Author: Arnold Snyder

A New Roulette Strategy from an Old Play

by Arnold Snyder
(From Blackjack Forum Vol. XXIX #1, February 2013)
© 2013 Arnold Snyder

Roulette Culture

The most surprising thing for a blackjack player about the transition to roulette is the culture shock. Blackjack has had a few staid decades now since the transition to corporate casinos and shoe games and auto-peek devices. (It’s getting less staid again with all the new hand-held games that have been launched since the invention of bad payouts on naturals).

But roulette is like going back to the days when dealers peeked and Mickey MacDougall had to be at the table to protect Thorp from card mechanics. The thing to remember is that those were also the golden days before Beat the Dealer had been published, when the handful of people who had figured out how to beat the game had the opportunities all to themselves.

One way that roulette differs from modern blackjack is that at most U.S. casinos there is either no fixed dealing procedure for the game, or whatever procedure is in place is never enforced, except when it comes to payouts. Casinos are universally strict about procedures for roulette payouts. But everything else is up for grabs.

In recent weeks of looking at roulette in Las Vegas, I’ve seen dealers who were clearly trying to aim for players give the ball less than three spins. Most dealers have been giving the no-more-bets wave 3 to 4 spins from the time the ball falls off the wheel track, but it hasn’t been unusual to see dealers waving off bets ten or twelve spins out, even when big action players are still trying to get their bets onto the layout.

These dealers seem to have no awareness at all that many players prefer to bet after the spin and that this is a long-standing tradition in the game. The reason it’s a long-standing tradition is that players believe dealers can steer the ball and worry that dealers will try to steer against them. Dealers who wave off right after the spin seem to have no clue that action roulette players will keep chunking out chips on the layout as long as they are allowed to, and that dealers who wave off such players ten or twelve spins out are costing the house money. Inexplicably, the house allows this.

Between spins, some dealers remove the ball from the wheel and set it on the table, while others leave the ball in the wheel and only pick it up again when it’s actually time to launch the next spin.

Some dealers spin the wheel by giving the rotor edge a light nudge with their fingertips; other dealers grab a fret (the divider between pockets) and shove it to boost wheel speed.

Dealers are allowed to spin the wheel at any speed they want, even when the wheel speed is causing the ball to fly out of the wheel and off the table repeatedly during their shifts. No one in the pit seems to care about delaying the game to hunt down the ball and respin. When players at the table (typically Europeans) complain about the wheel speed, they are ignored.

To summarize, in the U.S. every roulette dealer seems to have his or her own idea of how the game ought to be run, and, with the exception of payout procedures, every one of them is allowed to deal the game however he or she wants.

To get an idea of how bizarre the lack of procedural uniformity is for U.S. roulette, imagine a world in which blackjack games were dealt with a similar lack of standards. Some dealers would be dealing one deck of a six-deck shoe, others would be dealing three decks, and others would be dealing every last card to the bottom. Some dealers would deal the game face up, while others, at their own discretion, would allow players to pick up the cards with one or both hands.

Imagine playing blackjack at a table where the cards dealt regularly flew past the table edge and landed on the players’ laps or the floor, or at a casino where every dealer could deal from either the top of the deck or the bottom of the deck on any hand, and use the auto-peek device or peek manually according to her mood. Think of what it would be like if every dealer was permitted to design his own shuffling procedure, or just move the cards from the discard tray into the dealing shoe without a shuffle whenever he felt like it.

That would be the rough equivalent of roulette as it currently exists in the U.S., especially Las Vegas. At no other casino game that I’m aware of does the dealer wield such absolute control over dealing procedures. Blackjack players familiar with standardized dealing procedures and dealers whose attitudes toward the players don’t affect the outcome of the game will be amazed by the loose roulette dealing culture.

I haven’t scouted roulette in Europe or elsewhere in the world, except for parts of Canada, so it may be different elsewhere. This article will focus on roulette as dealt in the U.S.

Roulette Culture from the Players’ Perspective

I’ll discuss the debate over roulette dealer signature and section shooting (or ball steering) directly below, but for the purposes of discussing roulette culture from the players’ perspective, I want you to just assume, for the moment, that some skilled roulette dealers can steer the ball accurately enough, frequently enough, to alter the house edge on the game, either in favor of the house or player.

All hardcore roulette players believe that dealers can “section shoot”—meaning steer the ball to the area around a desired number. Squares who get to Vegas only once or twice in a lifetime bet birthdays, anniversaries and other lucky numbers—typically one chip per number. There’s often no way for the dealer to aim for a table of these types—if there are two or three of them at table, they may cover every number on the layout with a single chip. It never occurs to such players to team up on a number and give the dealer something to aim at. These players bet before the spin and sit there placidly, waiting for luck to strike them or not—the ideal casino customers. They play small and, when their buy-ins are gone, they leave the table.

Hard core roulette players, by contrast, not only stack up their chips on a sector for the express purpose of giving the dealer something to aim at, but expect the dealer to deliver. I get the feeling from European roulette players that dealers in Europe are more widely competent at steering than dealers in the U.S.. European players get really pissed off when a dealer seems unable to hit a big sector.

Sometimes you’ll see a conflict at the table—one action player betting a sector around zero, for example, while another action player bets a sector around the 17. But usually one of these players will concede the battle fairly quickly, so everyone can team up and avoid diluting their results. If they don’t team up on one sector, they all tend to start betting both sectors. They all want to be in on the win if the dealer should start to aim.

I don’t mean that action players at roulette actually expect to win—they don’t, because they virtually never do. One reason they never do is because they bet too big of a sector (or too many sectors) for any dealer, no matter how skilled, to hit frequently enough to be able to overcome the house edge on all those bets.

Another reason such players lose is because it never seems to occur to them that what the dealer giveth, either through luck or skill, the dealer will soon taketh away. After these players get a few hits, they stack the chips higher or spread out on more numbers, and they quickly give their winnings back, plus some. Action roulette players seem comfortable with this arrangement. All they seem to expect is that the dealer will give them a decent amount of play and excitement for their money—significantly more play and excitement than they are likely to get without help at a game with a house edge of 5.26%.

At casinos or on shifts or in pits where virtually all of the dealers do seem to follow a strict house dealing procedure, or where none of them can steer, a crowd of hardcore action roulette regulars never seems to form. That’s because hardcore action roulette players tend to test dealers before they’ll commit any significant amount of money at their tables. Action players who buy in, stack up their chips in a sector, and get wiped out in two or three spins leave the table or the casino in disgust for a dealer who knows what he’s doing.

The Debate Over Roulette Dealer Signature and Steering

As I mentioned above, hardcore roulette players tend to believe that dealers can steer the ball to the area around a desired number. This belief is shared by many dealers and casino personnel.

Casino management consultants and gambling experts, on the other hand, have almost universally denied the possibility. Let me first lay out the recent history of the debate, and then I’ll provide my response to each aspect of the argument.

As I pointed out in “The Roulette Debate Heats Up” (Blackjack Forum Vol. XII #1, March 1992—link at the upper left) John Scarne, in his 1978 revision of Scarne’s Guide to Casino Gambling, states in response to a reader’s question about roulette dealer steering:

The modern wheel, with its obstacles on the bottom track of the bowl, together with the fact that the croupier must spin the wheel and ball in opposite directions and must spin the ball from the last number into which it dropped, makes [steering or section shooting] an impossible feat even for the greatest of all Roulette croupiers. I once heard a lady friend of mine ask a croupier in a Las Vegas casino if he could drop the ball into any slot he wished. “Lady,” he said, “if I could do that I wouldn’t be working here. I’d have been worth millions years ago.”

In some makeshift casinos throughout the country you may, however, find croupiers who can drop the little white ball into any predetermined group of 12 adjacent numbers on the wheel. Doc Winters, an old-time crooked Roulette dealer, could even drop the ball into any six-numbered section he desired. But this can only be done by spinning wheel and ball in the same direction and using a wheel that has no obstacles. You can guard against this sort of cheating by not placing your bets until after wheel and ball are spinning, just before the croupier announces that no further bets are allowed.

Russell Barnhart, in his 1992 book Beating the Wheel, has a chapter titled “Can the Croupier Control the Ball?” He describes a meeting between himself and a couple of European gambling experts and says this:

There was one point on which we all agreed. No croupier can ever consciously influence, even in the slightest degree, the ultimate destination of the ball as it circles the roulette wheel. Indeed, no croupier can get it into even a predesignated half of the wheel, let alone into 1, 2, or 3 favored numbers.

Barnhart goes on to quote Alois Szabo, one of the people at the meeting:

In theory it seems not impossible that an adroit croupier with tremendous experience might succeed in directing the ball according to his wish, but he could do so only if he could spin as he wanted and if there were no obstacles to divert the ball on the border of the roulette machine.

Szabo goes on to say that numerous roulette dealers had told him that directing the ball was impossible. He wrote, “One of them told me that in the Roulette School he and his colleagues tried countless times not to spin but simply to drop the ball into a certain number, and even in this they succeeded very rarely.”

Szabo also wrote that the most experienced croupiers, after a lifetime of practice, agreed “that only under the following conditions could one possibly succeed in spinning the ball into a predetermined section of the wheel:”

1. If the spinner could throw from a fixed point in every instance.
2. If, each time he does so, the wheel were turning in the same direction as the ball.
3. If the wheel were always turning at the same speed.
4. If he could always spin with the same hand.
5. If, and this is most important, there were no obstacles [canoes or ball stops] on the machine.

Ironically, Barnhart says later in his book that he believes in dealer signature (the unconscious creation of an exploitable pattern in spin results, based on fixed dealing habits), which strikes me as illogical. If the obstacles and other factors that he identifies actually worked to randomize results too much to allow conscious dealer steering to a predictable number, they would also randomize results for ball spins that weren’t consciously aimed.

