Card Counting Analysis, 101

“The fourth way differs in that it is not a permanent way. It has no specific forms or institutions and comes and goes controlled by some particular laws of its own." – G.I. Gurdjieff

When blackjack card counting was first popularized in the early 1960’s, the idea that the edge changes as cards are dealt from the shoe was revolutionary. Casino management panicked at first, believing counters would soon overrun casinos and lay waste to a business model that had thrived for decades. Fortunately for the casino industry, the technique was too complex for the average player. Many tried and continue to try. Very few ever succeed. Even fewer view blackjack card counting as an ongoing source of income. In my opinion, blackjack card counting in one of smallest advantage play threats casinos face today.

The most commonly used blackjack card counting system is the hi-lo system (2,3,4,5,6 = +1, 7,8,9 = 0, T,J,Q,K,A = -1). What continues to surprise me is how ordinary card counters approach a potentially new card counting opportunity. As I read message boards, I often find blackjack card counters posing a question of this general form, “at what hi-lo true count should I play the xxx side bet?” Few blackjack card counters understand that the best approach to any wager is to examine it as a new and independent game. It has particular laws of its own.

Advanced advantage player: Why are you using that screw driver to try to hammer that nail?

Blackjack card counter: Because I already own a screw driver and don’t own a hammer.

Advanced advantage player: Why not just get a hammer? It’s a far superior tool.

Blackjack card counter:  Because I already understand how to use a screw driver and I've never used a hammer.

Advanced advantage player:  Why don’t you learn how to use a hammer, it’s the right tool for the job?

Blackjack card counter:  Why bother? A hammer wouldn’t work at all when I need to screw something in. This screw driver works for everything I want to do.

This conversation goes for both sides of the table. Any time a new wagering opportunity arises in a game that is dealt from a shoe with multiple rounds between shuffles, the wager is potentially vulnerable to card counting. This mostly occurs with blackjack and baccarat variants and side bets. What we need is a way of measuring the size and scope of a potential card counting vulnerability.

The goal of the card counting analysis I do is to compute a single number that gives an idea of a game’s potential vulnerability. The metric I developed is called the “win-per-100-hands.” I express this as dollars won per 100 hands, assuming the AP makes a $100 wager whenever he has the edge and otherwise makes no wager. That is, I assume the AP is watching the game, counting from behind, without making bets on the main game. Whenever the count indicates, the AP makes a $100 wager, no more, no less. I assume the AP is playing the game perfectly according to the counting system. This AP’s earnings give an absolute upper-bound on what is possible against the game using card counting.

Because the card counting world revolves around blackjack, it is worthwhile knowing the win rates for perfect blackjack card counting:

• Blackjack, six decks, cut card at 260, (rules: H17, DOA, DAS). Win-per-100-hands = $33.58.
• Blackjack, two decks, cut card at 75, (rules: H17, DOA, DAS). Win-per-100-hands = $66.29.

(Note. H17 = dealer hits soft 17. DOA = player can double on any first two cards. DAS = player can double after a split).

It usually shocks casino management when they see these values for the first time. There is disbelief. How can blackjack card counting be so small, yet the industry-wide focus be so large? However, these numbers are correct. They show that blackjack card counting is a small problem. But the historical momentum created by decades of fear of blackjack card counters is not lessening any time soon.  Today, however, there are much bigger card counting problems to worry about. There are hammers.

For example, the Slingo Bonus Bet 21 side bet in blackjack gives a win-per-100-hands of over $1,700. In baccarat, the UR Way Egalite side bet gives a win-per-100-hands of over $600. These wagers have 10-to-30 times the vulnerability of ordinary blackjack card counting.

Baccarat and blackjack side bets are not the only card counting issue. As the market evolves, new versions of baccarat and blackjack are being released. For example, 7-Up baccarat has a win-per-100-hands of $68.70, making it a stronger opportunity than card counting two-deck blackjack.

I have analyzed dozens of card counting opportunities in this blog. The method I use is fairly consistent between these articles, but I am concerned that many readers may still be scratching their heads. For that reason, I am going to go through the process one step at a time.

When analyzing a new wager for a card counting vulnerability, I follow these steps:

1. I first compute the baseline house edge and standard deviation. This is done using combinatorial analysis. I determine every possible way the hand can play out and count the number of those that correspond to different payouts for the wager. This is done off-the-top, assuming no other cards have been dealt. This is just the standard house edge computation, nothing more.
    
2. I then determine the effect- of-removal (EOR) for each card. To do this, I run the same analysis as in step 1, only I do it 13 more times, with individual cards removed. One at a time, I remove an Ace, 2, 3, …, Jack, Queen, King from the shoe and re-compute the house edge. The new house edge with the single card removed is compared to the original house edge. This allows me to determine the change in house edge that the card gives when it is removed from the deck. The EORs are used to see how important various cards are to the wager to help create a card counting system.
    
