Category: Blackjack

The Gambler’s Crimp, Shading Decks, and Other Methods for Marking Cards

by Howard Collier, with photos by Christi Collier
(From Blackjack Forum Vol. XIII #4, December 1993)
© 1993 Blackjack Forum

In 1902, S. W. Erdnase wrote, “A perfect understanding of the risks that are taken may aid greatly in lessening the casualties.” This is extremely profound. Blackjack table or no blackjack table, Erdnase laid out the truth about cheating and marked cards in his turn of the century publication, Artifice, Ruse and Subterfuge at the Card Table.

My life as an amateur magician changed forever late one summer evening when my mentor gave me a copy of the Erdnase book re-titled, The Expert At the Card Table . It was my fourteenth birthday and it broke me out of my Houdini phase forever, much to the relief of my mother.

The Expert at the Card Table is every good sleight of hand artist’s primer. If you gamble for money, it should be your primer too, particularly now with casinos popping up all over the place. North American natives, privileged with “nation within a nation” status, can now open casinos. Revenue-poor cities and states are also turning to casino gambling in an effort to refill their treasury coffers. Up and down the Mississippi, 24-hour, permanently moored riverboats have recently opened, much to the delight of everyone.

So, now that we can gamble in bars, tents and riverboats, everything is great. Right? No, not really. If you open a bunch of casinos, you’re going to need a bunch of dealers. Experienced ones would be nice, but truth is you’ve got five riverboats with 40 blackjack tables each. Assuming your 40 tables are always busy, you need 200 dealers for just the first 8 hour shift and there are three shifts a day. Suddenly, anyone who can count to 21 and shuffle a deck of cards is looking good to you.

Now think about this. If you were a grifter hustling blackjack, would you rather work in Nevada where the patrons are guarded and house security knows the tricks of your trade, or would you rather grift someplace where no one has ever grift before?

My point here, folks, is that it’s good to know thy game. Educate yourself. There are vultures. Vultures are everywhere. The majority of cheating in private games is accomplished with marked cards. Some methods are very sophisticated and some are very simple. There are lots of different ways cards can be marked. Some of these methods might work in a casino staffed by amateur pit and security personnel.

Easy Methods for Marking Cards or Exploiting Existing Marks

You can buy a factory marked, factory sealed deck of Bicycle brand readers from most magic shops. Factory marked Bicycles are examples of shade work. To shade a deck, an artist will sit with a paint brush and trim the thickness of some part or other of the design work on the back of the card. Note the flower petals in the upper right hand corners of this rider back. (See the illustraton above.)

If you have talent with a paint brush, you can find the proper shades of red and blue inks (also at the magic stores), and create customized decks for your own specific needs. The ever-popular “Bee” back design (the standard deck in many casinos) lends itself beautifully to shade work. (See the illustration below.)

Could a cheat get his own deck(s) into a casino? It depends on how amateur the operation is… (See Steve Forte’s article, “Don’t Be a Mark for Marked Cards,” for more information on cardroom controls.)

No artistic talent? Got an iron? Heat it up and press the backs of all the 10 count cards, deadening the shine. This ploy makes ready identification possible — if you know what to look for.

Don’t own an iron? Put a single drop of water on the back of a card. Wipe it off in a few minutes and it will leave a small dull spot. You can see it if you hold the deck at the proper angle to reflect light. (These marking techniques are too subtle to illustrate with photos.)

Go back to the magic shop and ask the kid behind the counter for a magician’s “card marker.” (See the illustration directly above.) A card marker is a handy little device to own. The card is inserted between the jaws of the marker and pressure is applied. The resulting mark can be felt, which means an accomplished second dealer could know without peeking whether the top card should be dealt or not.

Batch variations in the printing run can also be taken advantage of by cheats. You will find that you can buy two identical decks of cards (with borderless backs, such as Bee decks) and the ink on the edges will create a specific edge shading for each deck. This shading varies from deck to deck because of the cutting and printing process. Remove the tens from one deck and replace them with the tens from the other. The variance in the two decks’ edge shading makes “reading” the edges of the tens possible.

Daub, a sticky substance smeared onto the backs of cards, is undoubtedly the most worked subterfuge in private games. This is because you don’t have to bring in your own cards, but simply do your work on the spot. You can actually buy daub, but nobody does. Most folks make their own. Green or blue eye shadow works well on blue back decks, as does pencil graphite, ash from the ever present cigarette ashtray or dirt from the sole of the shoe.

Classically, daub is placed into the recessed head of a button sewn under the edge of the cheat’s jacket or vest. To mark a card with daub, he simply touches the back of any card with daub on his finger. The mark is faint but good enough to be seen later — if you know what to look for.

With the exception of daub, the work discussed here has been of a mechanical nature (i.e., there is a physical, permanent mark on the card). This is why daub is good (for the cheat). Along comes the last few rounds of the game and the daub artist begins wiping the backs of the cards along the table to remove the daub as he folds. This would be impossible in a normal casino environment, and would probably be difficult for a blackjack dealer to accomplish anywhere. Every cheat worries about covering his tracks — except for the sleight of hand artist.

The Gambler’s Crimp for No Permanent Mark

A sleight of hand artist uses the gambler’s crimp — a method that will leave no permanent mark on the cards. If you’re a cheat, this is a very good thing. If there are no marks on the cards, cheating is virtually impossible to prove. There are about a million different ways to do the gambler’s crimp. Some ways are better than others.

The natural act of picking up a playing card from the surface of the table lends itself to the gaff. Simply pick up a card (see the illustration directly above.) between your index finger which is on the back of the card and the thumb and middle which are on the face. By pressing the index finger down onto the back of the card, you create a little “V” shaped crimp in one corner of the card. There are dozens of crimping methods a dealer might employ in the “natural” handling of the deck.

Education doesn’t have to be costly. Learn the tricks of the trade to protect yourself from marked cards and other forms of cheating. I recommend Erdnase’s The Expert at the Card Table, especially if you play much blackjack in boats that go nowhere, roadhouse bars, decorated warehouses, and converted fire stations.

[Editor’s Note: Howard Collier is a 5th generation Texan and a sleight of hand expert. He resides in the Chinatown district of Los Angeles County. He wrote and produced the video, The Business of Blackjack (out of print).] ♠