In any case Barnhart refers to Stephen Kimmel as a recent proponent of the idea of unconscious dealer signature. In fact, he says Stephen Kimmel came up with the term “dealer’s signature.” The basic idea of dealer signature is that a person who deals roulette eight hours a day 50 weeks a year will tend to develop routines that would create predictable results for the dealer’s ball spins. Kimmel published his views on roulette in an article titled “Roulette and Randomness” in the December 1979 issue of Gambling Times.

Kimmel suggested scouting for dealer signature by looking for a series of results a similar number of pockets apart. If you’re walking by a roulette wheel, for example, and you see on the reader board that the last five results are 1-19-23-30-34-27 (on an American wheel), you are looking at results that are all eight pockets back from the previous result (an extremely unlikely occurrence). Kimmel said you’d need to see at least 50 spins before you started to bet, but he tracked 199 spins before betting on a dealer in Las Vegas.

Barnhart says many more spins than that would be required. I will quickly point out that the number of trials required to ascertain exploitable dealer signature would depend on how strong the signature was—a weaker pattern of results would require more results to confirm an exploitable signature than a strong, consistent pattern of results.

Ed Thorp on Dealer Signature and Steering

Ed Thorp addresses Kimmel’s dealer signature article in the roulette chapter of his book, The Mathematics of Gambling (Gambling Times, 1984). He writes that for dealer signature to be exploitable three factors must be consistent from spin to spin: rotor speed, ball revolutions, and the position of the rotor at ball launch.

Thorp also provides a formula for statistical analysis of roulette that has since become key in the arguments of all gambling experts who maintain that dealer steering and signature cannot occur. Thorp instructs analysts to gather data on the number of revolutions the ball makes between release and crossing onto the rotor, and to state the results as the “average number of revolutions plus an error term.”

Next, the analyst should count the number of revolutions the rotor makes during each of these ball spins, and again compute the average number of revolutions, plus a second error term. Finally, Thorp instructs roulette analysts to “count how far the ball travels on the rotor after it has crossed the divider between the rotor and the stator.” These results should again be summarized as an average distance (in pockets or revolutions) plus an error term.

To tie it all together, Thorp says that for a dealer’s “signature” to be exploitable, “it is necessary that the square root of the sums of the squares of the error terms be less than 17 pockets.” In other words, if the dealer’s average ball spin is 10 revolutions, plus or minus 10 pockets, and his average rotor spin during each ball spin is six revolutions, plus or minus 19 pockets, and the average ball roll is 19 pockets, plus or minus 19 pockets, you square each error term and add them up: (10 x 10) + (19 x 19) + (19 x 19) = 100 + 361 + 361 = 822. Now take the square root of 822, which is 28.7.

Thorp provides a table that shows the rate of return given various root mean square errors (“Typical Error E”). This table is reproduced below. If you look at Thorp’s table, a Typical Error E of 28-29 pockets would have you betting at a disadvantage of 5.26% (the house edge on an American wheel) if you were trying to exploit dealer signature in this situation.

	Percent Advantage Betting on Best:
Typical Error E (No. of Pockets)	Pocket	Octant
0	3500.00	620.00
1	1278.53	611.06
2	610.69	467.86
3	376.52	328.65
4	258.12	236.98
5	186.76	175.71
6	139.09	132.62
7	105.00	100.89
8	79.41	76.65
9	59.54	57.60
10	43.77	42.38
11	31.19	30.18
12	21.24	20.52
13	13.54	13.03
14	7.73	7.37
15	3.47	3.24
16	0.46	0.30
17	– 1.62	– 1.72
18	– 3.01	– 3.07
19	– 3.90	– 3.94
20	– 4.46	– 4.49
21	– 4.81	– 4.82
22	– 5.01	– 5.02
23	– 5.13	– 5.13
24	– 5.19	– 5.19
25	– 5.23	– 5.23
26	– 5.24	– 5.25
27	– 5.25	– 5.25
28	– 5.26	– 5.26
29	– 5.26	– 5.26
30	– 5.26	– 5.26
31	– 5.26	– 5.26
∞	– 5.26	– 5.26

Thorp goes on to explain that his own observations of roulette dealers in action indicate that the error terms at each stage will be too large to make dealer signature exploitable.

My own observation is that the dealer error in the number of revolutions for the ball spin is about 20 pockets for the more consistent dealers; it is much larger with a less consistent one. I also noticed that the rotor velocity is not nearly as constant as Kimmel would like…

It is also true that the deflecting vanes on the sides of the rotor add considerable randomness to the outcome [of the ball travel after it has crossed onto the rotor], as do the frets or spacers between the pockets. The upshot is that I don’t believe that any dealer is predictable enough to cause a root mean square error of less than 17 pockets.”

Thorp uses the same formula to dismiss the idea of roulette dealer steering. He believes, based on his own observation and data, that the variance in dealer ball spin, wheel speed, ball launching point and ball roll upon entering the rotor will be too high for the dealer to either steer the ball or have an exploitable dealing signature.

Steve Forte, in his excellent book Casino Game Protection (Las Vegas: SLF Publishing, 2004), relies heavily on Thorp’s formula when he concludes that it’s impossible for dealer steering to exist (see his subchapter titled “The 17 Rule” in his chapter on roulette). He argues that it will be impossible for any dealer to meet the requirement of Thorp’s formula:

I don’t believe that section shooting exists on any level, and here’s why…Many gamers believe that section shooting is an acquired skill, a manipulative skill that one can learn through arduous practice. Juggling, golfing, typing, what so different about learning to section shoot versus many other difficult eye-to-hand manipulative skills? The answer lies in the limits of human capabilities. Has anyone ever learned to juggle 100 balls, hit nothing but holes-in-one, type 500 words a minute?

Although there’s no reason to doubt that muscle memory may result in dealers releasing the ball with similar velocities, that’s not enough precision to accurately control the speed of the ball. On the first revolution the error may only be a half-pocket; after the second revolution, experience has shown that the error can jump to one pocket, then two, then four, and so on. As you increase the revolutions, a significant margin of error compounds. With only ten to twelve revolutions, which is considered very slow, it’s not just ten to twelve times harder, it can quickly become impossible.

Another factor regarding the alleged skill, and, curiously, one that many seem to overlook, is that section shooting is an act comprised of three questionably attainable skills, not just one, and they must occur simultaneously. The dealer must push the rotor to a predetermined speed, spin the ball with a predetermined force, and then factor in the impact of the ball’s bounce, in order to have any chance of controlling the outcome. Compare these actions to those of professional bowlers, golfers, pool players, and similar athletes. They only aim once and can take as much time as they want to warm up, evaluate, and calculate their actions. A baseball pitcher might be able to throw a variety of pitches, all at different speeds, and all with incredible accuracy, but he doesn’t have to throw three strikes at the same time!

Forte further writes that steering and signature are made impossible by “the capricious nature of the wheel,” or purported changeability in drop and roll according to minute changes in conditions (experts have suggested humidity, barometric pressure, and phase of the moon as factors, among others). He stresses that this information is based on discussion with many experts who have told him that drop and roll and bounce characteristics of wheels are highly variable from day to day.

The final argument made by gaming experts against roulette steering and dealer signature is that they’ve simply never seen it, and further, that if such skills or traits existed, modern casinos could not survive.

Barnhart quotes Szabo saying that dealer steering would bring ruin to all the casinos of the world.

Darwin Ortiz, in his “Letter Regarding Nevada Roulette” (Blackjack Forum Vol. XI #4, December 1991—link at the left), says that dealer steering is “rather like Bigfoot or flying saucers.”

I’ve met people who know people who know people who can do it. I’ve met people whose brother-in-law can do it. I’ve even met people who could do it on every day except the day that I happened to meet them. But I have yet to meet one dealer face-to-face who could reliably do it when challenged by me.

Regarding the idea that skilled roulette dealers would bankrupt the casino industry, Ortiz says:

The fact is that if dealers could actually [steer], the game of roulette would have been destroyed long ago. Dealers would, indeed, have used their talents at every opportunity to bankrupt the house by helping agents win.

Regarding a claim by Laurance Scott, in his article “Nevada Roulette” (Blackjack Forum Vol. XI #3, September 1991—link at the left) that roulette dealers would use steering for job security, Ortiz responds: “The money-making potential of such a skill makes the whole issue of job-security irrelevant.”

And here is Steve Forte on this point:

I asked a close friend and triple sharp, all-around gaming executive … to help me find the top wheel dealers in Las Vegas. Our research led to a couple of Cuban dealers who worked together in a major casino. This was no surprise, since the best roulette dealers in the world come from Cuba, the Dominican Republic, and Puerto Rico.

Having spent time in the casinos of the Dominican Republic and Puerto Rico, and having watched the Cuban dealers work here in the states, I can vouch for their incredible mastery of and dedication to the wheel. With a combined 75 years of dealing experience between them, both in Cuba and in the U.S., they were asked for their opinions regarding the recent controversy. They laughed and said, “If we could do that, do you think we’d still be working?”

Laurance Scott is pretty much alone in the serious literature on the other side of the debate. In his article “Nevada Roulette,” which he says is based on his own experience at the tables, he portrays a roulette scene in which dealers are not only able to steer the ball to a sector, but do so routinely to bust out players for themselves and the house. He does take care to limit this claim to dealers on the old-fashioned, deep-pocket wheels, of which there were a number still in action at the time his article was written:

How can anybody cheat at roulette? Well, first of all let me qualify the statement by saying that not all Nevada casinos cheat. Some casinos run a fair game. They use modern wheels which tend to yield truly random results. However, other casinos still use older style equipment which is quite beatable. Why would a casino use beatable equipment when modern non-beatable equipment is available?

When I first published Scott’s article in Blackjack Forum I got quite a response, but the only correspondence I received in support of Scott’s assertion that it was possible for roulette dealers to steer the ball came from pit bosses and roulette dealers, none of whom claimed to be able to steer the ball themselves.

