest plays. This is not a new revelation, but a simple fact. Davis somehow never mentions this. I’m sure NBJ players do, in fact, win more of their double downs than basic strategy players. But I do not believe NBJ players win more money on their double downs, as Davis’ analysis—based on his private research—finds.

I consider it nonsense that an NBJ player could predict that highs or lows would continue “running,” but if an NBJ player uses his estimation of such factors to make double and split decisions, he will play a more bankroll conservative strategy than a basic strategy player or a card counter, he will win more hands, and he will win more of his doubled bets. I can almost see the above words on the cover of Davis’ next NBJ newsletter. “Arnold Snyder States That NBJ Players Win More Hands!”

Actually, Davis’ theories and explanations are bunk, but he has developed a fairly intelligent style of play for the betting progression system he is selling.

If you are going to play a betting progression system, and especially a Martingale progression like the ones Davis touts, you would want a system designed to win the greatest number of hands. To win a Martingale progression, it only takes one win. A large bet on the table is not indicative of a large advantage, as with card counting, but of a previous series of losses. It would be foolish to double your bets in risky situations with a Martingale strategy. You want to win your series so you can quickly revert to a single unit bet again.

Davis acknowledges this quite blatantly on page 93 of his Home Study Manual, when he explains the “most important reason” for always insuring your blackjacks: “…we aren’t just risking one hand. We are usually risking an entire progression…”

His logic is flawless. If you always take even money for your blackjack, instead of playing it out, you will win your series with certainty. Why play out the hand, risk pushing the dealer, and then risk losing the series on the next hand? A card counter can pull his bet back after pushing a dealer blackjack, but a Martingale man has to win as many series as possible, and abandon as few as possible. Davis’ Martingale strategy is very conservative, since he advises abandoning a series, and reverting to a one-unit bet, after only three consecutive losses. On a coin flip, a player would win 87.5 series out of a 100 with this betting progression. On his 12.5 abandoned series, his average loss would be seven times greater than his average win, and in the long run, he would break even.

Unfortunately, unlike a coin flip experiment, multi-deck casino blackjack is less than a 50-50 proposition without a betting strategy that is based on an intelligent analysis of the mathematical advantage.

Davis does not advise players to use NBJ in single-deck games, nor would I. My reason for not advising it is different from what he says. In single-deck games, a player who bases his strategy on “runs,” with the assumption that the “run” will continue, will be playing at total odds with count logic. My count goes down when I see a run of tens on the table, and such an occurrence would lead me to play as if I were less likely to be dealt a ten.

With six or eight decks, however, a current “run” of high cards or low cards will have a relatively minor effect on the hand probabilities. Davis is still at odds with count logic, but the count is less volatile in shoe games.

For many casual players who do not have the dedication to learn a card counting system, and who do not seriously entertain fantasies about professional play, NBJ is not that bad of a strategy. It might save some players from making some of the more expensive hunch plays, and could also discourage overbetting in players who have a tendency to “steam.” I feel sorry for players, however, who believe that NBJ will turn them into blackjack pros.

The system should appeal to those who prefer to embrace all of the “common sense” myths about blackjack which we know to be baseless. Insuring blackjacks. The third base player controlling the table. Bad players affecting other players’ expectations. Dealer hot streaks and cold streaks. Davis does not really explain the real logic of his system, as he does not admit that NBJ is just a complex progression system. He acts as if he’s discovered some magic blackjack secret based on the non-random shuffle.

Some of his explanations are a howl. On page 15 of his manual he insists that one of the reasons computers cannot be used to analyze casino blackjack is because you cannot program in “the humidity,” and also, there are “…certain player types who can change the odds of the entire game.” He has no shame about spouting such nonsense.

Many players who use NBJ might feel that they are winning more than they are losing. One reason is that they will win more hands. Another is that they will, in fact, win so many more of their betting series than they will lose, that it will feel as if they are winning much more. In fact, some will win, and some will lose. More will lose in the long run.In short runs of play, the positive and negative fluctuations will be pretty wild.

NBJ is not a cut and dried system. The NBJ player is encouraged to “educate” his “intuition,” and to play according to it. If you predict a hole card or hit card incorrectly, it’s not necessarily the system’s fault, it could be your fault for not reading the cards correctly. There’s a chapter on “enhanced card reading,” which is to card counting what numerology is to arithmetic. In my opinion, the Manual leaves a lot of questions unanswered.

Regarding Jerry Patterson’s TARGET system, Davis admits that TARGET and NBJ are “compatible” and “compliment each other.” There’s a chapter titled “Target and NBJ—The Perfect Marriage.” Now there’s a match made in heaven…

NBJ begins with the TARGET premise that the non-random shuffles make the game predictable and exploitable without card counting. On page 41 of his Manual, Davis provides “proof” that casino games are not random. Simple observation, he informs us, tells us that there are “good games and bad games.”

If the shuffles were random, he reasons, “…all games would be the same.” Using this same reasoning, if I were to flip a coin one hundred times in succession, always betting on heads, and then if I were to do this series of one hundred flips again and again, always betting on heads, I should never experience “good games and bad games” with an honest (random) coin. In fact, multi-deck games are not always well-shuffled, and certainly aren’t randomized. I have no argument with that. Professional players do exploit these games, but only with mathematically justifiable methods.

Basic strategy and card counting have been computer tested extensively in poorly shuffled games. Numerous articles have been published in Blackjack Forum in the past decade reporting on these findings. Davis simply ignores all of the literature on this subject.

One amusing note: throughout his Home Study Manual, Davis concludes many of his analyses with the words: “It’s a thinking man’s game!”

On page 56, Davis teaches us that traditional basic strategy assumes that the dealer’s hole card is a ten. However, he reasons, since only 30.8% of the cards in a deck are tens, he can improve on basic strategy simply by coming up with a hole card prediction that is correct more than 30.8% of the time! (Run that by me again…?)

Most serious players realize that basic strategy considers the distribution of all of the cards in the deck(s). No assumptions are made about the dealer’s hole card, other than its proportionate likelihood of being any one of the available cards.

In any case, even if we accept the betting progressions as bunk, and the strategy deviations as simply designed to win more betting series, and all of Davis’ theories about basic strategy and card counting as tongue-in-cheek humor, is there any possibility that Davis is suggesting anything of value to the player? Is it conceivable that a player might profit from playing his hands differently according to how the high cards and low cards are “running?”

To be honest, I’ve never seen a computer simulation of casino blackjack in which the player made strategy decisions by predicting that the short run pattern of the cards would continue. Personally, I tend to doubt that there would be any value to such a strategy. Disregarding the humidity factor, it seems to me that a computer-simulated casino-style non-random shuffle would suffice for the test.

It would also be necessary to define much more specifically than Davis does exactly how to determine the type of game, and what quantities and proportions of high, low and middle cards determine when something is “running” or stops “running.” Computers don’t have a lot of intuition. Some of Davis’ theories could be computer tested, but his NBJ system, as it is presented, could not. There is just too much guesswork.

If, despite my remarks, you believe Davis’ theories are worth investigating, then I’d be interested in hearing about your personal experiences with his methods. If anyone has tested card “running” strategies via computer simulation, we would be interested in your findings. I did talk with one other Blackjack Forum subscriber about NBJ, and he liked the system, though he acknowledged very limited casino experience with it.

