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Blackjack Card Counting Studies

This page contains links to and explanations of several charts which have been created with CVData and provide illustrations of many Blackjack principles. Note: Some of these studies are quite advanced. You do not need to understand these charts to count cards. Below is a quick index of sections describing the charts. Each section has one or more links to the chart images.


Ease of Use vs. Efficiencies of Various Strategies

In an attempt to visually illustrate the differences in ease of use and efficiencies between strategies, Iíve created a 3D Scatter Chart. The chart consists of 14 balloons suspended above an x-z grid. The x-axis is Betting Correlation. The z-axis is playing efficiency. The string from each balloon intersects the grid at the BC and PE for that strategy. The height of the balloon (y-axis) is the ease of use of the strategy. Thereby, each balloon indicates all three variables. The ideal system (impossible to obtain) would be at the top, right, back. Note, the two strategies in the center (Omega II and Uston APC), are very high PE, Ace-Neutral strategies. If Ace side counts are kept for these strategies, they would move substantially to the right placing them closer to the ideal combination of efficiencies. However, they drop in height as they become more difficult to use. Be careful of the parallax problem. Balloons closer to the front appear not to be as high as they are.


Advantage and Units Won/Lost vs. True Count

The count is better for you at extremely high counts with two esoteric exceptions (described at the end.) Iíve attached a combination chart which shows Advantage and Units Won/Lost vs. True Count. The green area shows the losses at negative values and gains at positive values. Of course, the big gains and losses are at relatively low plus or minus counts, because this is where the majority of hands exist. The red line shows advantage. It is very smooth for the majority of counts, but goes wild at the very high and low counts. This is despite the fact that this is a simulation of one billion hands and the data has been smoothed (with a quadratic B-spline algorithm.) Problem is, there just arenít that many hands at the extreme counts and the variance is obscene. Of course, if you play long enough, you will experience a few wild counts. Your results at those counts are essentially random. Unfortunately, the human mind is more likely to remember such events, even though they have no meaning. This is why people watch X-files and other silly TV shows.

Esoterica

  • 1. If you are playing single deck, and you and all other players play without any variation whatever, then certain wild TCís will only occur with certain dealt card sequences. This will result in automatic wins or losses at specific extremely high or low counts. The odds of running into this situation are approximately zero.
  • 2. If you are side-counting Aces and there are none left, youíve got a problem with a high count.


Advantage by Type of Hand

I've been experimenting with topology maps in an attempt to better show statistics by type of hand. The attached Advantage Surface Chart shows advantage for the various first two cards. X-axis is type of hand (all hard hands, soft hands and pairs). Z-axis is dealer up card. Y-axis is eventual advantage given six deck, Hi-Lo, 1-8 spread.


Time Spent in Advantage Situations Balanced vs. Unb.

Comparing the percentage of time that two systems indicate specified advantages is problematic because the counts are not continuous. Different systems result in different levels of advantage percentage. However, I took a shot at it:

The first chart displays the time spent at certain advantages for K-O and Hi-Lo.

The second chart shows the advantage at each count (Running Count for K-O and True Count for Hi-Lo.)

The third chart shows the frequency of hands at each count. Here, the red and green are charted using a logarithmic scale. The blue ribbon in the back is the same data as the green ribbon plotted with a standard scale. It is there to show why I had to use a logarithmic scale and to show the huge number of hands at a TC of zero in a system that truncates instead of rounds.


Cost of Errors

The Cost of Errors Chart is an attempt to show the cost of various types of errors. Five columns are provided using data from five multi-deck, multi-player sims. The height of the columns represents advantage.

  • Column 1 displays the effect of playing errors. The total height of the column represents the advantage with no errors. The green segment indicates the penalty resulting from one error per hundred hands. The blue section is the additional penalty of another error per hundred. And on through five errors. The red pedestal is the advantage with a 5% error rate. The errors are serious; but not idiotic. Insure is reversed, surrender is reversed, split or double is changed to hit, hit and stand are reversed. But, a hard 18 up is never hit, an eleven down is never stood, and a double or split is never taken when it shouldn't.
  • Column 2 displays the effect of betting errors for a non-cover bettor. Again, succeeding circular slices of the bar show the effect of errors. The player is spreading 1 to 8. Errors are: if should be a one bet, bet two; if should be a two, bet three; if should be a three through eight, but two.
  • Column 3 displays the effect of miss-estimating the remaining cards for TC calculation. The green shows the effect of a 10% error and the blue shows the additional effect of an additional 10% error.
  • Column 4 shows the effect of using no indexes at all. The green section is the penalty when using perfect BS vs. -10 to +10 indexes. The count is still used for betting purposes.
  • Column 5 is the effect of errors for a cover-bettor. The betting is very conservative not allowing large increases or decreases, no increases after losses, no decreases after wins and no change after pushes. The error scenario is too complex to explain here. (OK, Iím too lazy.)