So, to sum up the case against dealer steering or signature we have the argument that there will always be too much variance in ball speed, rotor speed, and ball drop and roll for steering or exploitable signature to be possible.

According to section-shooting detractors, part of this variance will be due to the design of the wheel itself, with its deflectors in the path of the ball as it crosses the apron to the rotor. Part of this variance may be caused by dealing procedures that require the dealer to launch the ball in alternating directions or vary rotor speed on each spin or perform a blind spin without looking at the wheel during the launch. Part may be due to simple human physical limitations. More variance may be due to the effects of weather and other physical conditions on wheel and ball behavior.

And finally there is the argument that none of the experts have seen a dealer with such skills, and that if such skills actually existed the casinos would all be broke.

Laurance Scott, on the other hand, believed that such skills could exist for the same reason that wheel predictors could exist—if there’s not too much variance for a roulette computer or visual predictor to get an edge (and all of the experts agree that roulette prediction by computer or visual methods is possible, at least on some few deep-pocket wheels), there’s no reason to assume there’s too much variance for dealer steering to exist.

Further, Scott and the pit personnel, dealers and players who believe that dealer steering and/or exploitable signature is possible usually believe they have seen it.

My Response to the Debate on Roulette Dealer Steering and Signature

I will first address the claims that variance is inevitably too high to allow dealer signature or steering. As Thorp pointed out, variance in ball landing results can come from variance in dealer ball launch, variance in launch location relative to the rotor, variance in rotor speed, and variance in drop or ball roll after the drop. I will address each type of variance separately.

Variance in ball launch location relative to the rotor

Thorp’s biggest objection to the idea of dealer steering or signature was his belief that a dealer would be unable to launch the ball at a consistent place relative to the moving rotor because the dealer would be unable to respond precisely enough to the visual cue of the targeted launch number. There are related casino procedures mentioned by Steve Forte in both his book and his article “The Myth of Roulette Dealer Signature and Section Shooting” (Blackjack Forum Vol. XII #2, June 1992—link at the upper left), described as follows in his article:

There are two procedures that effectively stop any possibility of the roulette section shooting or steering myth from becoming a reality. They are the “blind spin,” where the dealer spins the ball without ever glancing into the rotor, and the “last pocket spin,” where the dealer picks the ball out of the winning pocket, waits one revolution and spins from the same position the ball last landed.

My response is that a dealer needn’t rely on a visual cue to launch the ball. Instead a dealer may rely on timing for his launch, or a fixed set of ball pick-up and launch acts carried out the same way, with the same timing, every spin.

At one casino on the Las Vegas Strip where I believe we observed dealer steering to agents, there was a house policy in place that required dealers to do a blind spin. (The policy was apparent in the dealers’ behavior, and I was able to confirm it with casino personnel there.) The dealers adjusted the wheel speed, then looked away from the wheel while feeling for the ball with their fingertips. When their fingertips made contact with the ball, they picked up the ball and launched it with a fixed routine. They were able to maintain a remarkable consistency of launch position relative to the ball pick-up point with no visual cue, based solely on consistent timing.

Regarding Scarne’s idea that steering requires the ball and wheel to be rotating in the same direction

As I pointed out in Part 1 of this article, “How to Win at Roulette: Traditional Visual Prediction” (Blackjack Forum Vol. XXVIII #1, October 2012—link at the upper left) the length of ball spin merely sets the length of time the wheel will be rotating before the ball drops onto the rotor. The speed of the rotor determines how much of the wheel will pass any given point during that amount of time. There’s no requirement that the ball and wheel be going in the same direction to calculate how much of the wheel will pass in a given amount of time.

Regarding house procedures that require alternating direction for each spin

A dealer’s only got two hands, and a roulette wheel only has two directions. If a dealer is required to change hands or directions for each spin, and there is a difference between the number of ball spins when launched by the right and left hands, or in the rotor speed when spun in one direction versus the other, all you’d have to do is adjust for the difference in your prediction, or bet only on those spins launched in the preferred direction, or by the preferred hand.

Regarding variance in roll due to deflectors

It’s true that the ball deflectors (obstacles) on the apron have an effect on how far the ball will roll after dropping off the track, and cause variance in ball roll that leads to variance in results. This makes it impossible for a dealer, no matter how skilled, to hit a single target sector of consecutive numbered pockets every spin.

But because wheels tend to have drops (caused usually by tilt, sometimes by wear), the variance in results is often not too great to eliminate the possibility of gaining an edge from steering.

I don’t agree with the assertion that small changes in weather and other conditions lead to significant changes in the wheel itself, or a wheel’s drop. Based on my data, I believe that the changes that experts say they have observed are mostly due to normal, ongoing variance in the drop unrelated to weather and external conditions. The exception would be when there is a sudden rapid change in barometric pressure. It’s not that the wheel changes due to such barometric pressure changes, but that you can get extreme changes in normal ball behavior at such times.

Regarding inability to drop the ball into a specific number

My reply to Szabo’s story of the roulette dealers who had tried, unsuccessfully, to drop the ball into a particular number would be that simply dropping it from above would leave lots of energy to be dispelled by the ball in a very random manner. That’s very different from a ball launched into a particular direction in a spin, for which you’d expect inertia to carry the ball in basically the same direction until its energy was dispelled.

Regarding variance in rotor speed

Let me just say that we have clocked wheels—that is, taken measurements of their speed using reliable instruments—sufficiently to confirm that some dealers are able to maintain a consistent rotor speed if they choose, or hit a chosen rotor speed consistently.

Regarding house requirements to vary the speed of the rotor

First, as I have mentioned at the beginning of this article, very few U.S. casinos enforce any kind of fixed dealing procedures on roulette.

But even at casinos where a fixed dealing procedure seems to be enforced (at least on some shifts), we have found roulette dealers who are able to steer the ball using various combinations of wheel speed and ball speed. In other words, a dealer doesn’t need to maintain a single rotor speed to steer, because the multiple factors involved in steering a ball allow for multiple ways of getting a ball to a targeted number. If you are forced to change speed each spin, you can compensate for the changed speed by changing ball speed or launch location relative to the rotor.

In fact, one of the ways we confirm that a dealer is steering the ball is by watching for the dealer to use these factors in multiple combinations to achieve a specific goal.

I’ll go even further and say that a dealer who attempts to steer by always launching from a specific number, while maintaining a consistent wheel speed, is probably a dealer who can’t steer very well. We see this type of kindly dealer all the time in casinos, when the players have all joined together in betting the area around the 17, for example. The dealer strives to maintain a constant wheel speed and tries time and again to hit the 17 by adjusting his launch point, only to miss the target on every spin and quickly wipe out the players he’s trying to help.

But what about Thorp’s formula?

I don’t have any problem with Thorp’s formula per se, or with the data he provides (which are similar to my own), but I partly disagree with the factors used in the formula, I disagree with the way his formula has been applied, and I disagree with conclusions that have been drawn from these applications of the formula. I can’t discuss all of these disagreements within this article because casino personnel will be reading but, for example, I disagree with the way Thorp measures ball roll (as well as other factors). One problem I can discuss briefly in this article lies in the calculation of mean error terms.

Let’s say that a particular wheel has a drop which half of the time causes a ball roll of three pockets plus or minus two pockets, and half of the time causes a roll of 41 pockets plus or minus two pockets. (You will never find such a wheel—this example is simplified to make it easy to follow.) To assume that the 41 pocket roll is a 38 pocket error term would be unhelpful (if you look at a picture of a wheel, you’ll understand why).

It would also be unhelpful to calculate that the average roll was 22 pockets, with an error term of 21 pockets. Average roll might be a relevant factor if the roll was as likely to roll any number of pockets between 3 and 41, but if the actual rolls center around 3 pockets and 41 pockets, with a large gap of pockets mostly unhit, then using the average roll as a factor in this calculation is not helpful.

Or let’s say that half of the time the ball roll is 5 pockets and half of the time it’s 24 pockets, each plus or minus 2 pockets. You don’t have to regard the 5-pocket roll as correct and the 24-pocket roll as an error term of 19 pockets. Nor do you have to regard the average roll as 14 ½ pockets, with an error term of 11 ½ pockets. Instead, you’d want to adjust your betting sector and consider the error term of the ball roll very small.

As I mentioned in Part I of this series, there’s no reason a betting sector in roulette has to be comprised of consecutive numbers. A sector for a particular wheel could be a couple of adjacent numbers and another number twelve pockets away. Optimal betting sectors can take on many different configurations on modern wheels spun at modern speeds.

There are similar problems with error term measurement for other factors in Thorp’s formula.

And what about the argument that all casinos would go broke in a year?

After watching an awful lot of professional gamblers now for over three decades, many of them with enormous bankrolls, it’s never seemed to me that the casinos were in any particular danger from us. If the Hyland team, the MIT team, the Greeks, the Czechs, Al Francesco, Zeljko, and a few thousand other high stakes blackjack pros couldn’t kill off the casino industry, I don’t see how a few steering dealers could do it.

For one thing, there truly aren’t that many dealers who can steer well enough to get an edge steering to an agent, and it’s unlikely many will learn because of the difficulty of understanding the natural variance in spin results. For example, it would be difficult for a dealer who was trying to learn to steer to know whether a bad result was due to variance or to his poor skill. He would lack the kind of consistent feedback most people need to learn.

And any dealer who could steer would be restrained in the use of his skill by the rational desire to extract the maximum possible lifetime gain from his hard-won skill rather than burn out a great play in a couple of sessions for a much smaller total win. It’s true that a steering dealer without a big bankroll of his own could play at a much higher level if he involved an investor, but it’s also true that the investor would demand the bulk of the win for his cut, and involving an investor entails other risks, such as the risk of the investor training other dealers, or the risk of the investor revealing the play. I’ve never known a professional gambler to prefer giving away the bulk of his win when he could afford to keep it all to himself simply by taking a play more slowly.

