Use Caution When Evaluating a Blackjack Player’s House Advantage

The other day I was sitting in a marketing director’s office discussing player promotions and loss discounts. I expressed my opinion as to the problems of using the same discount of loss program for all customer discounts. Where most calculation regarding baccarat, is for the most part, accurate, when you use it for the other games such as blackjack, the discount of loss model becomes very imprecise. This is due to management’s inaccuracy in establishing a blackjack player’s true house advantage (H/A) maintained over the blackjack customer in the long-term.

In baccarat, the house advantage for any discount of loss model is set at 1.15 percent, or the average between the player H/A of 1.24 percent and the banker H/A of 1.06 percent. In blackjack, the player has a huge influence on their own personal mathematical disadvantage when making decisions on how to play their individual hands. Realistically, a blackjack player could use perfect basic strategy to play each hand, or could be somewhat naïve and play his hands through intuition and guess work. The mathematical advantage could range from 0.25 percent to somewhere around 2 percent. A number of experts agree that baccarat, with its H/A fixed at 1.15 percent, provides a good game for discounting loss; however, any time a game is subject to a H/A of under 1 percent, it’s subject to a much greater discounting loss cost.

Using Counter Catcher Software to Analyze a Blackjack Customer’s Mathematical Advantage
After I’d made my point regarding the varying house advantage of discounted blackjack players, the marketing director picked up a DVD off his desk and stated that the casino had just purchased software that would help analyze a blackjack player based on observed hand decisions. The software package the director alluded to is known as card counter “catcher” software, commonly used to determine whether or not a customer is counting cards. How does this software, primarily developed to determine if a blackjack customer is counting cards, return an accurate H/A percentage on the not-so-knowledgeable player?

Using card counter catcher software is a great tool for determining whether a player is counting cards or using advantage play technique. The different software packages are designed to spot correlations between wagering and player advantageous situations to determine if the suspected blackjack player is skilled to a point where they have a long-term advantage over the casino. Having this tool available for use in surveillance provides their operators with a “black and white” printout that can be given to management detailing the suspected player’s abilities. It provides the proof that a person should be allowed to continue play, or should be banned (in New Jersey, restricted) from play. Note: These software packages can also determine whether a player is gaining an advantage from other legal and illegal techniques.

These counter catcher software systems can also be used to determine how well the less-than-knowledgeable blackjack customer plays the game. The software determines if the customer is playing basic strategy on each hand and, if not, assigns a cost to the deviation. These costs are added up, and used to determine how much the customer gives back to the house in playing errors. For example, the software analyzes the input of the customer’s play over a number of hands. These errors are added to the game’s natural house advantage based on the number of decks and rules. If the basic house advantage of the game is 0.28 percent, and the customer exhibits an average error of 0.68 percent per hand, then the true house advantage subject by this customer would be 0.96 percent, or the house expects to win 96 cents for every $100 wagered. Using this calculated “actual” H/A, marketing can insert this number into their model to determine the true cost of the discount of loss. However, there is a big problem with this house advantage number, and it’s based on the limited number of observations used during the analysis process.

Analyzing a Customer’s H/A Subject to Variance
In most situations, the blackjack customer in question is run through the software analyzer based on a minimal number of observations. This minimal input is the result of either limited play of the observed customer, and/or the person conducting the analysis time constraints to record the observations. In any case, when the observation and analysis is limited to a few hundred hand decisions on a specific player, the software gives the analyst a H/A percentage that falls within a varying range of “actual H/A” possibilities.

Included is a sampling of observations conducted on a higher limit blackjack player (Table 1). The customer in question was observed for 100 hand decisions during the analysis. Each decision that did not coincide with basic strategy, was recorded by the analyst. It’s noted that the customer made 12 decision errors during the observation. In addition to the listed errors, the analyst also input the cost for making each error. For example, the customer was noted failing to hit his 12 versus the dealer’s up card of 2. This decision error cost the player 4 percent of his wager. If he would have been wagering $100, in theory, he just gave the house an additional $4 above and beyond the games’ natural advantage of 0.54 percent (six-deck game used in this example). Note that each error receives its own cost. This includes each decision where the customer elected to insure his hand against the dealer’s ace up-card.

