Card Counting Analysis, 101

If you are not familiar with card counting (for example, in blackjack), you should first learn the topic before reading this post. For example, my book “The Blackjack Zone” (available on Amazon) is a good introduction.

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 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 “units won per 100 hands.” The “unit” is the size of bet the AP is playing. This wager can range from $1 (e.g. on Super Sevens) up to $10,000 or more (Pairs bet on baccarat). Without specifying the unit size, this metric gives a win-rate that allows wagers to be compared apples-to-apples.

I often rephrase “units won per 100 hands” in dollar-terms by assuming a unit size of $100.  That is, I assume the AP makes a $100 wager whenever the count indicates and otherwise makes no wager. 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 with a maximum wager of $100. I will sometimes use the terminology “win-per-100-hands” when referring to this dollar amount.

Because the card counting world revolves around blackjack, it is worthwhile knowing the win rates for perfect blackjack card counting.2 The world’s best High-Low card counter, no matter how good he is, can do no better than the following with a maximum bet of $100:

• Blackjack, six decks (H17, DOA, DAS), cut card at 260 (52 cards from the end): Win-per-100- hands = $33.58.
• Blackjack, two decks (H17, DOA, DAS), cut card at 75, (29 cards from the end): 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. However, there are much larger card counting problems to worry about.

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. The method I use is fairly consistent. I am going to go through the process one step at a time. When analyzing a 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 ways 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 EORs 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. There is a way of measuring the quality of a card counting system called the “Betting Correlation” (BC). I compute this number for some wagers, but not all. 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 than 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:
   • Trigger true count: the minimum true count at which the card-counter has the edge.
   • 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.
   • 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, “win-per-100-hands” (in units) can be determined by simple multiplication. Here’s the formula:

Win-per-100-hands =

(100 hands) x  (average edge) x (bet frequency)

If this number is sufficiently large to indicate a substantial vulnerability, I may 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 tradeoff. 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 these time/motion costs.

There is one last detail that is part of the equation the AP considers when deciding 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 for the AP (the AP gets more than an 8% edge over the casino), 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 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.