Casino Surveillance Turns the Tables on Card Counters by Arnold Snyder (From Blackjack Forum Vol. XVIII #1, Spring 1998) © 1998 Blackjack Forum Online This article was initially titled: “The Two Faces of Camouflage.” My original outline had Part I dealing with the camouflage techniques card counters and other advantage players can use to disguise their skills from the casinos. Various Blackjack Forum writers and reporters — including G.K. Schroeder and Jim Taylor — contributed their thoughts on this subject. Part II went into the methods the casinos have been known to use to hide their surveillance efforts from players. A month ago, I realized that this feature had grown too large for a single issue; I had to split the article into two issues. As I have continued to work on both articles, I’ve also recently decided to publish Part II first, in this issue of Blackjack Forum, with Part I to follow in the next issue. This was not a coin-flip decision on my part, but a decision based upon what I see as exigency. Most of the books on card counting published in the past 3+ decades have provided varying amounts of advice on camouflage, always from the blackjack players’ perspective. Little has been written, however, on the devious tactics used by the casinos to hide their own surveillance spies. Many players are unaware of the fact that the casinos’ espionage efforts often go well beyond the eye-in-the-sky. This article provides no information whatsoever on how to play blackjack. There are no strategies, no system evaluations, no game recommendations, no tips on how to win. Yet, I believe this to be one of the most important articles we have ever published in this magazine. I suspect that many players will become angry reading what is contained herein. Not as angry, however, as the casino spies who find themselves, and their tricks, exposed. Longtime friends I have on both sides of the table — and I do have friends within the casino industry — may write me off their Xmas card lists this year, but I will let the chips fall where they may. My obligation, as always, is to my readers. The information herein is what you, my faithful flock, need to know. And, I think you will agree, many card counters need to know this now. In the next issue of Blackjack Forum, we’ll look at camouflage from a more comfortable perspective. Knowing that many casino spies read this magazine, we’ll focus on the art and science of camouflage, rather than on specific ploys that would educate the spies as well as you. For now, let’s delve into the dark side of casino surveillance, beyond the hidden cameras . . . Harrah’s Sneaks In Every coin has two sides. Never forget that the casinos are every bit as devious as you are when it comes to camouflage, and they will sometimes go to great lengths to disguise their spies. I learned firsthand about how serious the casinos took this business twelve years ago. (See Blackjack Forum, Dec ’86, “A Spy in the House of Zen.”) At the time, I was still a mailman, but once a month I was hosting “Blackjack Roundtables” in my Oakland apartment. I publicized these meetings in Blackjack Forum, which at that time had fewer than 1,000 subscribers. The Roundtables were informal discussion groups where mostly local Bay Area card counters and blackjack aficionados could meet and trade stories. Occasionally, out-of-town Blackjack Forum subscribers would show up if they were vacationing in San Francisco, but these meetings were generally small (12-36 attendees), always held in my livingroom. Admission was $12. Unbeknownst to me, one fine evening Harrah’s Tahoe Casino in Stateline, Nevada, sent a pit floor supervisor down to one of the meetings. I learned about this when a local Berkeley doctor, who had long been a comped guest at Harrah’s when he went to Tahoe, was barred on sight when he showed up at the casino to play one weekend. Harrah’s was up front with him about the reason for the barring. “You attended Arnold Snyder’s Blackjack Roundtable last month,” the pit boss explained. The dumbfounded doctor was shocked when the floorman then announced, “I was there.” When the doctor tried to play at Harvey’s across the street the next morning, another Tahoe casino where he had long been a comped guest, he immediately found himself surrounded by security guards. He was ordered to leave the casino. Harrah’s had apparently alerted the other Tahoe casinos that this high rolling doctor was a dangerous pro. The doctor called me from Nevada from a pay phone, in a very agitated state of mind, as soon as he had left Harvey’s Casino. “You’d better tell everyone who attended that Roundtable last month they can’t play at Harrah’s anymore,” he said. “Their floorman recognized me, and I recognized him. He was there.” The thought that a casino would send an employee 200 miles into a different state, posing as a player for the purpose of spying on card counters, opened my eyes to the lengths the casinos would go to in order to find card counters. Howie’s Angels Not long after this, I learned of another counter-espionage ploy being used in Las Vegas and Laughlin to identify professional blackjack teams. According to the late Paul Keen, my Las Vegas reporter at that time, ex-Uston teammate turned freelance counter catcher, Howard Grossman, had been training attractive women, who had become colloquially known as “Howie’s Angels,” to flirt with big players who were suspected of having team affiliations. According to Paul, the women would allow themselves to be “picked up” by the players in order to meet their friends and associates, get room numbers, license plate numbers and anything else they could gather to expose team operations. Paul said that when the big teams learned about this — some of them the hard way — they finally got very serious about enforcing team policies against associating with non-team members during playing trips. As Paul put it (paraphrased to the best of my recollection): “Howie knew what these guys were like. He had been there. He was one of them. Card counters have the same mentality as the casinos — they think women are dumb. If I had wanted to get myself into a private team meeting, I couldn’t do it to save my life. But some of these guys thought nothing of dragging along the bimbos they’d picked up that day by flashing wads of money around the tables — hookers, cocktail waitresses, any babe would do — women were like trophies to some of these guys. Remember all those photos Uston put in his books, where he’s got an unidentified babe hanging on each arm? The ‘Angels’ put an end to that kind of thinking!” Other Las Vegas counters told me stories about Howie’s Angels for a period of time in the mid-to-late ’80s. Unfortunately, Paul Keen — my original source — isn’t around anymore to provide any more information on this, and I lost contact with Howard Grossman years ago. (In the early ’80s, he contributed items to this publication occasionally. I’m not so sure he’d be interested in divulging much about this sordid episode in his life in any case!) I called Anthony Curtis, who knew both Paul and Grossman back then. I asked him if he’d ever heard of Howie’s Angels. He laughed. “Heard of them?” he said. “I dated one of them!” He couldn’t provide much in the way of juicy details, other than that he’d gone out with a woman back in the mid-’80’s who had been trained to count cards by Grossman, and that the term “Howie’s Angels” was well-known to the local Vegas counters. Internet Interlopers One of the easiest ways these days for the casinos to infiltrate the ranks of card counters is over the Internet. Every day in cyberspace, you can find card counters electronically congregated, sharing vast quantities of information. You may enter these discussions anonymously, and represent yourself as being anyone you want to be. In the Spring ’97 issue of Blackjack Forum, I stated: ” . . . casino personnel are lurking on every gambling website, and are specifically targeting the ‘players only’ forums and chat rooms . . . There are no ‘players only’ areas on the web! ‘Safe’ areas are a myth! Surveillance guys are just like writers; they go wherever the hell they want, and they believe they have a right to do this, by hook or by crook. It’s their job! They’re paid to do this . . . I know five players who have been added to the Griffin book . . . because of information initially published ‘anonymously’ on a popular blackjack web page. Casino personnel read the posts, put two and two together, and identified the players. Protect yourself. Always assume that anything you post on the Internet will be read by casino pit and security personnel, possibly at the very casinos where you play.” There were, in fact, five players I knew who found themselves with names (some with photos) in the dreaded Griffin Book as a direct result of posts on a public blackjack Web site. With respect for these players’ privacy, I will only say that a discussion about a unique playing opportunity in a specific location, with a veiled reference to a large blackjack team, led a casino lurker to investigate the possibility that this play might be going down in his casino. He struck pay-dirt. Five big money players who had been welcome comped customers found themselves booted from the blackjack tables in one fell swoop. Within days, they found themselves unable to play in any of the casinos where they had previously been welcome. They were not yet in Griffin, but — as with that Berkeley doctor in Tahoe — the casinos in this area were now faxing their names and photos around to each other. American casinos refer to these local agreements they have with each other to immediately fax information on suspicious or undesirable players as a S.I.N. (Surveillance Information Network), an appropriate acronym. I learned about this Web site fiasco from one of the players who called me and asked me to look on the Web site in question to see if I could find the posts that got the team busted. The player told me he’d asked one of the casino hosts whom he’d known for a long time why they had all been barred so suddenly. The host told him there were some Internet posts that made the casino suspicious. “We never posted anything ourselves,” he said, “but one of our guys called a friend about the potential value of the promotion we were playing. He made no mention of the specific casino. Unknown to us, the guy he’d called posted a message about it on that site, and someone else who we don’t even know followed up with a message that contained a blackjack team reference. The promotion was unusual, so it wasn’t hard for the casino to figure out they were being discussed on the Internet. So, they went looking for a blackjack team and just like that, we were busted.” The following month, all five players learned they were in the Griffin Book, perfect examples of the old World War II adage: “Loose lips sink ships.” Many card counters are naive about casino surveillance. In the old days, surveillance was almost solely dependent on the intelligence and experience of the pit boss. If you could fool the boss, you had it made. Counters who still think this way are setting themselves up for a fall. Surveillance in the major casinos these days takes place behind the scenes. Pit bosses are no longer expected to be detectives and bouncers; they are glorified clerks and bookkeepers who specialize in employee and customer relations. It’s their job to continually record player betting levels, making sure the dealers’ check racks are replenished as necessary, the cocktail waitresses are keeping the glasses full, the VIPs are getting the comps they want, the dealers are rotating and taking their breaks properly, the number of open tables is appropriate for the crowd conditions, and any disputes that arise in the games are settled quickly and fairly. Keep everything running smoothly; that’s the prime job for a pit boss. Surveillance is a different department. These are real detectives, not former dealers who rose up through the ranks. Surveillance departments today have lots of full-time employees and multi-million dollar budgets. Back in the mid-’80s, Harrah’s Tahoe physically sent a floorman 200 miles into California to infiltrate my blackjack roundtable. Today, the surveillance cops don’t have to travel so far; they surf the net. When I go into many of the blackjack web sites today, I am astonished at the amount of sensitive information posted on public message boards by players. There seems to be no