This article addresses a problem posted by Bill Zender on Richard Reid's Mathematics of Blackjack discussion board. It concerns dealer cheating by manipulating the choice of upcard and holecard. Ordinarily, the dealer always exposes deals the first (or always the second) card as the upcard. In this way, the selection is essentially random. Conventional strategy is based on the assumption that this is a random process. That is, when the upcard is a "6", the conventional assumption is that the dealer has an almost 1/13 chance of having an Ace in the hole, 4/13 of having a 10 in the hole, etc. A player utilizing this strategy will suffer an unbridgeable disadvantage. The mathematics of this are explained in the accompanying tables. There are two provided, for slightly different rules sets:
To understand the meaning of the table, let us consider an example. Suppose the dealer has a Ten ('X') and a 6. Half the time the dealer will show the 6, and the basic strategists will adopt an aggressive strategy, doubling and splitting very often and never breaking. The other time the dealer will expose the Ten, and the player will take a more conservative approach. Here the players gain considerably. They are using the "6" strategy. The difference between these is about 34% and is listed in the table as the "conditional cost."
Now suppose the dealer cheats and always selects the 10 as the upcard. This particular cheating could occur approximately 2.4 % of the time. (The dealer will get this hand twice as often; the cheating will involve altering the normal selection one-half of the time.) This is listed in the table as "cheating frequency." On these hands, players will be tricked into playing the hand incorrectly and it will cost them 33%. We multiply this conditional cost times the frequency to obtain the absolute cost of cheating on this particular play.
The table indicates which up card would be selected in order to hurt the player the most. For example, when the dealer has an 8-3, the dealer would select expose the "3", in order to bluff the players into thinking the hand was weak. However, if the ealer had an 8-4, exposing the "8" would hide the dealer's weakness. From the bottom of the chart, we see that the total possible gain is almost 5%, if the dealer cheats "optimally" on the 42% of the opportunities which exist This is the most gain that the house could realize through this chicanery; it would be attained if they the dealer could look at both cards before selecting the upcard.
Simple But Dangerous
Of course, the "optimal" cheating described above is difficult ot Implement without being obvious. However there are more subtle schemes that can attain almost the same results. One such scheme would be for the dealer to always expose a "10" if it was possible. Indeed, in the post mentioned above, Zender pointed out that such was actually implemented by one casino. They accomplished this by marking the backs of the 10s. An equally effective way of accomplishing this would be for the dealer peek at the first card, expose it if it were a "10"; if it is expose the second card.We can evaluate the overall cost of this type of cheating using the data in the table. Here, we can add up the Abs. Costs for X2,X3,X4,X5,X5,X6, and X7. However when the dealer holds X8 and X9, this cheating saves the players money, because the cheating strategy is to expose the 8 or 9. We adjust for this by subtracting the costs for this play. These calculations show that this cheating costs the player more than 3%.
It is possible, via judicious selection of the cards to mark, for the House toraise this to over 4%.
Disclaimers
First, these calculations assume that the player is using perfect basic strategy. The majority of players do not play perfectly, and this cheating may hurt them less. In an extreme case, a poor player who does nto even look at the dealer upcard will not be affected at all by this cheating. Of course, such a player is already giving the house a big advantage.
Second, I do some hesitation about using publishing this data. I realize that one application of it would be to aid cheaters in evaluating and refining their methods. Obviously, this is not my goal. Cheating casinos are already aware that this "works" and undoubtedly have various ways of measuring its effectiveness. My hope is that this data my help alert the playing public to the dangers and costs of this chicanery.
Last Revision: Tuesday, 14 April 1998