Also, card mechanics of varying skills have worked at blackjack tables for as long as the game has been dealt, and many have used these skills for decades without putting the casinos where they work out of business. Cheaters who have spent years developing special skills have no desire to kill the golden goose.

Another huge factor in protecting the casinos from bankruptcy would be the steering dealer’s fear of going to jail for cheating—I would think that fear would be huge in the case of a casino employee who was steering a roulette ball for the benefit of an associate.

Another limit on how much a skilled dealer can take from roulette in a short time if he is concerned about avoiding discovery is the behavior of the average roulette player. No gambler at roulette who actually manages to get ahead ever seems to leave with his winnings. Instead, what they do is continue playing until their winnings and buy-in are gone, plus any other amounts they may have available for buy-ins. In other words, they play until they run out of money, and all they really hope for, if they believe in dealer steering, is for the dealer to give them a longer playing time.

Again, in our experience, roulette players don’t get angry because they’re leaving the table broke. (They’re accustomed to leaving broke.) They only get angry if they’re leaving the table broke after receiving a very short period of play. People want some fun for their money.

So a dealer who is steering for an agent, and who is concerned about getting enough money out of other players to keep up appearances, probably has natural limits on how fast he can extract money from the game. While a dealer who was steering against players might take their money faster, I don’t believe that he’d necessarily get more money from them. Again, players who lose their money quickly at a roulette table tend to leave that table quickly.

As for a dealer who says, “If I could steer, why would I still be working here?” I would say, “Where else would you be working?” A dealer who can steer would probably choose to practice his skill as long as possible.

To those who say, “Why wouldn’t they take out millions?” my answer would be, “What makes you think they haven’t?”

Response to Scott’s claim of widespread dealer cheating

Scott’s claim of widespread dealer cheating at roulette (that is, steering against players) doesn’t fit what we have actually seen at the tables. But Scott was describing conditions he encountered over two decades ago, on wheels that are seldom seen on today’s roulette tables.

That isn’t to say we’ve never seen dealers steering against players. We have seen what we believed to be dealers steering against players in three kinds of situations.

First, we’ve seen dealers steering against players after periods of steering for those same players, essentially taking back the house money they had passed out. This has occurred with a small percentage of the dealers we’ve observed.

Second, we’ve seen dealers steering against players they seemed to dislike. Again, this has occurred with only a small percentage of the dealers we’ve observed.

Third, we’ve seen dealers steering against players after steering for different players that we believed were agents. This has been extremely rare, and in each case, it has been a number of years since we’ve seen any of these dealers working in Las Vegas.

How I Came to Believe that Roulette Dealers Can Steer

I came to believe that a small percentage of dealers can steer well enough to get an edge at roulette by sitting at the table of a particular dealer and recording enough spin results to ascertain that his frequency of hitting a particular sector had gone beyond three standard deviations.

We were at a large Strip casino one week when we observed what we believed were possible signs of a biased wheel. Specifically, the ball was landing in a small sector of the wheel too often for it to be likely that it was due to chance.

We sat at the table placing even-money bets and recorded the results of enough spins to reliably calculate that the edge on betting the sector would be roughly 20%. All of this play was on the same shift, and almost all of it was against the same dealer. Some of the play was against a relief dealer.

Then we returned to the casino for one last session of data collection—this time on a different shift. To our surprise, the results for that session, when taken alone, indicated no bias whatsoever. We went back to record more data against dealers other than our original dealer, and none of it showed a bias.

We went back to observe our original dealer over the next year or so, and we learned a lot about roulette by observing him. (Thanks, man.) We also began to notice that we would observe certain behaviors of his used by other dealers who appeared to be steering, while we didn’t see these behaviors used by average dealers.

With time we found a collection of dealers we believed could steer. We could often tell whether a dealer was steering before we even got to the table just by observing her behavior as we walked around the casino. Occasionally it would seem to us as we returned to one of these dealers that we had arrived in the middle of a play.

One skilled dealer at a Strip casino had unusual and distinctive facial features—black eyebrows shaped like sharp upside-down v’s and earlobes that were fleshy and elongated. (All details have been changed to protect any players and dealers described in this article.) Sitting at this dealer’s table, however, was a guy about the same height and weight, with the same black sharply pointing eyebrows and the same elongated fleshy earlobes. He was betting a small sector and doing well, with the ball landing in his sector at an unusually high rate. My first thought was, “This guy’s steering for his brother.”

After the next spin, we placed some bets in the same sector and had a win, but right after the dealer had paid us all off, the other player asked to be colored up. On the next spin the dealer changed to a wheel speed different from the ones he had used on the other spins we’d observed.

Other times we believed we’d stumbled onto a play just because it had that play look that you recognize quickly if you’ve ever been part of team gambling efforts yourself. Participants make eye contact for no seeming reason, and then sustain it for an unusual length of time. You hear certain words repeated when they don’t really fit the conversation, or the conversation seems strained. Strangers at the table get angry with each other, or look surprised with each other, when the emotion doesn’t seem to fit.

On one dealer/agent steering play, the agent was posing as a square who always bet the same adjacent numbers, always before the spin. He had the seat next to the wheel, but he never looked at the wheel. Then, late in the play, toward the end of the dealer’s shift, after a bit of a drawdown of his profits, he stared at the wheel intently as the ball drew near the end of its spin. On the final spin before the ball fell off the track, as the ball passed the drop-off point for the last time, this “square” pumped his fist and yelled “Yes!” The ball came around to hit the drop deflector, rolled beautifully, he got a hit, and the dealer got tapped out by his replacement. The player left a few spins later.

At another casino, after observing what we believed were multiple occasions of dealer steering for agents, we saw a steering dealer engaged in what we believed was a deliberate bust-out of a player. This player was a regular at the casino—a high-roller, a classic action player, and a memorable character, both because of his fashion sense (Steve Martin’s “wild and crazy guy”) and his unusual companions (Jodie Foster in Taxi Driver). We had observed him playing many times, always using the same system. He seemed to have a few preferred dealers at the casino, and every time we saw him playing one of them, he would get a long play on his buy-in.

But on this occasion, the dealer appeared to be steering against him from the very first spin. The reason we believed this dealer was steering against this player was because he used a different set of wheel and ball speed combinations than he used during the plays when we believed he was steering for a player. Based on my wife’s read of the speed, we would have been betting an area of the wheel where this player had no bets at all. This player bought in multiple times for large amounts and lost it all quickly. It was a massacre.

We’ve even run into dealers who openly discussed with us what they were doing. Early in our roulette learning process we ran into a dealer off-Strip who appeared to be steering for the players on his table. We started to bet on the target numbers, but only after we had a read on the spin to confirm that the dealer was still steering. The dealer watched us watch the wheel, then after we had bet he said something like, “You sure about that? I’m going for the six this time.”

That was a shocker. He made these kinds of comments for the next few spins. I was not comfortable with the situation. After one spin, our read indicated that the optimal bet was outside of the former target sector. I bet and he said, “Really? You sure?” My wife answered, “I’m sure.” He stared at the wheel a moment and said, “Looks like you’re right about that.” He and my wife openly discussed the next few spins, then I cut the play short. When we left, he said, “Come back any time. That was fun.” We never went back.

Again, though I could tell you more stories of this type, this kind of thing is not the norm. And, although we ran into other situations in which dealers appeared to be steering, we did not rely on the appearance of steering to determine whether steering was possible and exploitable in casinos. To determine whether or not a particular dealer could steer, we collected data.

How to Win at Roulette Using Dealer Tell Play

It’s virtually impossible to beat modern roulette as dealt in the U.S. with the old-style visual prediction methods I wrote about in Part I of this series. Even computer play (which is illegal in Nevada anyway, as well as in every state and country that has copied Nevada’s cheating statutes) is problematic on modern roulette, because of roll variance and other factors. All other methods of beating the game are either illegal or very difficult, requiring much research, scouting and practice.

Betting on dealers who are steering is potentially the easiest way to beat the game if you can learn to identify dealers with that capability and learn how to tell when they are turning the steering on and off. Further, observing dealers who are steering is the best way to learn visual prediction methods that will actually work on the modern game. The easiest way to do these things is by learning to read dealer tells.

The first person I’m aware of to suggest exploiting dealer steering and emotions at roulette was Laurance Scott, once again in his article “Nevada Roulette.” He wrote:

I also believe that it may be possible to get an edge at Nevada roulette without any predictive skills just by using an applied psychology approach. First, assume that the game is rigged (which it is) and that an experienced dealer can hit a section with alarming accuracy.

Next, develop an act such that the dealers can’t stand the sight of your face and start pre-betting sections with 25-cent chips. Have a co-conspirator bet $5 chips (at the opportune times) in the opposite section while hiding at the end of the table.

Now, though I do not recommend the specific tactics suggested by Scott, I can confirm that roulette tell play works.

In early 1986, I served as an expert witness for the defense in a case where a blackjack dealer had been accused of colluding with a player. This was back when blackjack dealers still manually peeked under tens, and I had just learned about tell play from Steve Forte’s manuscript of Read The Dealer.

When I reviewed the surveillance tapes of the dealer, it was clear to me that the dealer was not colluding with an agent, but simply had a strong unconscious tell that any player could exploit. I explained tell play to the judge and jury, and taught them how to read the dealer’s tells on the tape. Everyone in the courtroom spent the next ten minutes watching video of the dealer and correctly guessing her hole card. The verdict was “not guilty” but that tape and trial transcript got around. I’d just published Forte’s book, and soon auto-peek readers were being installed on blackjack tables all over Nevada.

So I don’t feel it would be wise to talk about specific tells in this article. Tells tend to vary a lot from dealer to dealer anyway—they did in blackjack, and they do at roulette. Suffice it to say that a dealer who is attempting to steer for players and feels confident in his ability to help them win money tends to exhibit behaviors that are different from the average roulette dealer. The average dealer who can’t or won’t steer goes through his work life stripping people of their money while giving them very little in return. It’s depressing, shameful work for any dealer with any feeling for other people, and a dealer’s demeanor may reflect that.