consciousness whatsoever that the casino surveillance folks are right there, and in many cases may be the ones who are asking the questions, joining the discussions, and attempting to milk players for whatever they can get out of them. How Casino Spies Sneak In I know for a fact that there are casino personnel with passwords even to restricted areas of blackjack Web sites. One casino executive was present when the subscription price of a restricted board was budgeted for the surveillance department. “Let’s just say I was present during a very amusing conversation,” this casino person said, “when a couple of casino execs were explaining to the president of the hotel why he had to give them the money to join.” This, in fact, describes the entire process of infiltration. Instead of just cutting a check on an official casino account, surveillance personnel must use personal checks or credit cards, then get reimbursed by the casino. No false ID, fake mustaches, or computer hacking are required. Last Spring, when Blackjack Forum did its first exposé on casino surveillance systems, we published the name of the president of Casino Software & Services — Oliver Schubert. Casino Software & Services is the Las Vegas company that markets the Blackjack Survey Voice counter-catching software. Stanford Wong must have been chagrined to learn that Schubert was affiliated with this company, because Schubert was also a subscriber to his Black Chip page. Schubert, we might assume, did not send in his subscription payment on the Casino Software & Services checking account. But (luckily?), he had used his real name. So, Wong immediately canceled his Black Chip subscription. But, is this an effective protection technique? Wong knows it is not, and he has told his subscribers that it is not. Yet, for some reason, this makes them all feel safer, like their privacy has been protected. Half of the big money card counters I know have credit cards in names other than their own. It is perfectly legal and easy to obtain such cards. Does it not occur to players that if Oliver Schubert wants to read and participate in the Black/Green Chip discussions, all he has to do is call in his credit card order under a different name? You cannot keep casino people out of restricted areas by simply stating: “No Casino People Allowed.” Most casinos have policies that professional card counters are not allowed at their blackjack tables, but do counters still try to sneak in? Counters, even when specifically ordered to leave, often sneak back in, using fake ID, disguises, even risking arrest for trespassing. If Stanford Wong discovers that some girlfriend of an employee of Casino Software & Services has subscribed to the Black Chip, and he suspects that Schubert may have given her the money to do this, and that she may have given Schubert the password, could Wong have Schubert arrested for trespassing? Is there any legal deterrent to keep casino spies out of the so-called restricted areas of players-only web sites? And even if such a deterrent existed, how on earth would Wong discover that someone had a credit card in another name, or a friend who subscribed, or whatever? I asked Steve Forte — who has been a casino consultant for the past ten years or so — if he knew any casino personnel with access to restricted blackjack Web pages. He said he’d heard casino personnel discussing items they’d read on one of these sites many times, but that he did not know of any specific casino execs who were on it. “But people in surveillance have that spy mentality,” he said, “They get into that kind of stuff.” A few days later, Steve said, “You asked if I knew any casino people who had passwords to that Web site. Well, I asked one guy I know yesterday, and he says he’s been on it from the beginning.” The Futility of Spy Hunting Other players, who understandably yearn to meet fellow blackjack enthusiasts, attend gatherings of card counters sponsored by various Web sites. With regards to the possibility of casino spies at these types of parties, all I can say is: 12 years ago Harrah’s sent a floorman, posing as a player, 200 miles into a private home in Oakland, California; so what are the chances a casino today would send a spy into a public restaurant just down the block from their casino? To me, it’s not a question of if there are casino spies at these events, but which of these events have been infiltrated, and which attendees are the spies? When I quit the roundtables in 1986, many of those who were regular attendees complained. They were nickel bettors with little fear of being barred. They simply loved talking blackjack with other card counters, probably much like many of the party-goers today. “We’ll take our chances,” they pleaded with me. Some suggested that attendees be required to sign statements that they were not affiliated with any casino. I talked with my attorney about what I might do to legally restrict admission to players only. I felt such an immense responsibility for the misery and embarrassment I had caused to this doctor who had just wanted to meet with other blackjack players for camaraderie. But there was no satisfactory solution that I could see. Although I enjoyed the company of nickel bettors, it was the players at the higher end of the spectrum that made the Roundtables such a joy for me, and a real learning experience. It was at a Roundtable where I first met Al Francesco — he just showed up one night. Gustav Shoe, the Mad Russian, Sam Case — all were regular attendees. None of them would ever return. Two years after they’d started, and at the height of their popularity, the Roundtables were dead. For 12 years I have grieved over the loss of those living room blackjack discussions. They were so much fun, so informative and edu- cational in so many ways. For years I had hungered for this face-to-face contact with other players who shared my passion. The main lesson I learned from the Roundtables, however, was the hardest one of all. How to Counter Counter-Camouflage In the Casinos: 1. Big money players should always be wary of other players who are overly friendly — especially attractive, flirty females. 2. If you do pick up a date while on a playing trip, do not tell this person that you are a card counter; and, 3. Do not introduce this person to any of your companions or teammates, if you are involved in a team. I do not know if any casinos still use “angels” to identify counting teams, but it would not surprise me. Also, do not assume that an angel will necessarily pick you up at the tables. If you are suspected, you could just as easily be followed and picked up later in a bar or lounge. At Public Events: 1. Black chip card counters should avoid all public gambling events that might indicate greater than average intelligence, including U. of N. Gambling Conferences, the B.A.R.G.E. convention, etc. Tournaments are probably okay. Lots of high-rolling jerks enter tournaments. 2. Green chip counters who attend public gambling events that might indicate intelligence should remember that many casino personnel attend these events. Be careful what you say to others, and know whom you’re talking with. 3. Be especially careful about being seen with known blackjack authors and experts. I knew a moderate stakes counter who was barred at Caesars Tahoe during the 1981 Gambling Conference because he was seen talking with Ken Uston. 4. “Card Counters” events (like Blackjack Roundtables!) should probably be avoided by any player who puts enough action on the tables to qualify for RFB. Also, it is probably safest to attend these functions in areas where you don’t play. If you play in Las Vegas, it would be much safer to attend a party in Mississippi or Reno, etc., if you never intend to play there. And be suspicious of anyone who seems to be trying to get personal information out of you. Do not provide your real name to the group. On the Internet: 1. Always assume that casino spies are lurking. They are, and they may be posting messages and asking questions that make them seem like players. 2. If you provide trip reports, do not include information that could identify you to casino spies. If you provide the time(s) and date(s) of your play/observation, do not include your win/loss result, betting spread, etc. 3. If you discuss your methods of camouflage, a playing style that works, etc., do not be so specific that the next time you play you could be recognized by your unique approach. 4. Never relay precise events, conversations, etc., that took place while you were playing. Always alter your descriptions of these events sufficiently that you could not be identified as the person telling the story by someone else who was present (like a dealer, floorman, etc.). Unless there is some good reason for doing so, do not mention the name of the casino where the incident took place. All you need is one surveillance agent who reads about an event that took place in his casino to start asking questions among the employees who might remember this event. Naturally, if you were standing in an aisle, not playing, and you witnessed something unusual that you were not personally involved in, then go ahead and be specific. But remember that if you are identified via a website post, you may never know why you are getting heat when you return to some casino weeks or months later. Also, there will be casino spies who will know who you are every time you post under the same name. 5. If you are contacted via e-mail by another “player,” as a result of some message you posted, be aware that this may not be a player. Be extremely careful about divulging personal information about yourself to strangers in cyberspace. 6. It is probably safe for small stakes players to attempt to make connections with other small bettors. Casino spies are unlikely to waste their time with small fry. But public message boards are no place for big bettors to attempt to form teams or make connections. This is extremely dangerous. 7. If you post your e-mail address, be aware that you can often be identified by this address. America Online users, for instance, can obtain the names/addresses of other AOL users from AOL’s directory. If you use AOL, be careful not to supply your real name and address where other users can look you up. 8. Remember to protect the games, as well as yourself. Stating that a specific casino has great penetration is probably fine. The casino knows this, and knows that card counters look for deep penetration, but made a conscious business decision to deal deeply, usually to increase the action on the tables (and thus the drop and hold). Posting information about specific dealers who deal deeply if you toke them, however, is best left off the boards where casino spies have access. Likewise, reserve information about specific dealers who inadvertently flash hole cards or accidentally overpay bets, specific pit bosses who seem particularly stupid, etc., for your personal friends and acquaintances. Are Public Blackjack Web Sites Too Dangerous? Public blackjack Web sites can be a worthwhile meeting ground for serious blackjack enthusiasts. But players who visit these sites must use the sites for what they are, and they are not “players only.” Accept this fact, and be careful what you say. Whenever you see others posting dangerous messages, tell them. I often suspect that some of the “loose lips” posts that appear on many blackjack web sites may be placed by casino personnel. If you play at the comp level, it is reckless to publicly announce the dates you will be in Reno or Las Vegas for the purpose of setting up meetings with strangers. When someone on the web appears to be trying to get personal information out of another subscriber, be suspicious. It is not necessary to fling accusations at others. Give all the benefit of the doubt; assume that some players are naive rather than evil. The casino surveillance departments already buy every book and software program that comes on the market. I know this because they contact me with questions about these products. The casino spies are no longer all sitting in the surveillance video rooms looking at monitors. They are actively seeking out the players who threaten their tables. And they are seeking them out in their homes. Don’t find yourself barred because of “anonymous” words you posted on the Internet from the comfort of your living room. Get your head out of the sand. Take this business as seriously as the casinos do. Protect yourself. ♠