I did not include a column for flat betting, because there wouldnít be one. Youíd lose all of your advantage.


Double Diamond Blackjack

A question was raised as to the advantage of a new game called Double Diamond Blackjack. This game pays extra on a Diamond BJ, but less on other BJ's. Also, several other fancy rules are added. The problem with such a game is the huge penalty of reducing the BJ payoff. I've created a Surface Area Chart which displays the difference in winnings between a normal single deck game and a game like Double Diamond (DD). The DD game I used was 6 card charlie, 5 card 2:1, Diamond BJ pays 2:1, Normal BJ pays even, but is automatic win, Double on any number of cards even after splits. The chart has all two card hand types on the x-axis, dealer upcard on the y-axis and difference in winnings on the z-axis. The z-axis is winnings on the Diamond game minus winnings on a normal game.

Looking at the chart, the games are equal where the blue and green meet. Green is a slight advantage for DD, red is a serious disadvantage for DD. The green/blue splotchiness is due to the small number of hands run (160,000,000). It indicates that the variance at that number of hands is actually greater than the difference in results between the two games. The solid green with upcard combinations totalling 5, 6, 7 and 8 indicates a very slight gain in using the DD rules. This is due to the gain from double on any number of cards, 6 card charlie, and 5 card 21. The huge slice through the stack shows the loss due to most BJ's paying even money. This is somewhat less at BJ vs. Ace because of the BJ automatic win rule.

The point of the chart is to show the enormous penalty of the BJ rule change versus the very slight gains by the oddball rules.


Components of Advantage

This chart provides two rows of Stacked Bars. There are 14 pairs of bars representing the advantages that can be gained using various strategies according to Griffin calculation techniques. The y-axis (advantage) is not quantified as it is relative. The rectangular columns in the back row indicate the relative gains when playing multi-deck. The dark green signifies gain from betting and the red indicates gain from using indexes. A 1-8 spread is assumed. The circular columns in the front row indicate the relative gains when playing single deck. A 1-2 spread is assumed. Here, three components are displayed. Again, betting and playing gain are shown. The, additional, blue segment indicates the gain from playing SD vs. MD.

So, what does this chart illustrate? Nothing new; but a few concepts that should be kept in mind:

  • Playing gain is equal to or more important than betting gain in SD as opposed to MD where betting gain is substantially more important. However, both are important in MD.
  • Spread can make up for the loss in MD advantage, or for the pessimist, spread is necessary to make up for the loss in MD advantage.
  • The differences between systems are dwarfed by the difference in spread. That is, we spend altogether too much time thinking and debating about which system is best and not enough time talking about how to maximize the spread without getting tossed. This is the simple point of the chart.

Disclaimers: No simulations were run. Results are calculated from Griffin formulae. Side counts, number of indexes and cover plays are ignored. PE calculation is questionable for unbalanced counts.


First Base Penalty

For some time, we have been aware that it is better to sit at third base in single deck, face down games. Common sense tells us that we get to see more cards and can make better playing decisions. In an extreme case (seven players), the advantage difference between seats 6 and 7 is about 0.05%. You lose another .05% per seat as you move toward first base. However, the difference in advantages between first and second seat is much worse. First seat can be as much as .16% worse than second seat. As this is a severe penalty, I decided to take a look. First, I looked at the winnings by true count. I created a chart which shows the winnings for first seat and second seat by true count. [link] The chart shows that the winnings are identical for all counts below 4. But, at a TC of 4, second seat does better. At 5-8, better and better. After that,. It evens out. OK, we now know that there is something about these particular counts that we should examine. I then decided to look at hand types. I took the winnings for the second seat and broke them up into an array of all possible two card hands vs. dealer up cards. This is an array of 330 values. I also created the same array for the second seat. I subtracted the second array from the first array and charted the remainders. The result is a combination surface area/contour chart that indicates the hands where the first seat has a problem. [link] Eureka! First seat has a serious problem with two tens against a 3, 4, 5 and 6. Tens vs. 6 is particularly severe. All other hand results are about the same. Common thinking would have expected many differences along the lines of the Illustrious 18.