From what I’ve seen, Davis is simply selling a betting progression system, and calling it the road to riches. Same old baloney. ♠

When to Rebuy in Multi-table Poker Tournaments

by Pikachu
(From Blackjack Forum , Fall 2006)
© Blackjack Forum 2006

In a recent post on rebuy analysis on the Poker Forum of this Web site, I ended with the conclusion that a player without skill should not rebuy in a percentage payback tournament, and a player with skill should rebuy if he’s got a certain (undefined) edge. While this served to illustrate the point that a winner-take-all example is not an appropriate model for multi-table tournaments, it wasn’t a very useful post. Now I’ll provide some examples from a multi-table tournament format that take a player’s skill level into consideration.

For these examples I make the following assumptions:

• 100 player field
• $100 buyin gets 100 in tournament chips
• A $100 rebuy gets an additional 100 tournament chips
• All 99 opponents have the same number of chips (makes the calculations easy)
• The “edge” translates into the increased probability of taking 1st place, and a lesser increased probability of lower finishes. So an unskilled player will have a 1% chance of finishing in 1st, while a player with a 10% edge will have a 1.1% chance of finishing in 1st. Note that this players ROI will be less than 10%. And yes, I might be defining it strangely.
• The following payout structure:

Place	% Pool
1	29.0%
2	18.5%
3	12.0%
4	10.0%
5	8.0%
6	6.5%
7	5.5%
8	4.5%
9	3.5%
10	2.5%

For the base case, an unskilled player who hasn’t rebought:

Rebuys 0
Edge 0.00%
Place	Prob	Return
1	1.00%	$ 29.00
2	1.00%	$ 18.50
3	1.00%	$ 12.00
4	1.00%	$ 10.00
5	1.00%	$ 8.00
6	1.00%	$ 6.50
7	1.00%	$ 5.50
8	1.00%	$ 4.50
9	1.00%	$ 3.50
10	1.00%	$ 2.50
EV $0.00

If he rebuys we get:

Rebuys 1
Edge 0.00%
Place	Prob	Return
1	1.98%	$ 58.00
2	1.96%	$ 36.63
3	1.94%	$ 23.52
4	1.92%	$ 19.40
5	1.90%	$ 15.36
6	1.88%	$ 12.35
7	1.86%	$ 10.34
8	1.84%	$ 8.37
9	1.82%	$ 6.44
10	1.80%	$ 4.55
EV ($5.04)

As expected, an unskilled player should not rebuy. He doubles his buy-in but does not double his chances of finishing in the money.

Now, consider a player with an 10% edge:

Rebuys 0	Rebuys 1
Edge 10.00%	Edge 10.00%
Place	Prob	Return	Place	Prob	Return
1	1.10%	$ 31.90	1	2.18%	$ 63.80
2	1.10%	$ 20.33	2	2.15%	$ 40.21
3	1.10%	$ 13.17	3	2.13%	$ 25.77
4	1.10%	$ 10.97	4	2.10%	$ 21.21
5	1.10%	$ 8.76	5	2.07%	$ 16.76
6	1.09%	$ 7.11	6	2.05%	$ 13.45
7	1.09%	$ 6.01	7	2.02%	$ 11.23
8	1.09%	$ 4.91	8	2.00%	$ 9.07
9	1.09%	$ 3.82	9	1.97%	$ 6.97
10	1.09%	$ 2.72	10	1.94%	$ 4.91
EV $9.71	EV $13.38
ROI 9.71%	ROI 6.69%

Without rebuying that 10% edge translates into a 9.71% ROI, or $9.71 in EV. Rebuying increased EV (though it lowers ROI). That 10% edge should not be at all difficult to attain (as anyone who has played a low-limit fast tournament can attest to). In all cases where the player has an edge, his ROI when rebuying will be lower than when not rebuying, though the EV will be higher. This assumes no juice on the tournament.

While edges far in excess of 10% are easily had, it may be interesting to see what edge is required to overcome the intrinsic disadvantage of rebuying in a percentage payback tournament. Brute force says:

Rebuys 0	Rebuys 1
Edge 5.77%	Edge 5.77%
Place	Prob	Return	Place	Prob	Return
1	1.06%	$ 30.67	1	2.09%	$ 61.35
2	1.06%	$ 19.56	2	2.07%	$ 38.70
3	1.06%	$ 12.68	3	2.05%	$ 24.82
4	1.06%	$ 10.56	4	2.02%	$ 20.45
5	1.06%	$ 8.44	5	2.00%	$ 16.17
6	1.05%	$ 6.85	6	1.98%	$ 12.99
7	1.05%	$ 5.80	7	1.95%	$ 10.86
8	1.05%	$ 4.74	8	1.93%	$ 8.78
9	1.05%	$ 3.68	9	1.91%	$ 6.75
10	1.05%	$ 2.63	10	1.89%	$ 4.76
EV $5.61	EV $5.61
ROI 5.61%	ROI 2.80%

In the case of a player with a 5.77% edge, he hasn’t gained or lost EV by rebuying, but merely increased his variance. It is worth noting that a player wanting to maximize his EV/VAR should not rebuy unless his edge is greater than 103%. This suggests that a skilled player with a sufficiently small bankroll would not want to rebuy in many cases.

For a decently bankrolled player with a 30% edge our payback table looks like this:

Rebuys 0	Rebuys 1
Edge 30.00%	Edge 30.00%
Place	Prob	Return	Place	Prob	Return
1	1.30%	$ 37.70	1	2.57%	$ 75.40
2	1.30%	$ 23.98	2	2.53%	$ 47.33
3	1.29%	$ 15.51	3	2.49%	$ 30.20
4	1.29%	$ 12.88	4	2.45%	$ 24.76
5	1.28%	$ 10.27	5	2.41%	$ 19.48
6	1.28%	$ 8.32	6	2.37%	$ 15.56
7	1.28%	$ 7.02	7	2.33%	$ 12.95
8	1.27%	$ 5.72	8	2.29%	$ 10.41
9	1.27%	$ 4.44	9	2.25%	$ 7.96
10	1.26%	$ 3.16	10	2.21%	$ 5.59
EV $28.99	EV $49.65
ROI 28.99%	ROI 24.82%

Rebuy!

Consider what happens when we add a $9 entry fee. Assume rebuys are juice free. Playing with a 10% edge (pre-juice) is not so great anymore:

Juice 9	Juice 9
Rebuys 0	Rebuys 1
Edge 10.00%	Edge 10.00%
Place	Prob	Return	Place	Prob	Return
1	1.10%	$ 31.90	1	2.18%	$ 63.80
2	1.10%	$ 20.33	2	2.15%	$ 40.21
3	1.10%	$ 13.17	3	2.13%	$ 25.77
4	1.10%	$ 10.97	4	2.10%	$ 21.21
5	1.10%	$ 8.76	5	2.07%	$ 16.76
6	1.09%	$ 7.11	6	2.05%	$ 13.45
7	1.09%	$ 6.01	7	2.02%	$ 11.23
8	1.09%	$ 4.91	8	2.00%	$ 9.07
9	1.09%	$ 3.82	9	1.97%	$ 6.97
10	1.09%	$ 2.72	10	1.94%	$ 4.91
EV $0.71	EV $4.38
ROI 0.66%	ROI 2.09%

The rebuy is a must in this situation. Because there is no juice on the rebuy, the rebuy chips are essentially sold at a discount. Even the small discount makes a large difference in EV and ROI.