Similarly, a dealer who has steered some hits to players but who has decided to take the money back has significant changes in appearance and behavior. He’s changed himself from a host and benefactor into an executioner, and it affects how he feels about himself and the players at his table. These feelings often show up as tells and can be used as indicators of when to get in or out of the game.

Dealer tell play at roulette is similar to the old dealer tell play at blackjack, except that blackjack dealer tell play didn’t involve conscious help from the dealer. Roulette tell play only works if a dealer is consciously steering for you, but if he suspects even a little that you are aware that he can steer, he will usually end the steering immediately.

If you’re actually watching the wheel to ascertain whether the dealer is steering, he will usually recognize that very quickly. (Camouflage is essential at all times!) And if a dealer suspects you of trying to get an advantage over him that he hasn’t chosen to bestow, he can and will make it very difficult for you to play.

The other thing to remember is that even if a dealer is rooting for the players and steering for the players, at some point the dealer will stop. Friendly dealers seem to enjoy giving customers a few big payoffs. But when your chips start stacking up, or the pit boss starts hanging around, they seem to feel you’ve been given enough, and they either go back to random or, more typically, start steering against the players they helped. Again, this only occurs with a small percentage of dealers.

I suggest you read “Stalking the Elusive Tell” by Dog-Ass Johnny (Blackjack Forum Volume VIII #2, June 1988)—to get an idea of how to keep dealers rooting for you, understand when they are rooting for or against someone else, and get the hell out when their good will disappears. Get your hands on a copy of Steve Forte’s long out-of-print Read The Dealer (Oakland, CA: RGE Publishing, 1986) if you can and read that as well. Some of the most useful information for roulette in Forte’s book will be the material on “ghost tells” and creating dealer tells, which are as valid at roulette today as they were at blackjack when the book was written.

A ghost tell is information you pick up from a dealer’s reaction to another player. Ghost tells are ideal for scouting purposes and, as Forte points out, they are simply easier to read than tells you pick up based on a dealer’s behavior toward you, because the dealer’s attention is focused elsewhere. Forte preferred looking for ghost tells between a dealer and a big player who had already established a friendly relationship with the dealer that usually included liberal tipping. That works at roulette as well.

If there’s no friendly big player and tipper around to provide you with ghost tells, you may have to encourage tells yourself. You can do this by cultivating the dealer’s good will. Be friendly, complimentary; toss him a tip on your first hit.

Don’t try to cultivate her ire. Be aware that a dealer’s hatred does you no good unless she can steer against you, which means your pre-spin bets are a goner, and even if you start seeing tells that a dealer is steering against you, that doesn’t tell you what numbers are most likely to hit, only what numbers are not likely to hit. Dealer hatred tells work best when they are ghost tells and the disliked players are really covering the felt, leaving only smallish groups of consecutive numbers without bets on them—something for the dealer to aim for—and where you can get some money down after the spin. (And tip when he hits you!)

For example, I once had a terrific opportunity for tell play at a large Strip casino where an experienced female dealer was trying to cope with a group of obnoxious French players. The French players were betting sectors, but their sectors were huge—there were only a few small sections of the wheel that they hadn’t covered with stacks of chips. The dealer disliked them because they all smoked and, despite the dealer’s hints that the smoke was bothering her, they refused to put out the burning cigarettes in their ash trays and took no care to prevent the smoke from drifting into her face. They even exhaled smoke directly into her face. You should have heard the outpouring of French when she started hitting the few numbers they weren’t betting and swept all those stacks from the table.

Here are some more tips on tells from Forte’s Read the Dealer. I’ve added my own adjustments to his comments (in parentheses) to make them more relevant for roulette:

• It is important to be natural when trying to read the dealer. Peripheral vision can be a great asset… You can’t just lean back in your chair and stare at the dealer.
• The first thing to always look for in a dealer is his base or normal behavior. Where do his eyes normally look… (immediately before and after spinning the ball? How about near the end of the spin? When does he normally wave players off?)
• When trying to analyze a new tell, I find it easier if I mentally ask myself questions. Would a dealer make a comment like that… (if he knew he was steering for me)? Absence can be a key word. Instead of asking myself what would I expect to see in this situation that might give me a clue, I say what didn’t the dealer do or what’s missing that I would expect to see in this situation? (Your first hint of an exploitable opportunity will always be some small behavior out of the norm.)
• If you play in smaller clubs, tokes will carry more weight…
• Sooner or later you will run into the dealer who is always smiling and seems to be extra courteous and helpful. In reality, he’s trying to win every dollar you have, your watch and your socks if they’re worth anything. This dealer wants the best of both worlds…It doesn’t make any difference how much you toke or how friendly you are. This is a perfect example of the “house dealer…” This dealer is not on your side and will attempt to deceive you by acting opposite to his true feelings …
• Learning to play tells can be difficult. For the most part, every tell you read on paper is clear and easy to learn. When you look for these same tells in a casino, they will be extremely subtle, disguised and at first seem almost invisible. The best way to begin learning tells is to get into the casino and observe. Walk up to a game and get a feel for the level of involvement. Then any time you see the dealer deviate from his normal behavior, force yourself to try to find the motive behind the action.
• Finally, [tell play] is 100% legal. So long as you don’t pay off the dealer to (steer for you)… or otherwise alter the “criteria of selection,” the law specifically states you can take advantage of any information that is available to all players. What exactly is the information that is available to everyone? The entire spectrum of subconsciously generated human behavior that takes place anytime you put a dealer behind a (roulette) table, pay him minimum wage so that he really must earn his living from tokes, treat him like dirt, have a pit boss sweat the game, then bring in the players who want to win so bad they can taste it. Skilled tell players simply exploit these facets of human behavior and turn this information into an advantage…

To learn more about reading dealer tells, even books on poker tells will be helpful. Two excellent books include Caro’s Book of Poker Tells, by Mike Caro, and Peter Collett’s Book of Tells. I also recommend Joe Navarro’s book on body language, What Every BODY is Saying: An Ex-FBI Agent’s Guide to Speed-Reading People. One important point made by Navarro is that while some tells are fairly common and usually mean the same thing, other tells are unique to a person, or can mean opposite things depending on how a person feels about a given situation.

And remember that you need two sets of tells—one that lets you know that the dealer is capable of steering, and another that lets you know whether or not he is steering right now, either for or against you. That first tell is critical, and you will be better off if you can confirm it over time with some hard data. Just because you get a tell that says the dealer is trying to steer for the players, that doesn’t mean you can automatically bet the numbers you think he’s trying to hit. Sometimes a dealer with poor steering skills will be exploitable, but usually not by betting the numbers you believe he’s targeting. I’ve often seen friendly dealers with poor steering skills wipe out the players they were trying to help.

Another thing to remember about any form of tell play is that you can expect your ratio of scouting time to playing time to be very high—in the hole card scouting-to-playing range, or higher. (Skilled hole card players may spend twelve hours scouting for every hour playing.) Once you have a collection of reliable tell dealers, your scouting time will go down, but you still have to find the dealers in situations that are good for tell play.

Lastly, you need to keep in mind that no dealer can steer the ball to a target number or sector on every spin. It’s impossible because of inevitable variance in the ball roll and other factors, and, again, that variance can make it difficult to discern in a short time whether or not that dealer can actually steer. That’s one reason it’s easier to play dealer tells at roulette than to learn to read wheels yourself. A roulette dealer who has spent many hours on a wheel will know a lot more about the normal variance in the drop than you can find out without many hours of clocking the wheel. ♠