Effects of Shuffle Tracking Errors on False Key Rates

by Radar O’Reilly
(From Blackjack Forum XXIV #2, Spring 2005)
© 2005 Blackjack Forum

In his book Blackjack Ace Prediction, David McDowell has made a serious error in his ace prediction hit rate calculation that has not yet been addressed. I am addressing it because I expect it to jump out at readers some time soon, and it may lead them to overestimate their advantage using his methods.

On p. 111 of Blackjack Ace Prediction, McDowell provides ace hit rate calculations as part of calculating an overall win rate estimate for players. He estimates, based on shuffle studies, that an Ace can be expected to land on the predicted betting spot 38% of the time.

I am not, at this time, going to address his estimate of 38%.

McDowell goes on to explain in Blackjack Ace Prediction that this 38% hit rate assumption must be reduced by the probability of broken sequences and false keys. There are serious problems with the probability he provides for false keys. I will address these in a moment. For now, I want to point out that the way he makes his adjustment for broken sequences and false keys is wrong. McDowell subtracts his overall probability of broken sequences (.15) and false keys (.10) from his .38 hit rate instead of multiplying and then subtracting the product. This is wrong because a share of the broken sequences and false keys properly belong to the aces that land on the other betting spots.

By subtracting .15 and .10 (.25) from .38 he comes up with an estimated 13% hit rate on his ace bets.

Instead, he should have multiplied .38 by .25.

.38 x .25 = .095

Then, he should have subtracted .095 from .38.

.38 – .095 = .285

If McDowell’s false key probability were correct, this would give him a 28.5% hit rate on his ace bets instead of 13%.

Since he specified that he is splitting the aces with the dealer, that would have given both the player and the dealer 20.8 additional aces each beyond the 7.7 accidental aces they will receive.

20.8 x .51 = 10.608 expectation for player when he gets the keyed ace on his ace bet

20.8 x .34 = 7.072 negative expectation for player when the dealer gets the keyed ace when the player has an ace bet out.

10.608 – 7.072 = 3.536% expectation for player on hands where keyed ace hits either dealer or player hand.

To this you must add the expectation on the 58.4 hands where the 7.7 per 100 accidental or random aces will be appearing (7.7 each for both dealer and player). This means 58.4 hands per 100 ace bets played at the house edge of roughly .5%.

58.4 x -.005 = -.292
3.536 – .292 = 3.244% edge on player bets for the keyed ace.
Again, this is assuming that the author’s false key probability is correct, which I will be disputing below.
But this is still not an overall win rate. Assuming the player is able to bet 3 aces per shoe (another assumption that needs to be challenged) in the 66% penetration game the author specifies, where 1/3 of the cards are cut out of play, the player will be playing roughly 33 hands per shoe heads up at the house edge. If he uses a spread of 100 to 1000:
$3000 x .03244 = $97.32 per shoe on his ace bets
$3300 x -.005 = -$16.50 per shoe on his waiting bets

97.32 – 16.50 = 80.82 player profit per shoe
80.82/6300 action = 1.28% win rate.

So, if McDowell’s false key assumptions were correct, and if his assumption that you could use visual tracking to locate three aces per shoe were correct, on this heads-up game where he split aces with the dealer, and used a 1-10 spread, he would have an overall win rate of around 1.28%. It’s less than 1/3 of the 4% win rate McDowell provides, and it would require more than three times the bank, but at least it would be a respectable card counter level of win rate.

Unfortunately, McDowell’s assumptions are not correct. I have already addressed the problem of visually tracking three aces per shoe in my Spring 2005 Blackjack Forum article. To summarize, I make the case that McDowell cannot visually track three aces per shoe in the shuffles he describes. A more realistic assumption is one ace per shoe in the more difficult shuffles (a highly skilled tracker might be able to visually track two per shoe in the simpler ones). I also show that McDowell needs to allow for a visual tracking error rate that depends on the size of the slug he is attempting to track. McDowell allows for no visual tracking error at all.

McDowell’s assumptions about false keys are also a serious problem. McDowell’s shuffle analysis on p. 72 has us tracking cards 50 to 62 in the preshuffle stack to positions 200 through 250 in the post shuffle stack. He also uses his analysis to track cards 80 through 92 to a full deck in the post shuffle stack. On p. 77, he says that “the ordinal post-shuffle position of a card is not easily obtainable in actual play,” and says “simply knowing where this slug of thirteen cards ends up after the shuffle is enough.” These statements are accompanied by a chart showing that a pre-shuffle slug of thirteen cards will end up spread over a full deck post-shuffle.

Yet, on p. 101, when it comes to the important calculation of the probability of false keys, McDowell suddenly has us visually tracking our ace to a half deck, rather than a full deck. He does this specifically to cut his false key probability in half. Then, on p. 103, he cuts his false key probability in half again, to one quarter of what it should be, by claiming that we can use “pointer cards” to halve the number of false keys we bet. Specifically, he claims that trackers can use asymmetric face designs (the number of pips facing up versus down) or the space between the index on a card and its edge to distinguish between a false key and our real key.

Let’s look harder at this assumption.

Regarding the 22 cards that have asymmetrical pip patterns, this accounts for 42% of the possible key cards (22/52). Assuming that false keys would be turned half one way and half the other, we could eliminate 21% of the false keys (those that were turned the wrong way). This would assume that the player is able to follow the orientation of the pip pattern through the dealer’s pick up, placing the discards into the discard tray, removing the discards from the discard tray, turning one half of the cards before the shuffle, as is the procedure in the majority of U.S. casinos, then continuing to follow the orientation through the tip-over, cut, and replacing the cards into the shoe. And this is assuming the player can remember the orientation of multiple key cards to begin with.

Regarding the asymmetrical gap between the index and the edge, unless there is a huge cutting error in the manufacturing process, most decks will not show any easily detected difference from the index on one corner to the index on the other corner. The prospect of a player having six or eight decks that are all this badly miscut is very slim. There may be a card here and there noticeably miscut, but there is only value if one of these cards happens to fall as a key card, and again, the player must follow that card orientation through the entire pick-up, shuffle, and placement into shoe procedure.

Then, on p. 104, McDowell says “Predicting Aces at the table is easier than the theory makes it look.”

McDowell’s automatic assumption that players will reduce their false keys by a full 50%, and his use of this number in his win rate calculation as standard, is naive at best.

Anyone planning to actually try McDowell’s ace location methodology at an actual casino table should use a false key probability of roughly four times the number he provides (roughly .40 instead of .10). Remember, McDowell is using only a single card to key each ace, not a two- or three-card sequence, which most ace trackers use in order to greatly reduce false keys.

What does a false key probability of .40 do to the expected win rate?

First, multiply the .38 theoretical hit rate by .55 (.15 probability of broken sequences plus .40 probability of false keys).

.38 x .55 = .209
br />Subtract the probability of broken sequences and false keys from the 38% hit rate.

.38 – .209 = .171

This means a 17.1% hit rate on his ace bets.

Since we have been analyzing a heads-up game, and McDowell specified that he is splitting the aces with the dealer, this would give both the player and the dealer 9.4 additional aces each beyond the 7.7 random aces they will receive.

9.4 x .51 = 4.794 expectation for player when he gets the keyed ace on his ace bet

9.4 x -.34 = -3.196 expectation for player when the dealer gets the keyed ace when the player has an ace bet out.

4.794 – 3.196 = 1.598% expectation for player on hands where keyed ace hits either dealer or player hand.

To this you must add the expectation on the 81.2 hands where the 7.7 per 100 accidental or random aces will be appearing (7.7 each for both dealer and player). This means 81.2 hands per 100 ace bets played at the house edge of roughly .5%.

81.2 x -.005 = .406

1.598 – .406 = 1.192% edge on player bets for the keyed ace.

And this is assuming that the player never makes a visual tracking error, and never has a shoe in which the tracked ace does not make it to the expected post-shuffle deck.

But this is still not an overall win rate. Assuming the player is able to bet 3 aces per shoe (again, another assumption that I have challenged) in the 66% penetration game the author specifies, where 1/3 of the cards are cut out of play, the player will be playing roughly 33 hands per shoe at the house edge. If he uses a spread of 100 to 1000:

$3000 x .01192 = $35.76 per shoe on his ace bets
$3300 x -.005 = -$16.50 per shoe on his waiting bets

35.76 – 16.50 = 19.26 player profit per shoe

19.26/6300 action = 0.3% win rate.

What can a player do to increase this win rate? For one thing, he has to address that 50/50 split on the aces with the dealer in this game. But in order to do this, he must either spread to multiple hands, and add in their costs, or play at a crowded table, and sharply reduce the number of aces he is able to bet per hour. In his p. 122 calculations of expected return, McDowell assumes 4 bets per hour at a full table. But to get this number of bets per hour, he assumes that the player will visually track 4 aces per shoe, and that none of these tracked aces will be cut out of play. Again, these are very unrealistic assumptions.