Take a look at the 30% edge player when juice is added:

Juice 9	Juice 9
Rebuys 0	Rebuys 1
Edge 30.00%	Edge 30.00%
Place	Prob	Return	Place	Prob	Return
1	1.30%	$ 37.70	1	2.57%	$ 75.40
2	1.30%	$ 23.98	2	2.53%	$ 47.33
3	1.29%	$ 15.51	3	2.49%	$ 30.20
4	1.29%	$ 12.88	4	2.45%	$ 24.76
5	1.28%	$ 10.27	5	2.41%	$ 19.48
6	1.28%	$ 8.32	6	2.37%	$ 15.56
7	1.28%	$ 7.02	7	2.33%	$ 12.95
8	1.27%	$ 5.72	8	2.29%	$ 10.41
9	1.27%	$ 4.44	9	2.25%	$ 7.96
10	1.26%	$ 3.16	10	2.21%	$ 5.59
EV $19.99	EV $40.65
ROI 18.34%	ROI 19.45%

The $9 entry fee comes directly out of our EV. Notice the ROI when rebuying is higher because of the discount.

Next, consider the situation where our player has managed to double his stack through play, and is considering an add-on. His 100 chip increase has left each other player 1.01 chips shorter.

Increased Stack 100	Increased Stack 100
Juice 9	Juice 9
Rebuys 0	Rebuys 1
Edge 10.00%	Edge 10.00%
Place	Prob	Return	Place	Prob	Return
1	2.20%	$ 63.80	1	3.27%	$ 95.70
2	2.17%	$ 40.20	2	3.19%	$ 59.64
3	2.15%	$ 25.76	3	3.12%	$ 37.78
4	2.12%	$ 21.20	4	3.04%	$ 30.74
5	2.09%	$ 16.74	5	2.97%	$ 24.01
6	2.07%	$ 13.43	6	2.90%	$ 19.04
7	2.04%	$ 11.22	7	2.83%	$ 15.72
8	2.01%	$ 9.06	8	2.76%	$ 12.54
9	1.99%	$ 6.95	9	2.69%	$ 9.51
10	1.96%	$ 4.90	10	2.62%	$ 6.63
EV $104.26	EV $102.32
ROI 95.65%	ROI 48.96%

With a 10% edge, having doubled his stack, the player should now decline the add-on.

If the player manages to triple up before the add-on, he needs to play with an edge over 20% to make the rebuy a +EV play:

Increased Stack 200	Increased Stack 200
Juice 9	Juice 9
Rebuys 0	Rebuys 1
Edge 20.54%	Edge 20.54%
Place	Prob	Return	Place	Prob	Return
1	3.62%	$ 104.87	1	4.77%	$ 139.83
2	3.52%	$ 65.12	2	4.59%	$ 85.77
3	3.43%	$ 41.10	3	4.41%	$ 53.48
4	3.33%	$ 33.32	4	4.24%	$ 42.82
5	3.24%	$ 25.93	5	4.07%	$ 32.90
6	3.15%	$ 20.48	6	3.91%	$ 25.66
7	3.06%	$ 16.84	7	3.75%	$ 20.84
8	2.98%	$ 13.39	8	3.60%	$ 16.35
9	2.89%	$ 10.12	9	3.45%	$ 12.19
10	2.81%	$ 7.02	10	3.31%	$ 8.35
EV $229.19	EV $229.19
ROI 210.27%	ROI 109.66%

From this, we can conclude that even a player with a moderate edge who is getting a small discount on his chips should not rebuy if he has managed to sufficiently increase his stack. A player an edge of 50% should still take the rebuy if his stack is less than 4.5 times greater than his initial stack. A 100% edge player should rebuy with a stack less than 7.5 times his initial stack. For very skilled players rebuys should still be made except in extreme cases.

These examples demonstrate that rebuys should not always be taken. The more skill a player has, the more often he should rebuy. The more chips the player has accumulated the less often he should add-on.

Always rebuying and adding on is a mistake. Most of the time, rebuying and adding on is a good move for the skilled, well bankrolled player. ♠

Note from Arnold Snyder on Rebuys and Add-ons

In The Poker Tournament Formula, I advise that unskilled players should not make rebuys. I advise that skilled players should always add-on or rebuy as soon as possible, in order to always have as many chips in front of you as possible.

In Appendix A of The Poker Tournament Formula, I advise players that it is futile to try to profit from poker tournaments with a small edge, and I give an example of 10% as too small an edge to play with. I also advise players that my win rate, using the exact basic strategy provided in The Poker Tournament Formula in Skill Level 2-4 tournaments, was consistently over 200%.

Higher edges will be available with fast play properly adjusted for tournament structure in slower tournaments with higher patience factors and skill levels, as defined in The Poker Tournament Formula.

And one other point. I consider Pikachu’s analysis to be of value to players in exceptional circumstances, which is why I’ve published it here. But it does not take into account the concept of an implied discount on chips due to the increased utility of chips in a big stack. Due to the implied discount, I would rebuy and add-on much more frequently than Pikachu suggests. In fact, I would almost always rebuy and add-on. ♠

The Ultimate VIC (Vision Impaired Count)

by Arnold Snyder
[From Poker World , February 1996]
© 1996 Arnold Snyder

Last year, a player wrote to me about a card counting system he had developed for his own use, which he had devised because he had less than perfect vision. He called it the “Senior Count,” because his vision problem was something that had developed with age. I wrote an article about it because the system was so ingenious, and so effective. Prior to hearing about the Senior’s card counting system, had someone told me that they had poor vision, so much so that a substantial proportion of the cards dealt, even when face up, were difficult or impossible to read clearly, I would likely have advised that person to just stick to basic strategy and waste little effort attempting to count cards.

After publishing the “Senior Count,” I received a number of letters from other players who had devised their own variations of card counting systems to adjust for their limited eyesight. The player who invented the Senior Count wrote that his specific difficulty in seeing was mostly limited to his inability to distinguish the 7s, 8s, 9s and 10s from each other. He had no difficulty discerning the court cards (J, Q, K) because of the paint, but the pip tens looked like the same blurry mess as the 8s and 9s. The low cards (2,3,4,5,6) were fairly distinguishable because there was enough white space on the cards to make the patterns of the pips stand out. All traditional card counting systems value the pip tens the same as the court cards because the game of blackjack values them the same.

The letters I received from others with vision difficulties reported similar difficulties reading the 7s, 8s, 9s and 10s. One player had no difficulty reading the card values of his own hand, or of the players seated adjacent to him on either side, but the hands of the players two or more seats away blurred. Also, one player – who was in his twenties but was severely nearsighted – said he even had difficulty reading the dealer’s upcard if he was on either end of the table. This player also objected to referring to these types of card counting systems as “senior” systems, as he was far from being a senior citizen. He suggested calling them “VIC” systems, for Vision Impaired Counting.

The idea behind the Senior Count is simply to ignore the pip tens, balancing the painted court cards (-1 each) vs. the 4s, 5s, and 6s (+1 each). Its a balanced card counting system, sort of a streamlined Hi-Opt I, with a surprisingly high betting correlation and playing efficiency for its simplicity of use. The more advanced variation of the Senior Count, which weíll call the “High-Low-Vic,” includes the ace along with the court cards as a -1 count, vs. the 3s, 4s, 5s, and 6s (all +1). This variation raises the betting correlation to a strong 90%.

The most unusual variation on this type of card counting strategy came from a player who had used the high-low count for many years, and when his eyes began to fail, decided that the best way to play conservatively was to count any indiscernible card as a ten, knowing that it was actually either a 7, 8, 9, or 10 – but feeling safer with the assumption that the card was a 10, since the removal of a ten had the most negative effect on his expectation.

This seemed to work okay for a while, until he noticed that in shoe games he continually went into double-digit negative counts. He also noticed frequent wrong assumptions, when players played out hands incongruous with his assumption that one or more cards in the hand were 10s.