The Roulette Debate Heats Up by Arnold Snyder (From Blackjack Forum Vol. XII #1, March 1992) © 1992 Blackjack Forum Can roulette dealers steer or “aim” the ball with enough precision to help or cheat players? Laurance Scott, author of How to Beat Roulette, says they can (see “Nevada Roulette,” by L. Scott, BJF XI #3). Renowned expert on cheating at gambling, Darwin Ortiz, author of Gambling Scams, says Scott is all wet (see “Letter on Roulette Dealer Section Shooting,” BJF XI #4). We’ve gotten an avalanche of mail on both sides of this subject since Scott’s article and Ortiz’s letter. Here’s a sampling of the pro-Scott, anti-Ortiz opinions: Letter from a Former Nevada Roulette Dealer I just wanted to comment on your letter from Ortiz. I agree 100% with all he has to say about the position of casinos and their management in regard to cheating. I really have no stake in whether he and Steve Forte believe a dealer can place a ball in a section, but I would ask these questions: How long do you suppose it would take to learn to spin a roulette ball so that it made exactly four revolutions before dropping? If the wheel head were spinning very slowly and you picked a number (say zero) and a fixed point (say the 12 o’clock position) and spun the ball so it made four revolutions and dropped, do you think if you picked up the ball and waited for the zero to be at 12 o’clock again and spun it four revolutions again that it might end up close to where it landed before? Do you think you could maybe land it in half the wheel head? If a player can clock a moving roulette ball, couldn’t a dealer? I believe that Mr. Ortiz could learn to spin a roulette ball to make four revolutions within one hour. In a few more hours I bet he could make it go five revolutions, or six or seven. If they are really interested in this, why not take an hour and give it a try? Letter from an Atlantic City Pit Boss “Section shooting” is not a myth. This was proven to me in a proficient demonstration by a person who had dealt the game of roulette for more than two decades. This person was many years my senior and my teacher. Section shooting is a common sought after skill in the realm of roulette dealers. For this reason dealers who are supervised in professional gambling casinos are not permitted to hawk stares into the roulette wheel. Mr. Scott’s methods of winning are valid. However, should a pit boss like myself identify such a skilled player, we would instruct the dealer to call “no more bets” prior to the ball launch, and Mr. Scott’s edge would go out the window. I can personally tell you that pit bosses and other casino management alike do not value “section shooting” dealers or any other skilled dealers of this degree because of (the possibility of) collusion. Mr. Ortiz asks for such a skilled section shooting roulette dealer… to come forward and demonstrate their ability to him. Mr. Ortiz has about as much chance of that happening as the trigger man in the JFK assassination coming forward. No one is going to display that ability to a layperson… Letter from Harry McArdle One of the tricks employed by casino managers in hiring cheats without admitting it even to themselves is to hire “experienced dealers” and to hold them responsible for the results at their table. Did you ever start winning a lot and see the dealer become nervous? That’s for real. His job is on the line. If you win big, they might just go out and hire a “more experienced dealer” unless they have one right there in the casino whom they can call over. Chances are they do. Laurance Scott was not the first person to suggest that a roulette dealer could place his ball in a section of the wheel. John Scarne in his Complete Guide to Casino Gambling, published in 1961, asserted that “A good croupier can place his ball to within six holes.” This is not an affirmation that I’ve blindly accepted out of faith in John Scarne. Rather, I’ve seen it reinforced repeatedly in my 33 years of casino gambling. In an afternoon, during the summer of ’66, I recall playing roulette at the table of a young female croupier who was practicing her skills. She would roll the ball slowly so that it would go around the wheel about two times before settling in a hole. Clearly, she was aiming for 0 or 00. I bet 1 and 27, which are next to 00, if she was aiming for 00, and 2 and 28 if she was shooting for 0. On one occasion I almost won. She looked with alarm at the pit boss who was standing at her elbow. He, in turn calmed her down and had her continue what she was doing. “You almost got her that time,” he said to me. Whereupon, he continued to root for me while I stayed for a few more bets. What Darwin Ortiz and others are thinking of when they say they are skeptical of a croupier’s ability to place his ball is probably the idea of the dealer achieving it while rolling the ball at top speed… I don’t know how many times a mechanic can roll his ball around a wheel and still place it with some reliability. This much I do know. It can be done and it is done. Thoughts from Arnold Snyder on Roulette Dealer Steering What it really comes down to is whether or not a professional roulette dealer can train himself to launch the ball such that it makes a specified number of revolutions prior to dropping. I don’t doubt that a dealer can launch a ball from a chosen point on the wheel, or that many wheels are biased such that the ball tends to drop onto the wheel more frequently from the same area of the track. But roulette dealers typically launch the ball so that it makes many revolutions (not 4 or 5) and the wheel, which is not spinning at a truly constant speed, is revolving in the opposite direction of the ball. In Scarne’s 1978 revision of his 1961 book, titled Scarne’s Guide to Casino Gambling, he states emphatically that no roulette dealer can section shoot with accuracy on modern roulette wheels. He also expounds some on the dealer who exhibited to him his ability to aim the ball accurately within a 6-number section, as mentioned by Harry McArdle. This dealer, according to Scarne, was able to do this only on an obsolete wheel that had no ball deflectors, and only when he launched the ball in the same direction as the ball was spinning. With both Darwin Ortiz and Steve Forte also of the opinion that roulette dealers cannot section shoot with accuracy, I still find the arguments that section shooting is a learnable skill to be less than compelling. Although Scarne was not the greatest mathematician, no one has ever refuted his vast knowledge of cheating at gambling. Darwin Ortiz has conducted literally hundreds of seminars on cheating for casino security personnel in the past decade. Steve Forte spent many years in Las Vegas as a dealer, then floorman, then pit boss, then casino manager, prior to opening his own dealing school, and then producing his card and dice cheating videos, and acting as a consultant to the industry on casino cheating techniques. The most convincing arguments that I have seen that section shooting may be more than a myth have come from Laurance Scott, who is attempting to arrange a confidential meeting with an industry insider who could not only explain to me the techniques to section shooting, but who could also demonstrate the ability to do it. I hope to follow up on this controversy soon.  ♠ [Note from Arnold Snyder: Laurance Scott was never able to arrange this meeting.]

Letter Regarding “Nevada Roulette”

by Darwin Ortiz

(From Blackjack Forum Vol. XI #4, December 1991)
© 1991 Blackjack Forum

[Note from Arnold Snyder: Darwin Ortiz is an expert on cheating at casino games. His works include Gambling Scams: How They Work, How to Detect Them, How to Protect Yourself; Darwin Ortiz on Casino Gambling: The Complete Guide to Playing and Winning ; and Strong Magic.

Ortiz’s letter is a response to Laurance Scott’s article Nevada Roulette, in which Scott asserted that there was widespread dealer ball steering at roulette games in Nevada casinos, chiefly to the disadvantage of players, and discussed methods of beating roulette.

I want to take a moment to disagree with one point Darwin Ortiz makes in this letter, where he states that “no casino management will keep a dealer who they believe has the ability to cheat” out of fear that the dealer will use the ability against the house. I know for a fact that lower levels of casino management have kept on dealers whom they knew could cheat–deal seconds, and the like–in order to use the talents of these dealers against players.]

Darwin Ortiz on Dealer Sector Shooting

I see that the latest issue of Blackjack Forum has an article (“Nevada Roulette,” by Laurance Scott) reviving the old myth that roulette dealers, through years of practice, gain the experience to place the ball in whatever sector of the wheel they wish. I’ve had enough experience in the field of gambling scams to have learned to be careful about labelling anything impossible.

However, dealers who can “section shoot” seem rather like Bigfoot or flying saucers. I’ve met people who know people who know people who can do it. I’ve met people whose brother-in-law can do it. I’ve even met people who could do it on every day except the day that I happened to meet them. But I have yet to meet one dealer face-to-face who could reliably do it when challenged by me.

However, placing aside the difficult question of whether there is any dealer anywhere who can section shoot, one thing that I can safely say is that Mr. Scott’s suggestion that such talent is widespread is utterly ridiculous. Hustlers who have worked at beating roulette through computer predictions by measuring wheel speed and ball speed can tell you that even something as minute as a breeze blowing across the wheel can negate the predictions. Yet, we are supposed to believe that dealers can take these countless variables into account and control the results.

Even more ludicrous is the claim that casinos highly value such dealers because they improve the hold. Perhaps most absurd is the claim that casinos will look the other way if a dealer with the ability to section-shoot helps a confederate win as long as he maintains a high enough hold.

Mr. Scott obviously knows nothing about the casino management mentality. First, no casino boss is going to under any circumstances look the other way while a dealer gives away even a dollar of house money (unless that boss is getting a cut–and that’s another matter, and one which Mr. Scott does not allege.)

Second, no casino management will keep a dealer who they believe has the ability to cheat because they realize he may at any time decide to use that ability to cheat the house. For example, I’ve met twenty-one dealers who were amateur magicians. They always keep that fact a secret for fear that they will be fired if management finds out they have even a modicum of sleight -of-hand skill. The house’s policy in these matters is, don’t wait for it to happen–get rid of the liability.

According to Mr. Scott, the casino has a dealer who they know can section-shoot, yet they keep him because he uses his ability to increase the hold. They’re not worried that he might decide to use that ability to help a friend win a huge score at one sitting that would more than outweigh whatever he might have helped the house win in the past, and his share of the profits would be enough so that he wouldn’t be the least concerned if the house fired him right afterwards.

Nor would management fears of such a scam be unfounded. The fact is that if dealers could actually do what Mr. Scott believes they can, the game of roulette would have been destroyed long ago. Dealers would, indeed, have used their talents at every opportunity to bankrupt the house by helping agents win. Mr. Scott is naive enough to believe that a dealer with this skill will use it for job security. The money-making potential of such a skill makes the whole issue of job-security irrelevant.

Some years back, Lance Humble discovered that he and his blackjack students weren’t winning as much money as his calculations predicted. He might have concluded from this that they weren’t really playing as well as he assumed. Instead, he concluded, and claimed in his book, that blackjack cheating was rampant in Las Vegas. (He didn’t win tons of money, therefore, he must have been cheated!)

[Note from A.S.–some big teams have kept records of their results at hand-held blackjack games vs. shoe games, and found their results at hand-held consistently below their results at the shoe games. At least one big team concluded that there was dealer cheating at hand-held games, and chose to limit their team play to shoe-dealt games.]

A close reading of Mr. Scott’s article suggests that he has gone through the same rationalization process. He hasn’t won tons of money. It couldn’t possibly be that his system isn’t as strong as he thinks. No, he must have been cheated! Once he settled on that theory it was, no doubt, easy to find many self-proclaimed insiders in Las Vegas willing to feed his fantasy.

I don’t know of any dealers who have ever made money for the house or themselves by their ability to section-shoot. But I do know of some who have made money off the willingness of players to believe they could.

Here’s a scam I ran across in a casino in the Orient. A dealer would approach a high roller and explain to him that, through years of practice, he had mastered the skill of placing the ball in any sector of the wheel he wished. It didn’t work every time, but often enough to affect the odds. All the player had to do to win was consistently bet a set of nine numbers in a particular sector of the wheel during the dealer’s shift. The dealer would do the rest by aiming the ball at that sector. All he asked in return was 50% of the winnings. A rendezvous was arranged to split the profits after the shift.

The dealer would then approach another high roller with the identical story but giving him a different set of nine numbers. This process was repeated with two other suckers, so that almost every number on the wheel was assigned to one player or another.

The dealer, of course, did nothing to help anyone win. Nevertheless, at least one player was bound to show a profit and he would be convinced that it was all due to the dealer’s aid. The dealer would then meet that particular player to receive his share of the winnings while avoiding the other players. The victims could hardly complain to the management, “One of your dealers was supposed to cheat to help me beat you guys and then he didn’t do it.” Who knows, maybe some of Mr. Scott’s ‘sources’ had some such idea in mind.

Snyder replies: I received a barrage of letters on Scott’s article, one of the most interesting from an alleged former dealer who claimed that section shooting at roulette was not only possible but was a common talent that most dealers possessed. He claimed that it did not take five years, but more like five hours, for a dealer to acquire this skill.