Once all errors and error costs are recorded, the costs are totaled to calculate a sum of all errors. This number (47.0 in Table 1) is divided by the number of total observations. The analyst wants to know what the average cost of decision errors is per hand played; in this case 0.0047 or 0.47 percent (47.0 / 100 = 0.0047 or 0.47 percent). The calculated average player error per hand is then added to the basic H/A (based on number of decks a rules), to determine the per hand “actual” theoretical win of this customer (Table 2).

A problem exists not in the calculations of the actual customer house advantage, but in the statistical range in which that “actual” H/A percentage falls. By taking the calculations further, the analyst can establish a percentage variance. By computing the “sum squared” of each error (see Table 1), the sum squared can be used to determine the variance or accuracy of a given result. Pollsters use the same formula when calculating the error in their predictions as to the popular vote for candidates. Based on the information gleaned from Table 1, the analyst was able to calculate the variance to be 0.32 percentage points either up or down from the previously calculated actual H/A percentage. Table 3 illustrates this variance. The 0.95 confidence indicates this variance occurs 95 percent of all possibilities.

Based on the 0.32 percentage point variance, the actual H/A percentage could fall between an upper or higher interval of 1.33 percent, and lower interval of 0.69 percent. These calculations indicate that the analyzed blackjack customer might be giving the casino only 0.7 percent, instead of 1.0 percent of his wager. When projecting discount of loss, the difference between 1 percent and 0.7 percent is huge, and could result in marketing offering a player incentive that returns a “negative” theoretical win to the casino.

Gaining a Better Idea of the Customer’s Actual Mathematical House Advantage
In order to gain a better actual H/A percentage, more observations are needed. Management might consider a minimum of 1,000 hand decisions before running the analysis. Depending on how well the blackjack customer’s hand decision strategy works, the variance should drop below a +/- 0.20 percentage point range. Studies have indicated that it may take as many as 7,000 hand decisions to bring the +/- range to 0.05 percentage points. Seven thousand hands? The average customer plays about 1,000 hands per trip. It may take months before marketing is armed with a reasonable estimate of how a customer plays blackjack, and in most situations, that’s too long a time period. Player incentive decisions need to be made now, not the next fiscal quarter.

There are two reasonable alternatives. First, consider that all your higher limit customers use perfect basic strategy or some estimation close to basic strategy. Unless the customer is an advantage player, the absolute minimum H/A percentage is your casino’s high limit games basic house advantage. Second, use the software analysis to establish the range from the calculated actual H/A, but instead of using 0.95 confidence level, use 0.68 (one standard deviation). Using Table 3’s information, one standard deviation is equal to ½ of two standard deviations (0.32 / 2 = 0.16), giving Table 3’s customer an actual H/A of 0.85 percent. To err on the high side of the actual H/A doesn’t hurt the discounting loss model. Erring to the low side does. By splitting the difference on the low side, marketing can work with a much more accurate H/A number.
Note: I’ve found that discount of loss programs calculated using any H/A percentage under 1 percent are “iffy” at best. Anything under 0.5 percent is usually disastrous, returning a negative result. If your casino offers a higher limit blackjack game that is subject to a basic H/A of lower than 0.5 percent, management needs to be very careful calculating promotions. You may consider limiting promotions on low H/A games. One element of play the casino can’t dictate is how well the customer plays basic strategy.

Conclusion
Following is a recap of the material covered in this article:

• Many of the higher end casinos are relying on the different counter catcher software packages to provide them with an educated estimate as to specific blackjack customer’s “actual” house advantage.
• Using the software can be faulty since most customers are observed and analyzed subject to a small sampling of hand decisions. Many times the calculated H/A is in error because the range of possible actual H/A is too great or wide. In some cases, the calculated actual H/A could be as far off as 0.30 percentage points lower than first estimated.
• This small sampling is the result of limit play by the customer and the fact that the analysis is quite time consuming. A larger sampling might not be feasible since player incentive decisions need to be immediate, not next quarter.
• Two alternatives: (1) market uses the blackjack games basic house advantage as the actual house advantage; (2) marketing calculates the customer’s actual H/A along with range and establishes the adjusted actual H/A at one standard deviation below the calculated actual H/A.
• Any loss discount that is subject to an H/A of under 1 percent is “iffy” at best. Any loss discount that is subject to an H/A of under 0.5 percent can be disastrous.