His solution was to continue using all of the high-low values for the aces, court cards, and 2s through 6s, but to count all indiscernible 7s, 8s, 9s, and pip 10s as -¼ each. This device raises the betting correlation to an impressive 93%. Weíll call this the Ultimate VIC System. The method strikes me as rather cumbersome to use at the tables, but the player who devised it claims little difficulty. What he likes most about it is that he feels most comfortable continuing to use his long-memorized high-low indices, since the Ultimate VIC has the same number of plus and minus points as the high-low. Actually, if you do not have access to a computer program which will figure out strategy indices for a VIC system, I’d suggest just using a set of indices for the high-low (Wong’s) or High-Opt I (Humble’s), depending on whether or not you elect to count the ace or neutralize it.

In any case, the development of these types of VIC systems, which allow even those who have poor eyesight to extract most of the value from counting cards, provides testimony to the ingenuity of players who refuse to give up the game they’ve learned to beat, and who also refuse to allow a handicap to seriously affect their skillful play.  ♠

Missouri Law Prohibits Barring for Counting Cards

by Robert A. Loeb, Attorney at Law
(From Blackjack Forum XX #2, Summer 2000)
© Blackjack Forum 2000

A new Missouri rule prohibits barring for counting cards. Under this new rule, the Missouri Gaming Commission will specifically allow certain countermeasures by the casinos to try to minimize the advantage counters can get. The new rule has been adopted, and I am informed by the MGC that it goes into effect on August 30.

While there is nothing revolutionary about the new policy, I think that serious players may be interested in seeing the fine points of the new rule as it concerns Missouri casinos, as well as seeing the regulatory approach to governing casinos. What follows is the text of the new rule. I have added the underlining, to highlight certain portions that I will discuss.

The Text of the New Missouri Law

Title 11 — DEPARTMENT OF PUBLIC SAFETY Division 45 – Missouri Gaming Commission Chapter 5 – Conduct of Gaming 11 CSR 45-5.051 – Minimum Standards for Twenty-One (Blackjack).

(1) The following words and terms, when used in this rule, shall have the following meanings unless the context clearly indicates otherwise.

(A) “Bart Carter shuffle” means the shuffling procedure whereby approximately one deck of cards is shuffled after being dealt, segregated into separate stacks and each stack is inserted into pre marked locations within the remaining decks contained in the dealing shoe.

(B) “Determinant card” means the first card drawn for each round of play to determine from which side of the two compartment dealing shoe the cards for that hand shall be dealt.

(C) “Double shoe” means a dealing shoe that has two adjacent compartments in which cards are stacked separately and which permits cards to be dealt from only one compartment at any given time.

(2) A person who, without the assistance of another person or without the use of a physical aid or device of any kind, uses the ability to keep track of the value of cards played in blackjack (“21”) and uses predictions formed as a result of the tracking information in her/her playing and betting strategy shall not be considered to be cheating.

(3) A class A licensee may implement any of the following options at a Twenty-One table provided that the casino licensee complied with the notice requirements contained in 11 CSR 45- 5.060:

(A) Persons who have not made a wager on the first round of play may not enter the game on a subsequent round of play until a reshuffle of the cards has occurred;

(B) (Persons who have not made a wager on the first round of play may be permitted to enter the game, but may be limited to wagering only the minimum limit posted at the table until a reshuffle of the cards has occurred;

(C) Persons who, after making wager on a given round of play, decline to wager on any subsequent round of play may be precluded from placing any further wagers until a reshuffle of the cards has occurred; and

(D) (Persons who, after making a wager on a given round of play, decline to wager on any subsequent round of play my be permitted to place further wagers, but may be limited to wagering only the minimum limit posed at the table until a reshuffle of the cards has occurred.

(E) Use of double shoe with a determinate card that selected which shoe to deal from during a particular hand.

(4) If a class A licensee implements any of the options in (3) of this rule, the option shall be uniformly applied to all persons at the table; provided, however that if a class A licensee has implemented either of the options in (3)(C) or (D) of this rule, an exception may be made for a patron who temporarily leaves the table if, at the time the patron leaves, the class A licensee agrees to reserve the patron’s spot until his or her return.

(5) Immediately prior to the commencement of play and after any shuffle of the cards, the dealer shall require that the cards be cut in a manner set forth in the class A licensee’s internal controls as approved by the Commission. Such internal controls shall be subject to the following conditions:

(A) If the “Bart Carter Shuffle” is utilized and the cards in the discard rack exceed approximately one deck in number, the dealer shall continue dealing the cards until that round of play is completed after which he shall remove the cards from the discard rack and shuffle those cards so that they are randomly intermixed. After the cards taken from the discard rack are shuffled, they shall be split into three separate stacks and each stack shall be inserted into premarked locations within the remaining decks contained in the dealing shoe.

(6) After the cards have been cut and before any cards have been dealt, a floor supervisor may required the cards to be recut if he or she determines that the cut was performed improperly or in any way that might affect the integrity or fairness of the game. If a recut is required, the cards shall be recut, at the class A licensee’s option, by the player who last cut the cards, or by the next person entitled to cut the cards, as determined by the class A licensee’s internal controls.

Discussion of the New Missouri Law Regarding Barring for Card Counting

The first point that needs to be made is that Missouri is to be commended for even attempting to make an official rule governing barring and what countermeasures are to be authorized. Most state laws do not specifically cover these areas, leaving casinos free to make their own policies without governmental regulation. By way of background, it should also be noted that this rule is an amendment to Missouri’s existing regulatory scheme.

Some of the references in this rule are to other sections of existing law. For instance, existing law already allows for barring, and one of the grounds for barring is when someone is caught cheating. Thus, the language in this rule basically says that card counting is not cheating.

This eliminates card counting as a basis for barring in other sections of the law. The word “cheating” appears in other parts of the statutes as well, including criminal offenses; it is confusing that the anti-barring rule should be based on the word “cheating” because card counting was never considered by Missouri to be criminal cheating.

In any event, this is how Missouri precludes barring for card counters. The phrase “without the assistance of another person” is also important. Under this regulation, if one is deemed to be counting cards with the assistance of another person, both people can be barred.

Realistically, there is no due process; in other words, if a casino arbitrarily decides that two people are counting cards together, or signaling each other in some way, both people can be barred, with little recourse to “appeal” the casino’s actions. In conversations with officials with the Missouri Gaming Commission and officials in other states, I have learned that these regulators truly believe that team play (at the tables, as opposed to merely sharing a bankroll) should be considered improper.

The underlined phrase “licensee may implement any of the following options” begins the list of countermeasures which the casino may take in response to card counters. This suggests, and Missouri officials confirm, that these are the authorized countermeasures, and that other countermeasures should not be implemented.

The Missouri Law Implies No Preferential Shuffling, or Flat Betting or Half-Shoeing a Player

Accordingly, Missouri is allowing a casino to prohibit mid-shoe entry, to impose minimum table bets only on mid-shoe entry, to use a Bart Carter shuffle, and to use a double shoe. By inference, Missouri is not allowing preferential shuffling, flat betting, “half-shoes” or other movement of the cut card, or discriminatory restrictions on a single player at a table (all of which are allowed by New Jersey).

I am curious to see how often Missouri invokes these countermeasures after August 30, particularly the use of a double shoe. Regardless of how clear and fair these new regulations are (or are not), I think that Missouri is to be commended for taking a lead among the states by attempting to make the law clear for both players and casinos, and for attempting to treat all players in a uniform and non-discriminatory way.