On the other hand, prior to receiving the letter from Darwin Ortiz, I had a phone conversation with Steve Forte (author of the Gambling Protection DVDs), whose opinion of Scott’s article is identical to Darwin Ortiz’s. I highly respect both Darwin Ortiz and Steve Forte, Is there a dealer out there who can demonstrate the ability to section shoot?  ♠

Nevada Roulette
by Laurance Scott

Darwin Ortiz Responds to Scott
by Darwin Ortiz

Steve Forte Responds to Scott
by Steve Forte

The Roulette Debate Heats Up
by Arnold Snyder

A New Roulette Strategy?
by Arnold Snyder

The Davies Card Counting System for Beating the Royal Match Side Bet

by John Leib
(From Blackjack Forum Vol. XIV #1, March 1994)
© 1994 Blackjack Forum

An advertisement for the Davies System appeared in Blackjack Forum, December 1993, along with a short review by Arnold Snyder. The purpose of the Davies Card Counting System is to profit from the “Royal Match” proposition on some blackjack layouts.

Just What Is “Royal Match”?

The Royal Match wager is offered before a round is dealt, and is independent of the blackjack implications of the play of the hand. The amount which can be wagered is limited by the casino, but is generally related to the amount wagered on the blackjack hand.

The Royal Match wager wins when the first two cards dealt to the wagering hand are of the same suit; otherwise the wager is lost. If the wager is a winner it is paid off at odds of 3 to 1 (4 for 1), unless the cards are the king and queen, in which case the odds are 10 to 1.

Obviously, there are some situations in which the Royal Match wager has a positive expectation for the player. (A clear example would be 13 cards remaining, all hearts.) The Davies System relies on a counting scheme and attendant strategy table to select the rounds on which the Royal Match wager should be made.

Testing the Davies System for the Royal Match Side Bet

I was immediately interested in this new money-making opportunity so, at Arnold’s suggestion, I wrote to CCS and Associates, the publisher of the Davies System, offering to perform an independent, unbiased analysis, and to publish the results in Blackjack Forum if, and only if, they and I had agreed to this beforehand.

By return mail I received a copy of the Davies System and a letter signed by Lee R. Bakewell, whom I presume is a principal in CCS and Associates. The letter showed interest in my proposal, but did not agree at that time to publication of the results.

On January 5th I got their agreement to evaluate the Davies System and to publish the results. For system comparison purposes, I included a parallel evaluation of perfect Royal Match decisions (taking the wager whenever it had a positive expectation). Both faced the same situations at every card level from 47 remaining to 6 remaining, in 50 million randomly shuffled decks. Table 1 contains the results of the simulations used in that evaluation.

What Do The Davies Count System Simulations Say?

The first question I had was, “How efficient is the Davies System?” The meaning of “efficiency” in this context is the ratio of winnings using the Davies System to what would theoretically be possible using perfect play. This turned out to be a question which had 42 answers, one for each level of penetration.

I had decided that the best playing situation to hope for on a continuing bases would be head-to-head play, with seven rounds dealt before shuffling. So what were the meaningful levels in going through the deck? Clearly, the first round is out because we know the Royal Match expectation for this round is -3.77%. So I approximated the 2nd, 3rd, 4th, 5th, 6th, and 7th rounds as coming with 45, 39, 33, 27, 21, and 15 cards remaining, respectively. (You may not like my choices. If not, the raw data is in the table so you can select your own.)

One might suggest playing out the blackjack hands to get the specific penetration for each round, for each deck. I agree with the purity of such an approach, and welcome anyone to do their own 50 million decks of corresponding data. I doubt that the effort would be worth it, but I would be very happy to see how closely we agree.

It may surprise you how well the Davies System did on the question of efficiency. Averaged over the selected penetration levels, it extracted 75.49% of what would have been the reward for perfect play. This is impressive because the Davies System is quite simple, but yet is quite powerful.

Hourly Royal Match Win Rates

Is Royal Match for you? Well, what are your goals? As a curiosity, the Royal Match can be beaten. But for how much? Can you retire on your Royal Match winnings?

Let us return to “the best situation for any system: head-to-head play, with seven rounds dealt before shuffling”. Using my selected levels for Royal Match wagering opportunities and $50 wagers whenever a wager is called for, perfect play returns an average of only $3.24 per deck played, while the Davies System returns $2.44. You can, of course, select your own levels and calculate your own winnings per deck.

At $2.44 per deck, what does this imply in money-making terms? If we are fortunate enough to get 200 rounds per hour, with seven rounds per deck we would expect to get about 29 decks per hour, yielding about $69.84 per hour.

Enough to live on, of course, but will you generally get this kind of game? If you find yourself with six other players, the $50 Royal Match wager returns per deck (30 cards remaining) 24 cents with perfect play and 21 cents with the Davies System. With 35 decks per hour, this translates into $7.30 per hour. This may not be enough to live on, and you must contend with playing blackjack too, and may have a difficult time with the count for that.

Some methods of play can improve the average winnings per deck. For example, if the player always takes two hands when the Davies System calls for it on the sixth round (21 cards lef) this round would return an additional 48 cents per deck on $50 Royal Match wagers. This might come at a stiff price, however, by reducing the likelihood of a seventh round, which shows a profit of $1.55 (and there would be a strong correlation between wagering on the sixth round and wagering on the seventh round). If, however, you do get the seventh round after this two-hand sixth round, it will come significantly deeper into the deck, perhaps with 13 cards remaining and its attendant $2.12. If you are getting seven rounds per deck, it may be better to try spreading to two hands on the seventh round (15 cards remaining). If this works, it would increase the per-deck return by $1.55.

So much of the success of the Davies System depends on how deep the dealer goes before shuffling. This would appear to be more important than how many hands per hour you play because of the great increase in expectation with penetration.

How to Read the Davies Royal Match System Simulation Tables

The table has four columns: LEFT, TxS-BETS, TxS-EDGE, and EST-EDGE, where “x” indicates which decision criterion was used: “P” for “Perfect”, “D” for “Davies”. LEFT indicates the deck penetration, and gives the unseen cards at the moment of making the Royal Match wager decision. TxS-BETS shows how many times in 50 million decks that the x-algorithm indicated the wager should be made. TxS-EDGE shows the return on those wagers for each unit wagered, divided by 50 million, and is given in percent. EST-EDGE is the sum of the theoretical advantages for those wagers divided by 50 million, and serves as a check on the simulation. These values will be close to those in the preceding column if the simulation is correct.

For calculating effectiveness, actual results of the wagers (TxS-EDGE) were used, not the values in the EST-EDGE. To find the amount won per deck on $50 Royal Match wagers, add the values in the TxS-EDGE column at the levels of penetration expected when Royal Match wagers may be made. Divide this value by 2 to get the per-deck winnings in dollars.

Parting Comments on the Royal Match Side Bet

Royal Match will probably never make anyone rich at the table, but it does offer some positive expectation opportunities which can be tapped for additional profit. The Davies System provides a fairly strong approach to these profits, yet is quite simple. It does take work to prepare for casino play, and probably more concentration than most effective blackjack card counting systems.

The question of whether attention to the Davies System count will cost the blackjack counter more in blackjack win than that won from Royal Match remains unanswered at this writing. If you have mastered basic strategy and that is what you play, then you should be able to concentrate on the Davies System and add a few extra dollars to salve the wounds caused by your blackjack losses. (I note no casinos that are identified as offering Royal Match have blackjack rules such that basic strategy does not lose.)

A further question in my mind is will a casino deal to a player who is playing the game only for the Royal Match play? Somehow, I think a $2 blackjack wager with an occasional $50 Royal Match wager (never off the top!) would not be tolerated for long, especially if it becomes clear that the player is winning with this approach. It strikes me as a more-difficult problem to camouflage a winning Royal Match system than to camouflage a winning blackjack counting system.

On January 8, Lee Bakewell sent me written confirmation of his agreement to publication of the results of my analysis of the Davies System, along with some enhancements designed to improve the system’s effectiveness. I immediately incorporated these enhancements and ran a new set of 10 million decks to see how beneficial the enhancements, indeed, were. The results, shown in Table 2, were quite impressive.

(NOTE: You will see that the TPS-EDGE and EST-EDGE columns for PERFECT PLAY are different in some corresponding entries between the two tables. You may wonder why this is the case. It is because they are each the result of an independent simulation and represent a different set of samples, with the resultant difference for these average values. The difference is small in all cases.)

Using the same penetration levels as before for head-to-head play, I found the efficiency of the Davies System Enhanced to be 91.73%, when compared to perfect play over these ten million decks. With $50 Royal Match wagers, this translates to an average of $3.01 per deck, or about $85.90 per hour in the fast, head-to-head game, a 23% improvement over the non-enhanced version. In the seven-player, two-round game, the average profit per deck (30 cards remaining) is about 23 cents. (Perfect play gives a profit of 25 cents per deck for this set of 10 million decks.)

This improvement comes at a price: The counting system is more complicated and would, therefore, increase the time to train for actual casino play, the concentration needed in the casino, and the difficulty in pursuing a point count playing strategy for the blackjack hand. In my opinion, only a rare few would have the discipline to accurately count for both Royal Match wagers and blackjack play, while the rest would have to rely on basic strategy for blackjack playing decisions. This would be ripe for a team approach to protect the blackjack wager while finding the lucrative Royal Match opportunities.

Below is a summary of things you should know as a result of these simulations. It shows six different playing conditions and gives the average profit per deck, profit per hour, and expectation on Royal Match wagers when called for by the Davies System. Both regular and enhanced versions are included. The columns labeled “EXP” is the average expectation on Royal Match wagers and is calculated as the total amount won divided by the total amount wagered, expressed as a percent.