3. The next step is to use the EORs to come up with a card counting system that is custom designed for the wager. This is a bit of both art and science. The EOR’s are decimals and are certainly not suitable for use in any realistic way. But, by doing some rough approximations and rounding, the EORs are scaled to give card counting systems to test. Sometimes several systems can be created and compared. Other times, the best card counting system is apparent from the EORs.
    
4. (Math geek moment #1) There is a way of measuring the quality of a card counting system called the “Betting Correlation” (BC). In some of my articles I compute this number. The closer the value of BC is to 1.0, the better the card counting system. If you like, BC is the cosine of the angle between the EORs and the count system, viewed as vectors in 13 dimensions. A cosine close to 1.0 corresponds to an angle close to 0 degrees; in other words, a perfect card counting system.
    
5. The next step is a huge simulation. In both blackjack and baccarat I have standard games that I simulate. In blackjack, I simulate both the two-deck version (cut card at 75) and the six-deck version (cut card at 260). In baccarat, I simulate an eight-deck shoe with the usual burn card rule and the cut card placed at 14 cards. These games are chosen because they represent the best circumstances a card counter will be able to find, under normal conditions.  It is rare for a casino to offer deeper penetration that these values. I try to simulate one billion (1,000,000,000) shoes. In practice, this is not always possible; some simulations are for far fewer shoes.
    
6. These simulations output a lot of different data for the game. I import the data for further processing using Excel spread sheet analysis. Here are the most important statistics I determine:
    
   1. Trigger true count: the minimum true count at which the card-counter has the edge. I have been inconsistent in the use of my language. Sometimes I refer to this as the "target" true count.
       
   2. Average edge: sometimes the counter has the edge, sometimes he doesn’t. Assuming the AP only makes the wager when he has the edge, this number gives the average edge the AP has.
       
   3. Bet frequency: the AP does not have the edge on every hand. This number gives the frequency that the AP makes the wager using the count system. In other words, this is the percent of time that the true count equals or exceeds the trigger true count.

Finally, after all these steps are completed, the win-per-100-hands can be determined. Here’s the formula:

Win-per-100-hands =

(100 hands) x ($100 per hand) x (average edge) x (bet frequency)

If the win-per-100-hands value is large enough, I typically run additional simulations to get win-rates for various cut-card placements. This is done so that a casino that offers the game can make an informed management decision about where to place the cut card. There is a time/motion trade-off. If fewer cards are dealt between shuffles, then income decreases because of rounds lost to the shuffling procedure. Each casino must assess whether the risk of advantage play outweighs additional fixed procedural costs.

(Math geek moment #2). There is a more sophisticated way to look at the problem of determining the upside win-potential of card counting. First, a distribution is obtained for the edges at various parts of the shoe. I do this by a massive simulation, where I collect house-edge distributions for hands being dealt at specific points in the shoe. By “integrating” under these curves, an absolute upper bound on the win-rate can be numerically approximated. This maximum win-rate is equivalent to using computer-perfect play. Since card counting obviously returns less than computer-perfect play, distribution analysis gives a way of seeing how much is out there to begin with. I used this method, for example, to show that baccarat card counting is a pointless endeavor.

There is one last detail that is part of the equation when an AP is deciding whether to go after an opportunity. This is the question of variance. The rule-of-thumb is that games with low win-rates should also have smaller values for the variance. The larger the win-rate, the more variance the AP will tolerate. Games that have a low win-rate and high variance are usually undesirable. However, this doesn’t mean that there won’t be anyone going after these situations.

The Dragon 7 side bet for EZ Baccarat is a good example of a high variance game. It also has a high average edge (over 8%), making it viable. A much better example of the edge/volatility dilemma is video poker. There are certainly professional video poker advantage players. However, it takes a unique type of individual to be able to play VP for months or years on end to squeak out a low return, while dealing with the extremely high volatility created by very rare top payouts. I personally prefer income without volatility, in other words, a day-job.

To finish this article, I will review card counting game protection. Card counting looks the same, whether it is ordinary blackjack or some new variation or side bet. But don’t expect the high-quality player to be using a screw driver. If the wager requires a hammer, the best APs will bring a hammer. From the casino side, the way you tell if someone is using a hammer is by knowing what hammers look like. Otherwise, you don't stand a chance.

Watch for the following:

• Players who have a large bet spread.
• Players who jump into the game and make large wagers.
• Unusual wagering.  For example, a larger wager on the side bet than on the main game.
• Side bet poaching. That is, playing a side bet on another player’s main bet.
• Groups of players who may be watching different games and signal the group.
• Groups of players at the same table wagering together, when the side bet has multiple options.
• Anything else that looks like card counting.