It is interesting to note that nine days after running an article about Missouri allowing card counting and prohibiting barring, the Kansas City Star ran an article entitled, “Card Counters Facing New Foe — Innovative Automatic Shuffling Machine Tilts the Odds Back to the Casinos’ Favor.” This article tells about Shuffle Master’s latest continuous shuffle machine. We’ll have to see how the future plays out.

Abuse of Casino Comp Card Information

There have been a few reports of casinos inexplicably providing player’s card information to outsiders. It’s inexplicable to me because casinos have no interest in sharing this information with others, and they have no interest in making their customers angry at the release of their personal information. Here’s what’s happened.

Casinos maintain computerized information of their customers who use player’s cards. That information will typically include name, address, date of birth, and the record of the person’s play each time he visits the casino. This will include how long he played and at what games, average bet, the time of day he played, and total wins and losses.

Usually, this information remains with the casino, and is not given out except perhaps to other casinos which are part of the same company. The casino is also required to obey subpoenas and court orders, so the player’s card information will be given out in response to a subpoena, such as in criminal (grand jury) investigations, or often in disputed divorce cases.

I have learned of three reports in which player’s card information has been released without a subpoena or court order. In two cases, it involved employers checking up on employees to see if they were gambling when they were supposed to be at work. The third instance was a private individual who was just checking up on a “friend’s” gambling habits.

I don’t expect this to become a trend. In at least one of the three instances, the casino executive stated that the release of the information was contrary to casino policy. The casino industry cannot desire to release this information to outside entities. However, it is worthwhile for serious and recreational players alike to be on their toes and be aware that there is no guarantee that your personal information will be kept confidential within the casino. ♠

For more information of laws and lawsuits that affect card counters and other professional gamblers, as well as info on how to protect your rights as a player, see Beat the Players: Casinos, Cops And the Game Inside the Game, by Bob Nersesian. Also see Bob Loeb’s excellent Blackjack and the Law.

The Non-Random Shuffle and Streaks

by Dr. John M. Gwynn, Jr. and Arnold Snyder
(Blackjack Forum Vol. VIII #1, March 1988)
© Blackjack Forum 1988

(This is a synopsis of a paper on non-random shuffles and computer simulation of blackjack presented at the 7th International Conference on Gambling and Risk Takng.)

[Arnold Snyder is the author of The Blackjack Shuffle Tracker’s Cookbook: How Players Win (and Why They Lose) With Shuffle-Tracking.]

It has been public knowledge for more than 25 years now that the game of blackjack can be profitably exploited by players who diligently apply strategies based on card counting. First popularized by Dr. Edward Thorp’s 1962 best-seller, Beat the Dealer, card-counting strategies have been expounded upon, refined, and extolled by dozens of authors since. The validity of card counting has been proven many times via computer simulation of the game.

In the past five years, however, some of the standard computer blackjack simulation techniques have been seriously questioned by a number of authors, players, and system sellers. Most authorities on the game believe that the blackjack simulation methods generally employed to test systems are applicable to the game as dealt in the casino, even though all casino procedures are not simulated. The prime complaint of those who question the simulation data concerns the methodology of computer “shuffling.” In a computer simulation, the most time-efficient method of shuffling is to place cards in order via a “random number generator” that assigns each card in the deck or decks a new randomly chosen position with each shuffle.

Detractors of this methodology complain that in the casino shuffle — as applied by less than perfect human dealers — “clumps” of cards often remain together from shuffle to shuffle due to “lopsided” cuts and sloppy riffles; other procedures, such as the order in which dealers pick up played hands, tend to clump high cards with high cards in the discard rack; etc. Indeed, casino shuffles are not random. The most recent work on shuffling indicates that seven riffles are required to randomly rearrange a deck of cards.

In the past five years, numerous new blackjack systems based on these non-random shuffles have appeared. Players are being advised by some system sellers to avoid certain types of shuffles while seeking out others. Non-card-counting systems for beating the game are also being extolled as valid by some system sellers due to these non-random shuffles. Strategies based on “streaks” of wins and losses, allegedly caused directly or indirectly by the non-random shuffles, are also being sold.

Unfortunately, none of these new systems and methods has been accompanied by any published computer simulation data that would support the system sellers’ claims. Nor is any logical mathematical theory presented that would support the validity of these new systems. Some of these systems inventors have claimed they have personally produced non-random blackjack computer simulations that have proven their systems to be valid, but none have published any hard data to support this claim.

Most of the non-random shuffle systems are primarily touted to be used in multiple-deck blackjack games. In this paper, we will limit our computer simulations and analyses to single-deck blackjack games. This is due solely to the time factor involved. It is our intention to continue testing many more of these non-random theories, claims, and systems in both single-deck and multiple-deck games; the results will be published as obtained.

Analyzing Live Casino Non-Random Shuffles

In order to write a computer simulation program that would mimic the shuffle of a human blackjack dealer, it was first necessary to analyze the types of shuffles that dealers in casinos employ.

The common casino shuffling procedures were all provided by Las Vegas author Steve Forte. Mr. Forte at one time owned and operated a professional casino dealing school in Las Vegas after having spent some years as a blackjack dealer, pit boss, and casino manager. In 1986 he personally surveyed all of the major casinos in Nevada; at this time he recorded each casino’s standard series of cuts, breaks, riffles, strips, discard procedures, etc. in an effort to devise various “shuffle tracking” and “card location” strategies.

“Shuffle tracking” and “card location” are advanced strategies, based on card-counting theory, and are most often employed in conjunction with card counting. These tracking and location strategies do take advantage of “non-random” human shuffles. A player applying one of these techniques might do so by first breaking down and analyzing a casino’s shuffling routine so as to learn how to follow the positions of specified cards in the discard rack when they are reshuffled. Although little has been written on card tracking of this type, this will not be a topic of study for this paper. In fact, the data from the simulations sheds no light on this type of strategy.

The purpose of this study is to simulate non-random shuffles and attempt to answer the following questions. Does a “poor” shuffle have a notable effect on the basic strategy player’s win/loss rate? Does it cause lengthier and/or more frequent “streaks” of wins/losses? Does it lower the effectiveness of card counting in determining favorable hands? Are there “biases,” independent of any conventional card counting information, induced by certain non-random shuffles, i.e., positive counts no longer indicate a player advantage because the deck is “dealer biased?” Are certain shuffles disadvantageous to players? Et cetera.

In short, how does nonrandom shuffling affect the game of blackjack? In this paper, we initially chose to simulate two of the most common single-deck shuffles, Riffle-Riffle-Strip-Riffle and Riffle-Strip-Riffle-Riffle. Mr. Forte indicated that most casinos require one of these as a minimum single-deck shuffling procedure. It was then decided to add Riffle-Riffle-Riffle, a procedure that would be used if the dealer omitted the intervening strip; this would mix the cards a little less.

In 1987, Anthony Curtis obtained data from four Las Vegas dealers to determine how they broke the deck for a riffle and how well their riffling mixed the cards. He asked each dealer to perform one riffle on a new deck; then he recorded data about the riffled deck. After each riffle, the deck was reordered and the dealer riffled again. Each dealer performed ten such riffles. Curtis reported the fol lowing data about these 40 riffles.