So, we have a problem with 10ís against 3-6 at TCís of 4-8. Guess what, the indexes for splitting tens at 3-6 are 4-8 (Hi-Lo.) So, why is there a major problem with splitting tens in seat one? Well, if you think about it, there is a quirk in seat one. Remember, we are playing SD, face down, seven seats. That means, two rounds. Only round two is important as that is where you are betting. To split tens, you must have two tens and the dealer must have a low card. If you are sitting at seat one, the only cards that you can see after the start of the round are two high cards and one low card. This means that the playing count will now be the count at the start of the round minus 1. If the round starts at a TC of +3, any seat has the possibility of splitting tens against a 3. That is, any seat except for seat one. Seat one cannot because the count will always be one less than the count at the start of the round or +2. 9% of the time, you will start round two at a true count of +3. 2.74% of the time, you will start at a true count of +4. This means that 6.26% of the time, every player has the possibility of splitting tens against a 6 in the second round, except for the player in seat one. (26% of all gain is in round two at a starting TC of +3 in this example.) The same holds for the other ten split opportunities, at reduced percentages. Therefore, seat one, and only seat one, has an automatic reduction in opportunity.

By the way, if you go through the same process between other seat pairs, you get the charts that you would expect. That is, the tens peak is muted and the other Illustrious 18 decisions start to poke out from the plane.

I donít consider this analysis complete and welcome comment.


Exact vs. Estimated TC Calculation

This section summarizes sims of nine billion hands with various methods of desk estimation. With the parameters that I used, TC calculation using exact (to the card) deck depth gave a .829% advantage and $17.29 win rate. When estimating the number of decks, generally, the worse the method of estimation, the lower your advantage, but the higher your win rate. This is due to overbetting. To show where this overbetting occurs, I chose a common method of deck estimation (287-312 cards=6 decks, 235-286=5 decks, etc.) and compared it to exact depth. Advantage is .810% and win rate $17.32 (very slightly higher than using exact remaining cards.) I created a chart showing the average bet on the Y-axis and deck depth on the X-axis. In general, average bet increases as deck depth increases because there are more high TC's. The average bet increases smoothly when TC calculation is performed with exact remaining cards. However, the increase is lumpy when the remaining decks are estimated. If you look at the chart (link is below) you will see how the sloppy estimate shows lumps of higher betting. The lumps increase in volume as deck depth increases because of the higher percentage of large TC's. These lumps in the graph signify the areas of overbetting. The area of the largest lump is the area of highest risk.

CHART

Conclusions

The better your deck esitmation the smoother and more accurate your betting, improving exposure to risk but not income.


Effect of a Back-Counter on your Play

Awhile back, I commented that Iíd leave a table if I thought it was being stalked by a back-counter. Thought Iíd sim the effect. Ran two sims. First sim had three players. BS players in seats one and two and a Hi-Lo player in seat three. We are interested in seat three. Second sim was the same, but a fourth player Wonged in at a TC of +4 and left at the end of the shoe. Again, we are interested in seat three. Six decks, five deck penetration. Each player played 150 million hands except the back-counter who played 13 million. The attached ribbon chart (link below) graphs the winnings by TC for the Hi-Lo player in each sim plus the back-counter. You will note that the red ribbon (seat 3 in the second sim) and the green ribbon (seat 3 in the first sim) run evenly through the negative TCís. At about +3, the green player pulls ahead. That is, the Hi-Lo player at the table with the back-counter won less money on positive counts. Overall, he lost about 0.15% advantage.

FIRST CHART - Winnings by TC.

OK, where is the lost advantage? The second chart has two series. The green series is the percentage of hands played by seat three at the back-counterís table of the hands played by seat three at the back-counter-free table. The chart shows that both seat three players played the same number of hands at negative TCís, but at positive TCís, the player disturbed by the back-counter played only 80% as many hands. This is due to the back-counter eating cards in positive TC conditions. So far, no surprise. However, there is another effect. The red series on this chart shows dollars bet instead of hands played. Again, the players at both tables bet the same per TC at negative TCís. But, at positive TCís the drop-off in units bet is more severe than the drop off in hands played. Only 75% as many units are bet at high TCís. That is, the average bet was lower at high TCís. Why is this? Well, the Hi-Lo player was using camouflage play. The spread was 1-8 on both tables, but the player would never make large hand-to-hand bet increases. Since the back-counterís interference tended to reduce the length of high TC consecutive hands, and reduced the number of hands dealt per shoe in favorable situations, the Hi-Lo player had fewer opportunities to win enough hands in a row to pump his bet up to the optimum level.

CHART TWO - Hands played and Units bet by TC

This shows an important point about running a sim exactly as you would play. It is not enough to show a simple 1-8 spread since realistic cover play may interact negatively with other characteristics of the sim.