Assuming a 17.1% hit rate on our ace bets, and a full table where the dealer gets nothing beyond his accidental aces, the player will receive 9.4 additional aces beyond the 7.7 accidental aces he will receive.

9.4 x .51 = 4.794% expectation for player when he gets the keyed ace on his ace bet

To this you must add the expectation on the 90.6 hands per hundred where the 7.7 per 100 accidental or random aces will be appearing (for both dealer and player). This means 90.6 hands per 100 ace bets played at the house edge of roughly .5%.

90.6 x -.005 = -.453

4.794 – .453 = 4.341% edge on player bets for the keyed ace.

Again, this is assuming that the player never makes a visual tracking error, and never has a shoe in which the tracked ace does not make it to the expected post-shuffle deck.

But this is still not an overall win rate. Assuming the player is able to bet one ace per hour (again, a generous assumption given visual tracking at the crowded game with 66% penetration the author specifies), the player will be playing roughly 59 hands per hour at the house edge. If he uses a spread of 100 to 1000:

$1000 x .04341 = $43.41 per shoe on his ace bets

$5900 x -.005 = -$29.50 per hour on his waiting bets

43.41 – 29.50 = 13.91 player profit per hour

13.91/6900 action = 0.2% win rate.

And again, this makes no allowance whatsoever for visual tracking error.

Note that by correcting McDowell’s arithmetic on false keys and broken sequences, he is credited for a higher ace rate than he himself estimated. But this is still not a high-enough win rate expectation for consideration by professional gamblers, especially after McDowell’s false key probabilities are corrected.

The problem for real-life gamblers who want to locate aces for profit in casinos is that David McDowell’s single key, visual tracking methodology is not a good one even in the one pass riffle and restack shuffle, with two riffles, that he uses for calculating his win rate (p. 111).♠

The Implied Discount: New Insights Into Optimal Poker Tournament Strategy

By Arnold Snyder
(From Blackjack Forum , Fall 2006)
© Arnold Snyder 2006

This article will dispel a number of unsound theories about poker tournaments that have been around for decades. These theories have led tournament authors to promote weak and even losing strategies that are the reason why so many smart and experienced poker players have found it impossible to make money in tournaments. Arguments occurring on poker discussion boards at various web sites, including this one, with regards to the rebuy strategy I propose in The Poker Tournament Formula were the impetus for this article. But the implications of this article will go beyond rebuy strategies and deal with the fundamental realities of how you make money in poker tournaments.

Among the topics I will address will be the logic of sound rebuy strategy, and specifically why rebuys are almost always the correct strategy for a skilled player, even when the rebuy chips must be purchased at the full price of the initial buy-in. I will also show the reason why a player should add-on even when he has a lot of chips, and why it is often wrong for a player to purchase the add-on when he is short-stacked.

Most importantly, this article will address the theory that the fewer chips you have the more each chip is worth, and the more chips you have the less each chip is worth, and show that this relationship is true only in very particular instances at some final tables, and is completely inadequate for understanding true chip value throughout a tournament, or devising overall tournament strategy. The peculiar idea that this theory can be applied throughout a poker tournament can be traced back to Mason Malmuth’s 1987 book, Gambling Theory and Other Topics. It has been embraced by other prominent poker authors as well as players, and has been used to guide players toward bad tournament decisions and strategies for many years. The concept is wrong because, as I will show, chip value is primarily based on chip utility (the various ways in which chips can be used), and, in the hands of a skilled player using optimal strategy, chip utility goes up with stack size.

The First Great Misconception

Much of the advice in The Poker Tournament Formula was written specifically to address errors in the existing poker tournament literature, especially a number of serious errors put forward over roughly a twenty-year period by Mason Malmuth and David Sklansky. One of the most widely held and erroneous concepts of poker tournament logic is the concept that the fewer chips you have, the more each of your chips are worth, and the more chips you have the less each of your chips are worth, and that as a tournament progresses, all chips lose value. David Sklansky presents this idea in his Tournament Poker for Advanced Players (p. 44-5), though he credits Mason Malmuth’s 1987 Gambling Theory and Other Topics as the original source of this notion. I am not aware of any reputable poker authority having ever disputed this claim, and it has come to be accepted as the common wisdom.

In a nutshell, the theory says that at the start of a $10,000 buy-in tournament, all chips are worth their face value. At the final table, however, since many of the smaller prizes have already been distributed to players who busted out in the money, the total payout to those at the final table will be less than the initial cost of the chips in play. As a recent example, the winner of the WSOP main event this year received a $12 million prize. When he beat his last competitor, however, he was holding more than $87 million in chips, for which $87 million had been paid by the 8,700+ competitors. So, the individual chips in his stack were worth less per chip to him than their initial cost.

Malmuth carries this idea further to conclude that the individual chips in a small stack are worth more per chip than the individual chips in a big stack. Let’s say a small buy-in tournament is down to the last two players, who are heads up at the final table. The remaining prizes are $3,500 for the winner, and $1,800 for second place. The player who is the chip leader at this point has $70,000 in chips, while the player in second place has only $10,000. Clearly, the value of each of the chips of the second place player is much greater than the value of each of the chips of the chip leader. Assuming the blinds are so high at this point that both players are simply all-in before every flop regardless of cards (making this a coin-flip situation for all intents and purposes), the player in second place has an 87.5% chance of taking the second place prize, and a 12.5% chance of doubling up enough times to take first place. This makes his $10,000 in chips worth more than $2,000 in prize money. So, the short-stacked player’s chips are worth more than 20 cents each. The chip leader, however, will find that his $70,000 in chips have a prize value of less than $3,300. The chip leader’s chips are worth less than 5 cents each. And ultimately, the more chips the second place player loses to the first place player, the greater the value of each of his individual chips. In fact, if he gets’ down to a single chip, that chip will have a value of more than $1,800, while the chip leader’s chips ain’t worth even a nickel each.

This is a true fact, and I don’t dispute it. But the problem is that Sklansky and Malmuth have gone on from this premise to devise rebuy strategies, and in Malmuth’s case, even entire tournament strategies, based on the idea that the chips in big stacks are worth less than the chips in small stacks. In Gambling Theory and Other Topics, p. 232, Mason Malmuth states: “As the poker tournament essays have stressed, in a percentage payback tournament, the less chips you have, the more each individual chip is worth, and the more chips you have, the less each individual chip is worth. This idea is the major force that should govern many of your strategy decisions in a poker tournament.”

I will show that extending this chip value idea to a dominant factor in devising overall poker tournament strategy is a humongous error in logic. And, because it has led to more bad tournament strategies than just about any other “truth” about tournaments ever revealed to the public, it has been a tremendously costly error in logic to players. It is the basis of the whole conservative, sit-and-wait-for-a-hand approach to tournaments, as well as bad advice on just about every aspect of tournament play from how to play a short stack to final table play to optimal rebuy strategy to satellite strategies, and more.

The reason it’s a humongous error in logic is because it starts from the assumption that all players have equal skill, and it ignores the value of a bigger stack in the hands of a skilled player. In the hands of a skilled player, all chips essentially have greater value. Indeed, as I will show, all chip purchases made by a skilled player using optimal tournament strategy are essentially made at an implied discount. This implied discount has a substantial impact on rebuy and add-on decisions.

Why More Chips Equals More Value per Chip: The Implied Discount

In order to make money at gambling, you have to actually gamble. That is, you must place money at risk on wagers on which you have an edge. The more money you can afford to wager with an edge, which is to say the more money you can put in action, the more money you will make. Sitting on your chips like a hen on an egg is not the way to make money gambling, especially in a tournament, which is, essentially, a race for all the chips.

It is incorrect to convert chips to dollar values with no consideration for how individual players might use those chips. It may seem logical that we could assign dollar values to chips since chips are initially purchased with dollars. But once the tournament begins, they cease to be dollars or even to represent dollars. You can’t cash them out. You can’t sell them, trade them, or buy anything with them. They are simply tools that are provided to players for competing in a contest. When the tournament director says, “Shuffle up and deal!” a battle has begun, and chips are nothing more nor less than ammunition.

The ground rules of this battle are pretty simple. If you run out of ammo, you’re dead. But if you outlast enough of your enemies, you will get some portion of the prize money depending on exactly how many of your enemies you bested. The player who survives the longest gets the biggest prize.

But you can’t survive by just hoarding your ammo. Your position will be attacked at regular intervals (the blinds), forcing you to spend some of your ammo even if you are attempting to avoid confrontations at all costs. These attacks on your position will start small, but they will escalate, costing you larger amounts of your ammo as the battle progresses. The only way for you to survive is to acquire more ammo, and the only way for you to do this is to engage in confrontations with your enemies and continually capture some (or all) of their ammo to add to your stockpile.