There were 906 singly interlaced cards, 358 2-card clumps, 72 3-card clumps (less than 2 per riffle), 28 4-card clumps (less than I per riffle), 10 5-card clumps (I per 4 riffles), and only two clumps larger than five cards (I per 20 riffles). On 33 out of the 40 trials, the break was within 2 cards of the center. Three breaks were 3 cards from the center, one break was 4 cards off center, two breaks were 5 cards off center, and one break was 7 cards off center (this break resulted in a 7-card clump after the riffle). Since dealers always break and riffle more than once during a shuffle, it is unlikely that any large clump would be preserved throughout a complete single-deck shuffle.

Richard Epstein, author of The Theory of Gambling and Statistical Logic, stated that when professional dealers riffle the cards, they drop one card 80% of the time, two cards 18% of the time, and three or more cards only 2% of the time. Applying Epstein’s distribution to the same 40 breaks that Curtis reported on, a little approximation yields around 1100 singly interlaced cards, about 250 2-card clumps, and nearly 30 clumps of 3 or more cards. The Curtis data, small sample that it may be, indicates less precise riffling that Epstein reported.

This research deals entirely with single-deck blackjack under the further assumptions that a single player goes head up against the dealer and plays a single hand. The authors are aware of the non-random shuffle theory that multiple players at the table have a sianificant effect on the subsequent non-random ordering of the cards. Again, it is our intention to test the effect of multiple players, and to publish these results as obtained. This initial study is a starting point, not a definitive evaluation of all non-random theories. All simulatians used Las Vegas Strip rules: dealer stood on soft 17; doubling was permitted on any first two cards (not offer splits); only one card could be drawn to each split ace; aces could not be resplit.

Basic strategy governed the play of all hands, and insurance was never taken. The High-Low count was kept for certain measurements, but it was not used to vary strategy. The number of split hands was limited to four because a single deck was used and tens were never split. In this description, the player is identified as male and the dealer as female and right-handed.

Every attempt was made to come up with a realistic model of actual casino play, where the order of cards in the deck is possibly nonrandom. Variables that might influence this order included: was the dealer’s up card dealt before or after her down card; in what order did she pick up the player’s cards when he did not split a pair; did she pick up the player’s split hands clockwise or counter-clockwise; were the player’s cards or her cards placed in the discard rack first; in what order did she pick up her own cards; what rules governed reshuffling; did she place the unplayed cards on top of or below the cards from the discard rack when preparing to reshuffle; how did the player cut the shuffled pack before she dealt from it. A myriad of combinations was possible if one ran the gamut of reasonable choices; hence, a single realistic choice was made for each of the above variables.

When the dealer picked up the cards used in any round, she placed all of the player’s cards in the discard rack before her own. In case a pair was split, she dealt the split hands left-to-right (clockwise) but picked up the unbusted split hands right-to-left (counter-clockwise). If any split hand busted, she placed its cards in the discard rack before play continued.

With these fixed, only the shuffling technique differed from simulation to simulation; for any one simulation, the same technique was used throughout. Seven different shuffles were selected for use in the simulations:

• Random shuffIling denoted RAN.
• One perfect riffle, denoted IPR.
• Two perfect riffles, denoted 2PR
• Riffle-Riffle-Riffle, denoted RRR.
• Riffle-Strip-Riffle-Riffle, denoted RSRR.
• Riffle-Riffle-Strip-Riffle, denoted RRSR.

Three of these shuffling techniques (RRR, RSRR, RRSR) attempted to simulate casino dealer behavior; three others were obviously inadequate (NOSHUF, I PR, 2PR).

The reshuffling rules reflected dealing through 3/4 of every deck. Immediately after each shuffle, the player always cut the cards close to the center of the deck; then the dealer burned the top card by placing it in the discard rack without the player seeing it.

To begin each simulation, the deck was randomly ordered before an initial shuffle; this guaranteed no dependency on the original deck order. Hence, the simulations would only measure the differences between random and other shuffles after the cards were thoroughly mixed. Many non-random shuffle system proponents claim that the continual introduction of cards in “new deck order” is a major cause of the subsequent non-random ordering of the cards via imperfect shuffles. The authors intend to test this effect, and to publish these results as obtained.

Eight simulations were run, one using each type of shuffle and an additional one using the random shuffle. To avoid confusion, the two random simulations were denoted RANI and RAN2. Each simulation dealt 2,650,000 freshly shuffled decks, resulting in over 20,000,000 hands played (a pair split into two or more hands only counted as a single hand played). During each simulation, numerous data were gathered for later analysis; hopefully, this data would provide enough information to determine if random and simulated dealer shuffling produced significant differences in the game of blackjack.

The reason for running two simulations with a random shuffle was as follows. If the gathered data seemed to indicate any significant “difference” between two shuffling methods, the data from RAN1 and RAN2 could be compared to see if the same “differences” appeared. If so, this observed “difference” could be deemed spurious and simply the result of random fluctuations.

Results of the Non-Random Shuffle Simulations

Table I shows the results of the simulations for a flat bet of one unit, giving the total number of hands played, expectation per hand played, and expectation per unit bet. All expectations are given in percent; in addition, all expectations are per hand played unless specifically stated to be per unit bet.

The results for no shuffling agree approximately with those of Stanford Wong, who found that no shuffling gave an increase of about 0.75% over basic strategy for the 6-deck Atlantic City game. The results for 2PR are a mystery. Two standard errors for the difference in expectations between two runs is about 0 07% and 2PR’s differs from others (excluding NOSHUF) by a little more than this. Extensive analysis of the voluminous gathered data led to the following conclusion: within the limitations of this study, point count basic strategy players cannot telI the difference between simulated dealer shuffling and random shuffling.

Point count systems presume that the unseen portion of the deck contains an essentially random mix of cards. The question here is whether non-random shuffles induce sufficiently many patterns in the unseen cards to perturb a count system’s ability to determine which hands are favorable, and, more precisely, how favorable.

Based on the three runs using simulated dealer shuffles (RRR, RSRR, RRSR) and the two runs employing random shuffles (RAN1, RAN2), the High-Low count was an equally effective indicator of player advantage.

HypotheticalIy, assume that the player can sit at the table and not play some hands, i.e., bet 0. If the player bets 0 on the unfavorable hands and 1 on the favorable ones, he will obviously win. The player uses a point count to help decide which hands are favorable; there is a number K such that a true count K indicates favorability and a true count

Table 1. Flat Bet Results for the Eight Simulation Runs

Simulation	Hands Played	Exp per Hand	Exp per Unit
RAN1	20094500	-0.1124	-0.0996
RAN2	20094022	-0.1169	-0.1036
NOSHUF	20114382	0.6054	0.5375
1PR	20146678	-0.1001	-0.0889
2PR	20095750	-0.1911	-0.1694
RRR	20110333	-0.1141	-0.1012
RSRR	20098374	-0.1078	-0.0956
RRSR	20095345	-0.1016	-0.0900

Table 2. Per Hand 0-1 Betting Expectation as a Function of K Simulation Run

K	RAN 1	RAN2	RSRR	RRSR	RRR	NOSHUF	1 PR	2PR
-1	0.90	0.86	0.88	0.87	0.88	1.40	0.87	0.83
0	0.92	0.90	0.91	0.90	0.80	1.38	0.90	0.85
1	0.85	0.85	0.85	0.85	0.83	1.09	0.84	0.82
2	0.81	0.81	0.81	0.81	0.80	1.00	0.79	0.77
3	0.73	0.74	0.74	0.73	0.72	0.89	0.71	0.70
4	0.63	0.63	0.64	0.64	0.62	0.73	0.60	0.60
5	0.56	0.55	0.55	0.56	0.55	0.63	0.53	0.52
6	0.47	0.46	0.46	0.48	0.46	0.53	0.45	0.45
7	0.41	0.40	0.40	0.41	0.40	0.45	0.39	0.39
8	0.32	0.32	0.32	0.33	0.32	0.35	0.32	0.30

For each integer K from -I to 8, the difference in per hand 0-1 betting expectations was examined for all ten pairs of the simulations RAN1, RAN2, RSRR, RRSR, and RRR. None of these 100 expectation differences was significant at the 95% level. These results strongly imply that a point count system is just as effective for casino shuffling as it is in theory, i.e., when random shuffling is employed. For all eight of the simulations, Table 2 shows the observed 0- I betting expectation. For NOSHUF, the 0-1 betting expectations showed the greatest differences when compared to the other runs.