Note: When just looking at the overall advantage, 150 million hands is OK. But, when you break this down into smaller groups of hands (e.g. by TC), then you have fewer hands per situation and need more total hands to give good results. However, there is a short-cut that was used here. All lines were smoothed with a 12 facet cubic B-spline formula. This takes information about neighboring data points (nearer points count more than farther points) and adjusts all points to produce a smoother graph. This requires several hundred million calculations, but thatís only seconds on a Pentium. If you are looking for exact data, this is not valid. But, if you are looking at trends, it is quite accurate and fast. To perform this on a CVSIM chart, double-click on a series (e.g. group of bars, a line, an area). The Format Series dialog box will appear. Click on the Options tab. Then, select a Smoothing formula at the bottom left. Click on Help to get information on the options.


Advantage at Very Low TC's

If you are using a huge number of indices, then your disadvantage at very low counts is slight. You have the ability to alter your play which makes up for part of your disadvantage. However, these days, few people bother with the negative indices. If you are using the Illustrious 18, then your advantage at very low TCís drops precipitously. The attached chart shows advantage by TC for two players at the same table. One uses the Ill. 18 and the other uses a full set of indices. Advantage at TCís below -14 barely changes for the full index player. Advantage for positive TCís continues to grow for both players. Does this mean that you should use a full set of indices? No, very little money is bet at those very low TCís.

Sim particulars: Single deck, three players, 1.6 billion hands per player, four rounds per shuffle, SE at TC -30 was .14. AO II was used as it has an excellent set of SD indices.


Cut Card Effect

Thought Iíd put together some charts to illustrate the Cut-Card Effect. I created four charts from 2.6 billion single-deck, basic strategy hands. About half of the hands were fixed at eight rounds per deck and the other half dealt to a 75% penetration (6 to 9 rounds.) The first simple chart shows the advantage by hand depth. The red bars show a even 0.2% advantage for the casino for all hand depths when dealing a fixed number of rounds. The green bars show the enormous increase in the casinoís advantage in the late rounds when dealing with a cut card. The advantage is so great, that I had to use a logarithmic scale (0.2% to 14%). Fortunately, there are not many hands dealt at the 14% casino advantage.

The following three charts each show hand dealt quantities. Each chart has as itís x-axis, all possible first two card player combinations. The y-axis shows the dealer up-card. The z-axis shows the number of incidents of each of the first two player cards vs. dealer up card..

Chart I: The first chart shows the normal distribution of hand types. That is, the number of times that you will receive each of the possible first two cards against each dealer up card.

Chart II: The second chart shows the distribution of hand types in the last rounds when playing with a cut card. In this chart, there exist more low cards since it is much more likely that you will see additional rounds when large cards are dealt in the earlier rounds.

Chart III: This is essentially the difference between the two previous charts. It shows the delta between the normal distribution of hands and the distribution of hands in the late rounds when using a cut card. This is a surface area chart with a projection of the colors to the base to more easily see the problem areas. Red and orange areas show the types of hands more likely to be seen in the late rounds. The chart shows a substantial increase in stiffs, particularly against dealer low cards. Also, more low hands (5-12) against a dealer ten. There is a corresponding decrease in BJís, twenties, and 17-19 hands against good dealer up cards.

I also have an old chart which shows the advantage at each of the above hand types. It can be seen that most of the hands where we have seen increases due to the cut-card effect are poor advantage hands.

Of course, all that Iíve shown with all of the above is what was already known. The cut card adds hands when the deck is lean in tens. So, does this mean that you should avoid SD dealt to a fixed penetration. Yes, if youíre playing BS. But, if youíre counting, itís not so clear. Iíve just started working on those charts, and it appears that counting overcomes the effect even in the late rounds. At least at the depths at which Iím currently testing.


The Effect of Number of Players with Cover Betting

Normally, the number of players at a table has no effect on your advantage. However, when cover betting, this can change. I ran a total of five billion hands with cover betting as follows:

  • No increase in bet after loss
  • No decrease after win
  • No bet change after push
  • Max increase or decrease two units
  • No cover plays
  • 1-8 Spread (1, 2, 4, 6, 8 at TC's of 1, 2, 3, 4, 5)
  • I allowed bet reset to one unit at shuffle as not resetting would clearly hurt a full table player.
  • Five/six deck, strip rules

Advantages:

  • 1 player: 0.60%
  • 4 players: 0.45%
  • 7 players: 0.33%

I created a Bet Size by TC Chart for the three players. X-axis is TC, y-axis is average amount bet (including double downs.) The red bars show the rapid increase in average bet size for the head-on player. It nearly matches the ideal. It drifts off very slightly at very high TC's because there are slightly fewer DD's at high TC's. The green and blue bars show the players' at fuller tables much slower and smoother increase in average bet size as they have more difficulty raising there bets quickly as high TC's occur at lower hand depths. It also shows them overbetting at +1 and +2 as they couldn't lower bets as quickly as desired.