If a chip is a bullet, and I have 500 bullets, and you have 4500 bullets, you can utilize your ammo in many ways that I cannot. You can fire test shots to see if you can pick up a small pile of ammo that none of your enemies are all that interested in defending. You can engage in small speculative battles to try and pick up more ammo, and you can back out of these little skirmishes if necessary without much damage to your stockpile. Most importantly, because all of your enemies can see your huge stockpile, you can get them to surrender ammo to you without fighting, even in battles they would have won, were it not for their fear of losing everything.

So, intrinsically, each of your bullets has a greater value than each of mine purely as a function of its greater utility. This is due directly to the fact that you have so much more ammo than me.

The more chips you have, the more each chip is worth.

The only time chips do not have more value in a bigger stack is when the bigger stack is in the hands of a player who does not know how to use them. For example, any player who plays according to Harrington’s M strategy will not gain the full available advantage from having a bigger stack of chips. When you are waiting for hands, primarily playing your cards, and taking so little advantage of the edges available from other types of poker moves, your bigger stack will not be in action enough to earn you this greater value.

So, when I say that the more chips you have the more each chip is worth, that assumes that the player will be deploying his chips in such a way as to extract their full potential earning value.

Even Sklansky, despite his mistaken advice on rebuys, clearly recognizes that the dollar value of chips can be greatly increased by skill. On page 44 of Tournament Poker for Advanced Players he provides an example of a bystander, at the start of a $10,000 event, who is a highly-skilled player but who has arrived at the event too late to buy-in. Sklansky comments: “It might be worth it for him to buy your original $10,000 in chips for $30,000 because of his great skill.”

With these words, Sklansky demonstrates that he understands the guiding principle of all of the strategy in my book, The Poker Tournament Formula. This guiding principle is the theory of the implied discount. What is an implied discount? An implied discount comes from the fact that a skilled player can earn more with his chips than an unskilled player can earn. And how do I arrive at an implied discount from this fact?

If the player in Sklansky’s example had arrived on time to buy-in for this tournament, then (according to what Sklansky is telling us) this player would have essentially purchased chips valued at $30,000 to him, based on his skill, for only $10,000. When a player is getting $30,000 in value for $10,000 in cash, it is the same as getting his chips for one-third the price of the unskilled player.

Another way of saying this is to say that the unskilled player will have to buy in three times as often to get the same amount of winnings (not profits, but prize money) as the skilled player. (In fact, an unskilled player will never be able to profit, and may never even be able to make the same amount of prize money as a skilled player, no matter how many times he buys in. The point is that the skilled player is essentially getting his chips at a discount.)

And what are the implications of this implied discount? In The Poker Tournament Formula, I show that there is a huge value to any player in making a rebuy or add-on at a discount in an even playing field. Sklansky agrees with this concept (see Tournament Poker for Advanced Players, p. 94.) The implied discount would mean that, for a skilled player, a full-price rebuy or add-on is never really full price, and is thus a great value, with some exceptions that I cover in The Poker Tournament Formula. In fact, the only exception for a skilled player would be when the skilled player is so short-stacked that the extra chip purchase would not provide enough chips to sufficiently utilize his skill advantage. Again, when skill cannot be used, the implied discount doesn’t exist.

Unfortunately Sklansky, in his rebuy advice, fails to take into account both the extra value of chips to a skilled player and the limits on how skill can be deployed when chips are too few. He says, “I think a decent rule of thumb would be to add-on if you have less than the average number of chips at that point, and not otherwise.”

In effect, Sklansky is saying it’s fine for a skilled player to pay $30,000 for $10,000 in chips, but not to pay $20,000 for $20,000 in chips (in a $10,000 rebuy event). Sklansky has failed to realize that extra chips add to the amount of skill a player can use, and the amount of action he can generate with an edge.

More Implications for Rebuy Strategy: When Opponents Rebuy

Which brings us back to Mason Malmuth’s error in using his evaluation of chips based on stack size to generate a rebuy strategy. As I have just shown above, the fewer the chips a skilled player has, the more likely it is that he will not be able to acquire enough chips to fully utilize his skill, and the less he should be inclined to rebuy or add-on. (Again, this is exactly the opposite of Sklansky’s and Malmuth’s advice.)

On p. 196 of Malmuth’s Gambling Theory rebuy chapter, he says: “If you are leading in a tournament and someone rebuys, the pot is not being ‘sweetened’ for you… Discouraging your opponents from rebuying when they are broke should be an important part of your overall tournament strategy.”

Malmuth bases this advice on the math in a model in which all players have equal skill, making the model inappropriate for poker tournaments, and making his advice absolutely terrible for skilled players. In any real-world tournament, assuming that any players at your table actually give a damn what you recommend regarding their rebuy decisions, you should encourage the poor (read “conservative”) players to rebuy and discourage the more skillful (read “skillfully fast and aggressive”) players from rebuying. If you believe you’ve got an edge on the whole table, encourage them all to rebuy and add-on as much as they possibly can. If they don’t know how to use chips when they have them, believe me, they will indeed ultimately be “sweetening the pot” for you.

Again, this is because the skilled player’s chips come at an implied discount. Whenever players of lesser skill are in effect paying more for their chips than you, as a function of your implied discount, you will profit. They are, in effect, buying chips for you.

This is not something that mathematical analyses based on players of equal skill will reveal.

In this section, we have dealt with the implications of the implied discount for optimal rebuy strategy. Now let’s look more closely at the implications of the implied discount for the validity of various authors’ overall poker tournament strategies.

Broomcorn’s Uncle

In SuperSystem, Doyle Brunson described a South Texas player, Broomcorn, whose uncle occasionally joined the game but never played a hand. He simply sat there until his chips were dwindled away by the antes. Whenever Doyle’s fellow poker players encountered a tight player in a poker game, they would needle him by saying, “You’re gonna go like Broomcorn’s uncle.”

To this day, poker tournament pros use this saying to make fun of tight players, and for good reason.

Mason Malmuth may be the foremost advocate of a tight style of play in poker tournaments. On page 210 of Gambling Theory and Other Topics, Malmuth advises players that it is incorrect for a player who is short-stacked to raise with a “marginal hand” or push all-in with a “calling hand” because, since “the less chips you have, the more (relatively speaking) each individual chip is worth… This means that going out with a bang is wrong. You should try to go out with a whimper. That is, try to make those few remaining chips last as long as possible.”

And on page 204 of Gambling Theory and Other Topics, Malmuth provides an example of two players who enter a small buy-in tournament where each receives $100 in chips. Player A plays very conservatively, so that he always has exactly $100 in chips at the end of the first hour. Player B, on the other hand, plays a very aggressive style such that he busts out in the first hour three out of four times, but one out of four times he finishes the first hour with $400 in chips. Malmuth asks the question, “Who is better off?”

He answers his question on the next page: “…because of the mathematics that govern percentage-payback tournaments, we know that the less chips a player has, the more each individual chip is worth, and the more chips a player has, the less each individual chip is worth. This means that it is better to have $100 in chips all the time than to have $400 in chips one-fourth of the time and zero three-fourths of the time. Consequently, A’s approach of following survival tactics is clearly superior.”

Malmuth tells us little about Player B’s strategy other than that it is “aggressive and reckless.” I have described in detail in The Poker Tournament Formula how fast strategy earns more chips than conservative strategy while actually tending to lead to fewer confrontations and less risk of bouncing out of a tournament early. So there is no reason to equate the higher earnings of fast strategy with recklessness and increased bust-outs. (Despite this, even if Player B was a poker neophyte who had very little understanding of the game, but just liked to get his chips into action and gamble, I’d put my money on Player B’s overall earning power in tournaments before I’d bet a nickel on Player A. That’s how strong the value of aggression is in tournaments, as opposed to conservatism.)

But there is reason to equate Malmuth’s approach of “following survival tactics” with players being worse off, not better off as Malmuth claims. Although we can see that a situation can exist at a final table in which individual chips have less value in a big stack than a small stack, assuming all players have equal skill, Malmuth’s advice assumes that this situation exists throughout a tournament, when it does not.

The reason it does not exist for skilled players is that, as I have shown, chips have increased earning power when they are put into action with an edge more frequently, thus creating an implied discount on the chips. Malmuth’s tight “survival tactics”, by failing to use chips for their full earning power, are in effect leading to a higher chip cost for his tight player.

Again, until the point in a tournament when the remaining players get into the money, at which time, in some but not all tournaments, equal-skill models may start to make sense, the real truth of tournaments is that the value of chips is based on what you can do with them. Malmuth’s theory about individual chip values is dead wrong through most of a tournament. Throughout 90+% of a tournament, any individual chips in a short stack aren’t worth squat. They simply represent a last desperate shot at survival for players who will almost certainly not make it into the money.