A significant observation can be made to support that the benefits of the enhanced version over the regular version lie in the higher expectations. (I understand that the Davies System now being distributed is the enhanced versions, and earlier purchasers have been updated.) Much of this comes from inhibiting wagers which would otherwise be called for. While making 23% more money, only about 80% as much was wagered. This makes it much safer from a “risk of ruin” point of view.

As you can see, this table does not form a smooth curve and interpolation/extrapolation might be dangerous.

What More Can We Do?

It would be interesting to see what the combination of Davies System and blackjack counting can do, with various approaches to the blackjack wager and number of hands, and with various depths of penetration. I hope to explore these questions for a future issue of Blackjack Forum.  ♠

Summary of Davies Royal Match Card Counting System Analysis

	REGULAR VERSION	ENHANCED VERSION
Players	Rounds	Hands/Hr	$/Deck	$/Hour	Exp	$/Deck	$/Hour	Exp
7	2	70	$0.21	$7.30	2.82%	$0.23	$7.88	3.21%
4	3	100	$0.43	$14.25	2.48%	$0.47	$15.67	2.96%
3	4	120	$0.91	$27.38	2.73%	$1.27	$38.14	4.73%
2	4	150	$0.35	$13.17	2.07%	$0.40	$14.92	2.53%
2	5	150	$0.83	$25.00	2.11%	$1.13	$33.83	4.48%
1	7	200	$2.44	$69.84	3.77%	$3.01	$85.90	4.33%

TABLE 1
ROYAL MATCH ANALYSIS

50,000,000 Decks Played	PERFECT	DAVIES
Per Deck Win on $50 *Bets:	$3.2380	$2.4444
	Efficiency:	75.4910%

PERFECT PLAY

LEFT	TPS-BETS	TPS-EDGE	EST-EDGE	LEFT	TPS-BETS	TPS-EDGE	EST-EDGE
47	0	0.0000%	0.0000%	26	10562306	0.8829%	0.9037%
46	16832	0.0003%	0.0003%	25	9810734	1.0417%	1.0344%
45	91650	0.0025%	0.0010%	24	9619151	1.1746%	1.1755%
44	146757	0.0034%	0.0023%	23	11710914	1.3514%	1.3520%
43	276459	0.0044%	0.0054%	22	11711140	1.5230%	1.5277%
42	402526	0.0118%	0.0106%	21	12496058	1.7238%	1.7361%
41	865461	0.0180%	0.0200%	20	12418021	1.9801%	1.9705%
40	1361867	0.0253%	0.0326%	19	13747058	2.2378%	2.2193%
39	1735815	0.0510%	0.0491%	18	12883330	2.5094%	2.5157%
38	2094195	0.0699%	0.0714%	17	13102034	2.8387%	2.8231%
37	2851369	0.0975%	0.1006%	16	14695124	3.2548%	3.2346%
36	3503400	0.1346%	0.1354%	15	15394673	3.6472%	3.6670%
35	4083919	0.1816%	0.1752%	14	16479167	4.1245%	4.1187%
34	4862287	0.2207%	0.2228%	13	15976203	4.7407%	4.7228%
33	5710462	0.2607%	0.2772%	12	17587821	5.4030%	5.4060%
32	6557465	0.3243%	0.3396%	11	18146368	6.1340%	6.1340%
31	6710487	0.4235%	0.4094%	10	18710903	7.3941%	7.4136%
30	7472002	0.4839%	0.4884%	9	14717051	8.3194%	8.3259%
29	7813685	0.5783%	0.5732%	8	12944131	9.4726%	9.4689%
28	8331258	0.6770%	0.6673%	7	17422067	12.2727%	12.2413%
27	9032900	0.7909%	0.7821%	6	24435568	14.1606%	14.1772%

THE DAVIES SYSTEM

LEFT	TPS-BETS	TPS-EDGE	EST-EDGE	LEFT	TPS-BETS	TPS-EDGE	EST-EDGE
47	0	0.0000%	0.0000%	26	13590557	0.6797%	0.7015%
46	16832	0.0003%	0.0003%	25	14428971	0.8323%	0.8385%
45	102973	0.0022%	0.0009%	24	15328227	0.9727%	0.9696%
44	356620	0.0022%	0.0009%	23	16885566	1.0489%	1.0466%
43	718453	-0.0006%	-0.0001%	22	19641568	0.9908%	0.9972%
42	1135989	-0.0016%	0.0005%	21	22509849	0.9637%	0.9597%
41	1607271	0.0019%	0.0058%	20	24362612	1.0763%	1.0774%
40	2203403	0.0100%	0.0158%	19	24376220	1.4059%	1.4032%
39	2784001	0.0227%	0.0284%	18	23855593	1.7579%	1.7626%
38	3299284	0.0453%	0.0468%	17	23135185	2.1845%	2.1714%
37	3771649	0.0669%	0.0743%	16	22761011	2.6146%	2.5867%
36	4309693	0.1095%	0.1085%	15	21646282	3.0950%	3.1219%
35	4777867	0.1503%	0.1430%	14	20644515	3.7246%	3.7313%
34	5175336	0.1782%	0.1847%	13	20905208	4.2307%	4.2300%
33	5509838	0.2141%	0.2386%	12	24082712	4.3297%	4.3337%
32	5875004	0.2813%	0.3013%	11	29575747	3.9535%	3.9621%
31	6411455	0.3723%	0.3618%	10	32428895	4.3252%	4.3423%
30	7400492	0.4172%	0.4198%	9	29099947	6.4594%	6.4693%
29	9021576	0.4726%	0.4666%	8	23998472	9.3691%	9.3665%
28	10740177	0.5164%	0.5153%	7	23875685	11.4091%	11.3733%
27	12300661	0.5910%	0.5877%	6	24018501	13.9370%	13.9517%

TABLE 2
ROYAL MATCH ANALYSIS
(ENHANCED VERSION)

10,000,000 Decks Played	PERFECT	DAVIES
Per Deck Win on $50 *Bets:	$3.2773	$3.0065
	Efficiency:	91.7349%

PERFECT PLAY

LEFT	TPS-BETS	TPS-EDGE	EST-EDGE	LEFT	TPS-BETS	TPS-EDGE	EST-EDGE
47	0	0.0000%	0.0000%	26	2111717	0.9011%	0.9037%
46	3383	-0.0001%	0.0003%	25	1961015	1.0119%	1.0345%
45	18353	0.0030%	0.0010%	24	1923584	1.1717%	1.1754%
44	29395	0.0007%	0.0023%	23	2342628	1.3226%	1.3514%
43	55223	0.0069%	0.0054%	22	2342291	1.5149%	1.5271%
42	80786	0.0099%	0.0106%	21	2500231	1.7312%	1.7363%
41	173071	0.0314%	0.0201%	20	2484415	1.9170%	1.9703%
40	272096	0.0311%	0.0326%	19	2748569	2.2354%	2.2189%
39	347388	0.0316%	0.0490%	18	2576641	2.5515%	2.5140%
38	418217	0.0779%	0.0712%	17	2619991	2.8304%	2.8222%
37	569621	0.1220%	0.1003%	16	2938812	3.2253%	3.2333%
36	700850	0.1308%	0.1352%	15	3077918	3.7214%	3.6654%
35	815824	0.1678%	0.1753%	14	3294481	4.0757%	4.1162%
34	972648	0.2258%	0.2229%	13	3194823	4.7808%	4.7220%
33	1141664	0.2774%	0.2773%	12	3516535	5.4100%	5.4016%
32	1310725	0.3056%	0.3396%	11	3628065	6.1460%	6.1280%
31	1341341	0.3993%	0.4096%	10	3740120	7.3129%	7.4044%
30	1493329	0.5048%	0.4883%	9	2940757	8.3377%	8.3151%
29	1560046	0.6066%	0.5729%	8	2585983	9.4554%	9.4556%
28	1665641	0.6758%	0.6668%	7	3483027	12.1492%	12.2308%
27	1805680	0.7901%	0.7819%	6	4887680	14.1385%	14.1776%

THE DAVIES SYSTEM

LEFT	TPS-BETS	TPS-EDGE	EST-EDGE	LEFT	TPS-BETS	TPS-EDGE	EST-EDGE
47	0	0.0000%	0.0000%	26	2381229	0.7913%	0.8018%
46	3383	-0.0001%	0.0003%	25	2550639	0.9312%	0.9318%
45	20589	0.0019%	0.0009%	24	2735606	1.0888%	1.0566%
44	71525	-0.0088%	0.0009%	23	2986979	1.1860%	1.1759%
43	150862	-0.0007%	-0.0001%	22	3335243	1.2898%	1.2869%
42	250891	0.0097%	0.0003%	21	3632284	1.4592%	1.4512%
41	369402	0.0194%	0.0044%	20	3737665	1.6570%	1.7130%
40	502906	0.0165%	0.0138%	19	3604193	2.0733%	2.0626%
39	622449	0.0090%	0.0279%	18	3397649	2.4518%	2.4193%
38	719314	0.0598%	0.0488%	17	3196603	2.7735%	2.7796%
37	802202	0.0901%	0.0787%	16	3083429	3.1543%	3.1544%
36	892655	0.1192%	0.1149%	15	3042799	3.6075%	3.5434%
35	964436	0.1332%	0.1522%	14	3212390	3.7979%	3.8684%
34	1024690	0.2054%	0.1959%	13	3517161	4.3052%	4.3008%
33	1075104	0.2500%	0.2506%	12	3787099	5.0876%	5.0407%
32	1137433	0.3071%	0.3124%	11	3758762	6.0972%	6.0692%
31	1232305	0.3604%	0.3728%	10	3447739	7.1186%	7.2093%
30	1402885	0.4500%	0.4349%	9	2931329	8.3254%	8.3024%
29	1669190	0.5104%	0.4992%	8	2498463	9.3508%	9.3534%
28	1939498	0.5979%	0.5751%	7	1994831	10.1246%	10.1801%
27	2178521	0.6854%	0.6738%	6	1334755	10.4211%	10.4522%