Royal Match 21 Blackjack Machine
The Davies Card Counting System for Beating the Royal Match Side Bet
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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.
Free Royal Match 21 Blackjack
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% |
| 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% |
21 Blackjack Free
| 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% |
| 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% |
For more information on blackjack side bets and professional blackjack card counting systems, see the Blackjack Forum Professional Gambling Library.
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