The fast strategies prescribed in The Poker Tournament Formula are essentially a blueprint for earning more chips by getting more chips into action with an edge than you can get into action with a conservative strategy.

In the PTF’s fast strategy, for example, there are three positions from which a player would raise if first in with any two cards. On the button, players are advised to call any number of limpers and even to call any standard raise with any two cards. Postflop, players are advised to always bet if they were the pre-flop aggressor, even when out of position, even when the flop does not hit them, even when the flop looks dangerous, and even when their preflop raise was just a position or chip shot with a trash hand. Likewise, if a player has position on an opponent postflop, and he checks, the player is advised to bet—regardless of his hand. Other more complex and dangerous plays—like pretending to slow-play a hand in order to steal even more chips from an opponent when you have nothing yourself—are also described. But the standard positional preflop and postflop bets and raises with any two cards are presented as a “basic strategy” that a player in a fast tournament should almost always follow. (There are discussions in PTF on proper violations of the basic strategy based on the type of opponent you are facing, your chip position, your opponent’s chip position, poker tournament structure and other factors, but the book’s general approach to tournament strategy is overall much looser and more aggressive than the conservative approaches that have been written about in the past.)

The fast play strategy in The Poker Tournament Formula will consistently out-earn conservative play because it keeps your money in action while your conservative competitors are sitting there waiting for stronger cards, and hoping to make back, with trapping hands, what you have taken from them. But, in fast tournaments, a trapping hand is unlikely to arrive frequently enough to help you, and in all tournaments, trapping hands are unlikely to get paid off by skilled players even if they do arrive.

The fast play strategy in The Poker Tournament Formula is specifically designed to earn you an implied chip discount. It is designed to take chips from conservative opponents and penalize them for their weaker strategy.

The Weakness of Using Equal-Skill Models for Devising Overall Tournament Strategy

Although I use coin-flip examples in the rebuy chapter of The Poker Tournament Formula to explain specific points of tournament logic, it would be incorrect for any player to think that coin-flipping contests (or Malmuth’s “equal-skill tournaments”) are analogous to poker tournaments. It is always dangerous to use a simple example to solve a complex problem, because it may tempt a less astute researcher to use a seemingly similar analogy to answer a question for which that analogy does not apply.

For example, in The Theory of Blackjack, Peter Griffin created a game he called “Woolworth Blackjack,” which consisted of a blackjack game where the only cards in the deck were fives and tens. He used this hypothetical and vastly over-simplified version of the game to test how well a statistical estimate of expectation based on approximations might correspond to a player’s actual expectation. For the problem he was attempting to solve, his analogy works well.

Some years back I received a letter from a blackjack player who had used Griffin’s Woolworth Blackjack deck to devise a unique betting strategy for a real-world casino blackjack game. His strategy was terribly misguided, although it would work fine if the player ever found a casino that would deal actual Woolworth Blackjack. Casino blackjack is not Woolworth Blackjack. There are many complexities to the game as dealt in the casino that do not apply in the five-and-ten version. Griffin’s analogy worked well for the point of logic he was testing, but he never meant to imply that all blackjack problems could be solved by Woolworth.

On p. 196-197 of Gambling Theory and Other Topics, Malmuth demonstrates that in an “equal-skill” tournament with a percentage payout structure (as opposed to winner-take-all), there will be situations in which it would be mathematically incorrect to make a rebuy. (In fact, this particular example is incorrectly used by Malmuth as “proof” that players should not rebuy when they have “a lot of chips”, meaning as many as or more chips than their opponents.)

In The Poker Tournament Formula, I too show that, in a coin flip tournament where every player is of equal skill, there is no mathematical advantage in making a rebuy. I have no argument with the mathematics presented by Malmuth for this specific “equal skill” situation. The math is the math. My problem is with his assumption that an equal-skill event provides an adequate model upon which to devise a valid strategy for rebuying. Although his math in this example is internally correct, it is irrelevant to rebuy decisions in real-life tournaments.

In an excellent post on the Poker Board at this Web site, Pikachu provides much better models in a post titled “A Closer Look at the MTT Format and Playing With an Edge”. To summarize, he finds “A player [with] an edge of 50% should… 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… The more skill a player has, the more often he should rebuy.”

Pikachu also shows that players with small advantages should rebuy much less often. Pikachu’s findings, by the way, validate the rebuy advice I give in The Poker Tournament Formula. (In Appendix A, I advise players that it is futile to enter multi-table poker tournaments without a very big edge, and say that with an edge as low as even 10% you should not enter multi-table poker tournaments.)

Pikachu’s models are superior to Malmuth’s because they take into account the fact that poker tournaments are not “equal skill” events. They are especially useful because they compare a wide range of realistic player skill levels (or edges) in poker tournaments.

In Closing

The Poker Tournament Formula specifically addresses winning strategies for fast tournaments. Many of the PTF concepts and strategies, however, will also have value in slow tournaments. These strategies will require adjustments for the speed of the tournament, but the concept of the advantage of fast-play over conservative strategy is still valid. Specifically, the PTF strategies will have to be adjusted to take advantage of the increased opportunities for profitable fast play action pre- and postflop, created by the slower blind structures and bigger starting chip stacks. Optimal slow tournament strategies will be very different from the conservative strategies recommended in many of the popular books today, as many of these books were written from the same erroneous perspectives on chip values as Malmuth’s book.

A number of the best professional poker tournament players believe that tight conservative play is already nearing obsolescence even in slow tournaments. Daniel Negreanu, interviewed by Peter Thomas Fornatale in the just-published Winning Secrets of Poker, says: “Basically, the math is behind an aggressive style of play. Books that have been written in the past simply didn’t have it all right as far as what hands to be playing and what hands to be folding… So without playing that [aggressive] style, you’re basically depending on getting really good cards. And when you’re depending on getting really good cards on a regular basis, you’re going to be disappointed because they don’t come often enough.”

Many of the most successful tournament pros do not follow tight, conservative strategies, even in the slowest major events. Malmuth remarks in the “Afterthoughts” on his tournament section in Gambling Theory and Other Topics, “…some people with what appears to be excellent tournament records have very little idea of what is correct… In addition, if it were possible to estimate the standard deviation for tournaments and a good tournament record was compared to a poor one, I suspect that a seemingly large difference would often not be significant. This means that some of the current superstars are probably not very good, just fortunate.”

I agree that there is a large standard deviation in poker tournaments. However, the combined win records of the aggressive fast players have moved well beyond the realm of normal fluctuations. I suspect that the fast-playing pros who keep making it into the money, keep making final tables, and keep racking up wins, know a lot about “correct” strategy.

Part II of this article deals with correcting the idea that chip value is based on stack size at a final table or non-skill tournament. See Chip Value in Poker Tournaments, With Implications for Winning Strategy.  ♠

Optimal Blackjack Strategy with a Wagering Requirement

by Arnold Snyder
(From How to Beat Internet Casinos and Poker Rooms, Cardoza Publishing, 2005)
© 2005 Arnold Snyder

When you are playing blackjack online to meet the wagering requirement for a bonus in an Internet Casino, or in any other situation where you have a wagering requirement, the best basic strategy for blackjack changes slightly.

For those who already know blackjack basic strategy who are surprised to learn that correct strategy with a wagering requirement would be different, here is the logic as it applies to a double down decision:

If I want to know whether I should double down on a total of 9 against a dealer deuce, I have to consider the return on getting double the money on the table with this strong total while giving up the option to rehit the hand if I am dealt a 2 or 3 on it. To double down on a 9 v. 2 and catch a deuce on it is a truly miserable result. Here I am with a total of 11, that I cannot take another card on, and I have double my bet on this hand!

As it turns out, this is one of those borderline decisions that changes according to the number of decks in play. In single and double-deck, the basic strategy is to double down on the total of 9 v. 2, because having that one deuce taken out of play (the dealer’s upcard), has removed a significant enough percentage of the remaining deuces to make the double down the optimal play.

In fact, with three decks, it’s correct to double down on 9 v. 2 if my total of 9 is comprised of a 7-2, since this would mean two deuces would have been removed from the remaining cards. But as soon as we get to 4 or more decks, the basic strategy for 9 v. 2 is to hit, and not double. That’s how the logic works.

But let’s look at how much of a borderline decision this is. In a shoe game (and it will be slightly different with 4, 6, or 8 decks), if I am dealt a total of 9 v. 2, I have a approximately a 7.85% advantage over the house if I just hit. In dollars and cents, this means that with a $100 bet, my average return on this hand with basic strategy (hit) is to make $7.85.