In the simulations, the player always rounded the true count to the nearest integer before using it to make a bet. In determining the player’s expectation per hand, the number of hands played included those on which 0 was bet.

If one plots the expectation per hand at integer values of K, the function increases to a maximum and then decreases. It is conceivable that the maximum might not occur at the same integer value of K for all of the simulations runs, especialIy since non-random shuffling (which includes no shuffling) was used in six of them. However, the function’s maximum occurred at the same value, K = 0, for every simulation except NOSHUF, for which it occurred at K = -1.

The hands were then partitioned into three classes: hands won, hands lost, and hands pushed. A Chi Square test with two degrees of freedom was used to compare the frequencies of hands in these three classes between pairs of simulations. At the 95% level of confidence, there was no significant difference between any two of RAN1, RAN2, RSRR, RRSR, and RRR. Hence, the frequencies of wins, losses, and pushes did not vary significantly between random shuffling and any of the three simulated casino shuffles.

Whereas the frequencies of wins, losses, and pushes seemed to be useful in measuring the differences between two shuffling procedures, the frequencies of hands dealt with positive counts, with negative counts, and with zero counts did not. Most important to the player are his expectations on hands in each of the three categories. If one considers only the hands dealt with positive counts, the per hand expectations did not differ significantly (at the 95% level) between any two of the simulations RAN1, RAN2, RRR, RSRR, RRSR; this was also the case for the difference in per unit expectations for these hands. Expectations for hands dealt with negative counts and for hands dealt with zero counts exhibited identical behavior.

Suppose the hands are partitioned into nine categories: wins with a positive count, wins with a negative count, wins with a zero count, losses with a positive count, losses with a negative count, losses with a zero count, pushes with a positive count, pushes with a negative count, and pushes with a zero count.

A Chi Square test with eight degrees of freedom was used to compare the frequencies of hands in the nine categories between each pair of simulations. For RAN1 versus RAN2, RAN1 versus RSRR, and RAN 2 versus RSRR, the difference was not significant at the 95% level; for the other 25 pairs of simulations, each difference was significant at the 99.5% level. It seems that considerable fluctuations are to be expected in the frequencies of hands in these categories.

As for expectations, however, there was little variation in each of the nine categories if one confined the discussion to RAN I, RAN2, RSRR, RRSR, and RRR. Clearly, hands in each of the three push categories always had expectation 0. For each of the other six categories, the differences in per hand and per unit expectations were examined for all ten pairs of these five simulations. Based on expectation per unit, none of these 60 differences was significant at the 95% level. If RRR was excluded from the list, the per hand expectations for hands in any category also did not significantly differ (at the 95% level) between any pair of the remaining four simulations.

Table 3. Percentages and Expectations of Hands Won and Lost with Positive, Negative and Zero Counts

	Win +	Lose +	Win –	Lose –	Win 0	Lose 0
RAN1 % Hands	15.33	16.30	16.95	19.43	11.23	12.31
RAN1 Per Hand	119.64	-107.26	121.98	-112.26	120.67	-109.55
RAN1 Per Unit	105.66	-99.69	103.31	-99.63	104.42	-99.66
RAN2 % Hands	15.31	16.29	16.96	19.33	11.23	12.33
RAN2 Per Hand	119.63	-107.26	122.00	-112.26	120.66	-109.57
RAN2 Per Unit	105.68	-99.69	103.32	-99.63	104.42	-99.66
RSRR % Hands	15.33	16.31	16.94	19.32	11.23	12.32
RSRR Per Hand	119.64	-107.22	122.00	-112.24	120.66	-109.55
RSRR Per Unit	105.69	-99.68	103.33	-99.63	104.42	-99.66
RRSR % Hands	15.38	16.36	16.89	19.26	11.23	12.32
RRSR Per Hand	119.60	-107.24	121.99	-112.24	120.69	-109.57
RRSR Per Unit	105.67	-99.68	103.30	-99.62	104.42	-99.66
RRR % Hands	15.20	16.18	17.07	19.47	11.22	12.30
RRR Per Hand	119.59	-107.19	121.94	-112.17	120.68	-109 51
RRR Per Unit	105.67	-99.68	103.32	-99.63	104.43	-99.66

For the five simulations using random and simulated casino shuffles, Table 3 shows the percentages (of all 20+ million hands) in each of the six non-push categories as well as the per hand and per unit expectations for these hands.

Some players contend that their casino observations indicate a higher percentage of player losses on positive counts than theory predicts. After examining Table 3, such players might feel vindicated by pointing out that RRSR showed a higher percentage of losses on positive counts than RAN1. If they ignored the fact that the expectations (both per hand and per unit) on these hands did not differ significantly at the 95% level, they would be disappointed that RRSR’s losses on these hands were less than RAN1’s in spite of RRSR’s higher percentage of such hands.

This example has additional implications concerning casino observations. Between RRSR and RAN1, the difference in percentages of hands lost on negative counts is only 0.06% For such a small difference to be significant at the 95% level, the player would have to observe over 2 million total hands, assuming RAN1’s percentage of losses on positive counts is the theoretical one. It seems doubtful that any player has made such extensive casino observations; even if some player has done so, it is even more unlikely that he kept accurate enough records to note this small difference.

The results for hands in the nine categories strongly imply that the simulated dealer shuffles RSRR, RRSR, and RRR do not perturb the per unit expectations of hands won or lost on positive, negative, or zero counts. They further strengthen an earlier statement that the count provides the same indication of the player’s expectation for these simulated casino shuffles as it does for random shuffling. Also, since the per unit expectations of the hands in each category do not differ significantly in spite of the variations in the frequencies, any perturbations in the frequencies of the nine categories probably cannot be exploited.

Non-Random Shuffles and Winning/Losing Streaks

Analysis of the data on winning and losing streaks indicates no significant differences between simulated dealer shuffling and random shuffling. The analysis will be confined to streaks of 1, 2, …, 9 consecutive losses and I, 2, …, 9 consecutive wins. For the purposes of this study, a streak of consecutive wins encompasses all hands from the first win and continues to (but does not include) the next loss. This means that embedded or following pushes are included as hands in the streak. Hence, Win-Push-Win-Win-Push is defined as a streak of three consecutive wins during which five hands were played. Analogously, a streak of consecutive losses begins with the first loss, continues to but does not include the next win, and includes embedded and trailing pushes.

The first test involved the frequencies of I, 2, …, 9 consecutive losses and of 1, 2, …, 9 wins. Each frequency was first divided by the total number of streaks to obtain a probability. Then a Z test was used to compare corresponding probabilities (one for each of the 18 different types of streaks) for all 28 pairs of simulations.

Comparing each of RAN1, RAN2, RRR, RSRR, and RRSR with all of the other four runs required ten sets of 18 tests each. Of these 180 tests, one streak differed significantly at the 99% level, another at the 98% level, and two more at the 95% level; these four differences occurred in testing four different pairs of simulations. The remaining 176 tests did not differ significantly at the 95% level. Hence, apart from expected statistical variation, the streak probabilities did not differ at the 95% level between any two runs.