I tried small sims with various number of players and no cover. There is no difference without cover. Also, the effect of cover when playing head-on is negligible. I also tried softening the cover by allowing a doubling or halving of the bet and allowing bet increases after a lost split or double and bet decreases after a won split or double. Didn't appear to change the results much, but I need to make more runs in that area.


The Effect of Cover on Advantage by Penetration

I put together an Effect of Cover chart to give some idea of the cost of various amounts of cover betting. The results are from one half-billion round sim. There were four players as follows:

Yellow: No cover Blue: No bet increases after a loss, no decreases after a win; but reset to one unit after a shuffle Green: Same as above but also no bet change after a push and no jumping bets up or down by more than two units. Red: Same as above but bet not reset to one after shuffle and Insurance Cover. (index of 4 for a BJ, 3 for a twenty and 2 for other hands.)

All players had a spread of 1-8. A two unit bet was allowed at TC of +1 Which is earlier than in BJ Attack's sims as the heavy cover player probably wouldn't have a chance with slower ramping. The y-axis is advantage. X-axis is penetration from 1% to 84%. Six decks, S17, DAS. TC accuracy was half-deck. All players played in all seats.

Note: The Red player had a disadvantage of .7% in the first hand. This is because he was not allowed to reset his bet after a shuffle. The other players all had .38% disadvantage of the first hand. (Which was fortunate as that's what my calculator says the BS advantage should be.)

I've also included a Percentage Chart This chart shows what percentage of the total loss due to cover can be attributed to each type of cover, by penetration level used by the Red (heavy cover) player. Red is the loss due to Insurance cover and not resetting your bet after a shuffle. Green is the loss due to no jumping bets or changing a bet after a pass. Blue is the loss due to no increases after a loss or decreases after a win. The Red area shows the large effect of not resetting the after shuffle bet for low penetration games. The Green area shows the effect of not being able to jump bets quickly at high penetration levels.

No surprises here. Cover is expensive.


Ameliorating the cost of cover

Given the high cost of cover play, I thought I'd look at one way of softening the blow somewhat. I ran five billion hands with three types of players as follows:

  • Red Players: No cover at all.
  • Blue Players: Never more than double or halve bet. No change after push. Except reset bet to one unit after shuffle.
  • Green Players: Same as above, but allow a bet increase after a Split or Double Down which lost or pushed.

The point of the sim is to see the gain from this one modification to cover play. The logic behind the modification is that after pushing a split or DD, you already have double the bet out. After losing double your money; it isn't unnatural to bet the amount that you lost.

Results (Initial Bet Advantage and Win Rates):

  • No Cover - 0.937%, $8.70/hr
  • Full Cover - 0.555%, $4.15/hr
  • Mod Cover - 0.643%, $5.10/hr

The gain in advantage from the change was .09% or about 23% of the cost of cover.

Chart - I've attached a Win Rate by Hand Depth Chart. The x-axis is the Hand Depth. Y-axis is the cumulative Win Rate for hands up to the Hand Depth and z-axis is the type of player.

Follow-up - These results beg a question. Most players do not bother with soft double indexes as it has been shown that the gain in advantage is minor. However, soft doubles may be more useful with cover when using this modification. The point is to increase the excuses to get more money on the table in positive situations without looking like you're jumping your bets. Of course you have to decide whether making unusual soft doubles makes you look more or less like a counter. I don't expect much gain here.

Sim details - Six decks, five deck penetration, S17, DAS, six players, Hi-Lo, 1-8 spread, quick ramping (two units at +2). With slower ramping, the effects would probably be greater than shown here.


Win Rate vs. Penetration vs. Hands/Hour

The question was, if you a game has less penetration, but is faster, will I make as much money. The game was double deck, H17, DAS. For this I ran 26 sims (actually one CVCX sim) for penetrations from 50% to 75% in increments of one card. The chart shows the win rate for each penetration. Five points are displayed for 100, 125, 150, 175 and 200 hands per hour. This makes it easy to compare different penetrations at different speeds. Note, the unit size and betting ramp are different for each penetration as they are calculated for maximum bankroll growth. See the chart here: Win Rate vs. Penetration vs. Hands/Hour

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