How much money do I lose if I double down? Well, not really that much. If I double down on this hand in a 4-deck game, my win expectation is about $7.45. Card counters who table hop and play only plus counts just about always double down on 9 v. 2 because with most balanced count systems the index number for doubling down on this hand is 0. If you have just the slightest positive count, doubling down becomes the correct play.

In any case, since a return of $7.45 is less than a return of $7.85, basic strategy with 9 v. 2 is to hit in all games with more than three decks, not double down.

But, consider an Internet 4-deck game where I have a wagering requirement to fulfill. Let’s say I have a total wagering requirement of $2000, and I’ve already played through $1800 in action. In other words, I have exactly $200 of action left to meet my wagering requirement. The casino allows a $100 max bet. I place a $100 bet, and I am dealt a 9 v. 2. How should I play it?

Consider:

If I hit, I have an expected return on this hand of $7.85. I then must play one additional $100 hand, and I must assume that the cost of this random hand will have the house edge of 0.50%. This second $100 hand that I must play to meet my wagering requirement has a negative return of -$0.50. So, for these two hands, my total return is $7.85 – $0.50 = $7.35.

If, however, I violate standard basic strategy and double down on my 9 v. 2, my total return on the $200 in action will be 10 cents higher, $7.45. So, when there is a wagering requirement, basic strategy for the 4-deck game changes.

But, with 6 decks, if I double down on 9 v. 2, it will cost me about 21 more cents than hitting and playing a second hand against the house edge, so with 6 or more decks, it is best to follow the standard multiple-deck basic strategy for 9 v. 2, and just hit.

The logic here does not require that you be down to the last two bets of a wagering requirement. As long as you are playing to meet a wagering requirement, and every additional bet (double or split) that you don’t place on a hand where you have this option will require another bet on a random hand with the house edge, you will be in a situation where the value of doubling down or splitting must include the value of eliminating a random hand that must be played at the house advantage.

In any case, the value of following a Wagering Requirement Basic Strategy as opposed to a standard blackjack basic strategy where no wagering requirement is imposed is negligible. But it does exist, and smart players may want to know about it. For those who are out there playing on bonuses with wagering requirements in Internet casinos, here are the changes:

9 v. 2 = double (4 decks or fewer)
A7 v. 2 = double
A6 v. 2 = double
8 v. 6 = double down in a 2-deck game with a 5-3, but not with a 6-2
11 v. A = double down in a 2-deck game

Normal basic strategy with 9 v. 2 is to double down in 1 and 2-deck games only. With a wagering requirement, we should also double down in 4-deck games. In a 6-deck game with a wagering requirement, however, this double down would cost us an extra 21 cents on a $100 bet, so we only make the altered double down in a 4-deck game, unless we’re looking for a cheap camo play in 6-deck.

Normal basic strategy with a total of 8 v. 6 is to double down in single-deck only. With a wagering requirement, we are correct to double down on 8 v. 6 in 2-deck games if our cards are 5-3, but not 6-2. With more than 2 decks, it is not correct to double down on 8 v. 6 with a wagering requirement.

Double down on 11 v. A in a 2-deck game. Normal basic strategy is to double down on 11 v. A in single-deck only, or in multi-deck if the dealer hits soft 17. In Theory of Blackjack, Griffin provided refinements to this rule, namely that if the player’s 11 is a 6-5 or 7-4 (but not a 9-2 or 8-3), it is also correct to double down in 2-deck. With a wagering requirement, it is optimal to double down with 11 v. A with 6-5, 7-4, and 8-3, but 9-2 is still on the other side of borderline. With more than two decks, however, do not double down on this hand.

Comprehensive Wagering Requirement Online Blackjack Strategy for Any Number of Decks

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STAND

Stand	2	3	4	5	6	7	8	9	X	A
17	S	S	S	S	S	S	S	S	S	S
16	S	S	S	S	S	H	H	H	H1	H
15	S	S	S	S	S	H	H	H	H	H
14	S	S	S	S	S	H	H	H	H	H
13	S	S	S	S	S	H	H	H	H	H
12	H	H	S	S	S	H	H	H	H	H
A7	S	S	S	S	S	S	S	H	H	S2

DOUBLE DOWN, HARD TOTALS

Double	2	3	4	5	6	7	8	9	X	A
11	D	D	D	D	D	D	D	D	D3	D12
10	D	D	D	D	D	D	D	D	H	H
9	D11	D	D	D	D	H	H	H	H	H
8	H	H	H	D5	D13	H	H	H	H	H

DOUBLE DOWN, SOFT TOTALS

Soft Totals	2	3	4	5	6	7	8	9	T	A
(A,9)	S	S	S	S	S	S	S	S	S	S
(A,8)	S	S	S	S	D5	S	S	S	S	S
(A,7)	Ds	Ds	Ds	Ds	Ds	S	S	H	H	S2
(A,6)	D	D	D	D	D	H	H	H	H	H
(A,5)	H	H	D	D	D	H	H	H	H	H
(A,4)	H	H	D	D	D	H	H	H	H	H
(A,3)	H	H	D5	D	D	H	H	H	H	H
(A,2)	H	H	D5	D	D	H	H	H	H	H

SURRENDER (LATE)

Late Surrender	2	3	4	5	6	7	8	9	X	A
17										¢6
16								¢7	¢	¢8
8-8										¢9
15									¢10	¢6
7-7									¢5	¢9

S = Stand, H = Hit, D = Double Down (if doubling not available, then hit), Ds = Double Down (if doubling not available, then stand),
в = Surrender

1 = Stand with 3 or More Cards
2 = Hit in Multi-Deck, or if Dealer Hits S17
3 = European No-Hole Hit
4 = S17 Multi-Deck or European No-Hole Hit
5 = Single-Deck Only
6 = With Hit Soft 17 Only
7 = Single Deck Hit
8 = Single Deck, X-6 Only
9 = With Hit Soft 17 in Multi-Deck
10 = Excluding 8,7
11 = 4 decks or fewer only
12 = Always double in H17 games. In S17 games, double in single and 2-deck games only.
13 = Double in single deck games. In 2-deck games, double on 5-3 but not 6-2. With more than 2 decks, do not double.

PAIR SPLITS
WITH DOUBLE AFTER SPLITS

Pairs	2	3	4	5	6	7	8	9	T	A
(A,A)	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y1
(T,T)	N	N	N	N	N	N	N	N	N	N
(9,9)	Y	Y	Y	Y	Y	N	Y	Y	N	N
(8,8)	Y	Y	Y	Y	Y	Y	Y	Y	Y1	Y1
(7,7)	Y	Y	Y	Y	Y	Y	Y2	N	N	N
(6,6)	Y	Y	Y	Y	Y	Y2	N	N	N	N
(5,5)	N	N	N	N	N	N	N	N	N	N
(4,4)	N	N	Y2	Y	Y	N	N	N	N	N
(3,3)	Y	Y	Y	Y	Y	Y	Y2	N	N	N
(2,2)	Y	Y	Y	Y	Y	Y	N	N	N	N

PAIR SPLITS
NO DOUBLE AFTER SPLITS

Pairs	2	3	4	5	6	7	8	9	T	A
(A,A)	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y1
(T,T)	N	N	N	N	N	N	N	N	N	N
(9,9)	Y	Y	Y	Y	Y	N	Y	Y	N	Y
(8,8)	Y	Y	Y	Y	Y	Y	Y	Y	Y1	Y1
(7,7)	Y	Y	Y	Y	Y	Y	N	N	N	N
(6,6)	Y2	Y	Y	Y	Y	N	N	N	N	N
(5,5)	N	N	N	N	N	N	N	N	N	N
(4,4)	N	N	N	N	N	N	N	N	N	N
(3,3)	N	N	Y	Y	Y	Y	N	N	N	N
(2,2)	N	Y2	Y	Y	Y	Y	N	N	N	N

INSURANCE: NO

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There may be a few other violations of standard blackjack basic strategy that would bring you an extremely small extra return in particular games, based on the exact number of decks in play and the precise rule set, when you are playing to meet a wagering requirement, but they will not be worth much to players in dollars and cents.

Also, some players have questioned whether correct online blackjack basic strategy would change again in a situation where you have a win target, such as when you are playing on a sticky bonus. It turns out that the online blackjack strategy for win target situations with a wagering requirement is the same as the regular Wagering Requirement Online Blackjack Strategy. I will explain why in a separate Blackjack Forum article.

Again, the value of following a Wagering Requirement Online Blackjack Basic Strategy as opposed to a standard blackjack basic strategy where no wagering requirement is imposed is negligible. But it does exist, and smart players should know it. ♠