These results strongly indicate that streaks occur with the same probability whether casino shuffling or random shuffling is employed. However, one fact seems obvious: the less random the shuffle, the more the differences in streak probabilities.

Table 4, shows some of the win-loss streak data from RAN1. This will give an approximate indication of what the player might expect to see during actual play. Negative streak lengths indicate losing streaks and positive streak lengths indicate winning streaks. The percent of streaks is the number of streaks of this length divided by the total number of streaks, which was 9,170,446 in this simulation.

Hands is the total number of hands played, including pushes, during streaks of this length. Hands per streak is column 4 divided by column 2. Expectation per hand is the net win/loss divided by the total number of hands played during streaks of this length (column 4). Expectation per unit bet is the net win/loss divided by the total money bet on all streaks of this length.

Table 4. Streak Data for RAN1

Length	Streaks	Percent of Streaks	Hands	Hands per Streak	Exp. per Hand	Exp. per Unit Bet
-10	6451	0.0703	70614	10.95	-100.26	-90.84
-9	12440	0.1357	122434	9.84	-100.28	-90.95
-8	23702	0.2585	207443	8.75	-100.23	-90.94
-7	45266	0.4936	346625	7.66	-100.22	-90.91
-6	86152	0.9395	565501	6.56	-100.29	-90.90
-5	163906	1.7873	896124	5.47	-100.33	-90.95
-4	315032	3.4353	1378864	4.38	-100.41	-90.91
-3	599285	6.5350	1966487	3.28	-100.42	-90.94
-2	1144036	12.4752	2502201	2.19	-100.52	-90.96
-1	2181809	23.7917	2384899	1.09	-100.62	-91.00
1	2404684	26.2221	2630072	1.09	100.40	95.71
2	1143846	12.4732	2501308	2.19	100.47	95.70
3	543899	5.9310	1784734	3.28	100.49	95.65
4	258802	2.8221	1131224	4.37	100.59	95.72
5	122860	1.3397	6714597	5.47	100.61	95.64
6	58107	0.6336	381254	6.56	100.52	95.61
7	27681	0.3019	212073	7.66	100.53	95.59
8	13284	0.1449	116154	8.74	100.81	95.64
9	6257	0.0682	61541	9.84	100.82	95.62
10	3074	0.0335	33521	10.90	100.48	95.82

Excepting streaks of length 1, there were always more losing streaks than winning streaks of any length; as the length increased, the ratio of losing streaks to winning streaks increased. In short, the player should expect considerably more losing than winning streaks. Since the table is truncated, the following data may be of interest.

There were 7144 streaks of 11 or more consecutive losses and 2729 of 11 or more consecutive wins. The longest number of consecutive losses was 23 and the longest number of consecutive wins was 21.

Similar to streaks of consecutive wins and losses, a streak of consecutive positive counts encompasses all hands from the first positive count and continues to (but does not include) the next negative count; embedded or following zero counts are included as hands in the streak. An analogous definition of streaks of consecutive negative counts also makes embedded and following zero counts part of the streak.

With minor exception, the authors could not find the same patterns in streaks of positive and negative counts that appeared in streaks of wins and losses. Even between two random runs there were significant differences (at the 95% level) between streak probabilities. These probabilities seem to show considerable fluctuation and are probably not a good measure of a shuffle’s adequacy. This is not surprising since the frequencies of hands with positive, negative, and zero counts also do not seem to provide a reliable measure.

Table 5. Numbers of Streaks with 30 or More Like Counts

	RAN1	RAN2	RRR	RSRR	RRSR	NOSHUF	1PR	2PR
NEG	1984	1975	2118	2000	2083	28596	5110	4262
POS	597	583	578	611	707	16679	1774	1283

The minor exceptions had to do with the numbers of streaks of length 30 or more. The numbers of such streaks were enormous for no shuffling and decreased rapidly to reasonable values as the amount of shuffling increased. This implies that grossly imperfect shuffling increases the probability of long streaks of like counts.

However, it is not clear that the long streak probabilities provide a sufficiently fine measure of the shuffle’s adequacy. There were significant differences (at the 95% level) in the frequencies of long negative streaks between RAN1 and RRR and between RAN2 and RRR.

In addition, there were significant differences in the frequencies of long positive streaks between RRSR and RSRR at the 98% level, between RRSR and RAN2 at the 99% level, and between RRSR and RRR at the 99% level. Table 5 shows the numbers of such streaks for each of the simulations. NEG indicates streaks of 30 or more hands with negative counts; POS indicates streaks of 30 or more hands with positive counts.

Data was gathered to allow computation of the correlation between outcomes of consecutive decks. After the play of each deck was completed, three accumulations took place. First, the mean expectation of this deck’s hands was computed and added to the sum of the deck means. Second, the square of this deck’s mean was added to the sum of squares of deck means. Finally, this deck’s mean was multiplied by the mean of the previous deck and added to the sum of consecutive deck products.

Intuitively, one would expect a statistically insignificant correlation for random shuffling and a statistically significant one for no shuffling; this was indeed the case. Table 6 shows the correlation coefficients for all seven shuffling methods; the note below each coefficient tells whether or not it was significantly different from zero (SIG or NOT, respectively) and at what level. The table values were computed using expectation per hand played; those obtained using expectation per unit bet were similar.

The major conclusion is that for each of the simulated dealer shuffles, there was no significant correlation between the outcome of one deck and the outcome of the next. The lack of any significant correlation between consecutive decks adds to the evidence that each of RSRR, RRSR, and RRR is an adequate shuffle.

Table 6. Correlations for Outcomes of Consecutive Decks

RAN1	NOSHUF	1PR	2PR	RRR	RSRR	RRSR
0.00032	0.00407	0.00208	-0.00098	-0.00013	-0.00053	-0.00015
SIG 99%	SIG 99%	NOT 95%	NOT 95%	NOT 95%	NOT 95%	NOT 95%

The correlation for NOSHUF and 1PR are perfect examples of values that differ significantly from zero in the statistical sense but are otherwise insignificant. Even for NOSHUF, whose coefficient is larger, the correlation between consecutive deck outcomes seems too small to be exploitable.

For single-deck play, two projects immediately came to mind as natural extensions of this research. The first would be to rerun the three simulated casino shuffles with a new deck being introduced every 100-200 hands. Of course, each new deck would be introduced in its usual order, and a simulated new-deck shuffle would precede dealing the first hand. It is conceivable that hands dealt after the initial and next two or three shuffles might have different characteristics than those dealt after more shuffIing has taken place.

A second interesting project would be to use a less precise riffle in RSRR, RRSR, and RRR. Such a riffle would be governed by a distribution with a decreased frequency of dropping single cards and elevated frequencies of dropping two or more cards; Anthony Curtis’ observations could be used to construct such a distribution. Simulations using this distribution for riffles would be run both with and without introducing a new deck every 100-200 rounds.

Acknowledgements: We owe much to Peter Griffin, who provided invaluable guidance toward proper statistical analysis of the gathered data and made several suggestions that significantly improved the paper.

We are also deeply indebted to Steve Forte, who isolated and explained each of the common shuffle techniques and procedures as employed in Nevada and Atlantic City casinos (even though we have not yet used the Atlantic City data).

In addition, we thank Anthony Curtis, who got four professional dealers to shuffle for him and provide new first hand data about how dealer riffling mixes the cards. ♠