## Optimum card counting

If you have done the demonstration in the Blackjack General article of this website you know that all shoe compositions are not equal relative to what a blackjack player can expect. The perfect way to count cards would be to remember each and every card as it is played and use that information to size bets and beyond that to arrive at the best playing strategy for the current shoe composition. It's possible to do this using a computer. In fact that's what the cdca program in the 'Programs' section of this website does. However an ordinary human wouldn't have this capability so the question is, "How can we go about evaluating changing shoe compositions in a blackjack game in a way that is practical to an ordinary human?" We'll try to answer that question in a way that makes understandable what a counting system is and is not before even considering any specific counting system.

## The Nuts and Bolts of a Counting System

Basic concept - If one of each rank (2,3,4,5,6,7,8,9,T,A) is systematically removed from a full shoe and the resultant effect on player's expectation recorded and compared with the expectation of a full shoe, then we come up with a set of data known as effects of removal or eor for short. The exact value of eor of a given rank is dependent upon the rules of the game and a player's playing strategy. Also the value of an eor is dynamic, meaning that if the eor is refigured using less than a full shoe as a starting point then the eor could compute to a different value.

Stating point - The most logical condition to refernce is a full shoe since player is confronted with that shoe composition 100% of the time following a new shuffle. As cards are played from the starting point we would like to know when the cards played cause an increased player advantage or an increased dealer advantage and by how much. The best player strategy to reference is basic strategy. At the start of a full shoe basic strategy is very close to optimal strategy. As more cards are dealt we may run into the problems listed in the Optimum card counting section, above, if we are tempted to use optimal strategy so it's best to be conservative and stick with basic strategy as a reference. Below are basic strategy eors for single deck. Rules are dealer stands on soft 17, double on any 2 cards, 1 split allowed on any pair, no doubling afer splitting allowed, one card to split aces, no surrender, blackjack pays 3 to 2. These are all calculated values.

```				     Single Deck EORs (in %)
2	  3	  4	  5	  6	  7	  8	  9	  T	  A
CD Basic 	.38241	.43594	.55079	.69885	.41436	.28541	.00822	-.16868	-.50291	-.59566
TD Basic	.3868	.4383	.5367	.6764	.4554	.2885	.0229	-.1613	-.5124	-.5945
Generic Basic	.3706	.4116	.5132	.6455	.4327	.2745	.0195	-.1556	-.4785	-.5975
```

CD Basic is composition dependent strategy as calculated by the cdca program in the 'Programs' section of this website. TD Basic is total dependent basic strategy as calculated by another program written by the author of this website. Generic Basic is a strategy that is a good approximation of TD Basic but applies to a broader range of conditions so it simplifies learning a basic strategy. Generic Basic is computed by the cdca program by inputting a large number of decks, removing cards such that the full shoe composition of the desired number of decks remains, and then computing using basic strategy for the large number of decks applied to the desired number of decks. When a large enough number of decks is input and resultant CD basic strategy used then the CD basic strategy is identical to TD Basic for that number of decks. When CD Basic is applied to a small number of decks created by removing the appropriate cards from a very large number of decks then TD Basic is replicated for the most part but absent are some of the closer EV plays present when TD Basic is actually computed for the same small number of decks. The greatest difference between Generic Basic and TD Basic is for single deck and it is not a large difference. As more decks are added then Generic Basic quickly approaches TD Basic and becomes a distinction without much difference. As even more decks are added Generic Basic and TD Basic are at some point identical.

## Considering multiple decks

In developing a counting system it should be applicable to any number of decks so let's next consider 6 decks as an example (using the same rules as above.) When one of any rank is removed from 6 full decks the eor is numerically much less than the eor from a full single deck. This is because removing one card has less effect on a shoe's composition the more decks there are. First we are going to list the actual eors for 6 decks. Following that we are going to normalize them so we can compare single deck eors to 6 deck eors. Normalization is a statistical method for doing comparisons. The formuls used is -

Normalized_EOR = EOR * (52 * (deck_size) - 1) / 52,
where deck_size is number of decks and EOR is eor of 1 card of a given rank

```				     Six Deck EORs (in %)
2	  3	  4	  5	  6	  7	  8	  9	  T	  A
CD Basic 	.06314	.07210	.09384	.11816	.06945	.04479	-.00235	-.03282	-.08223	-.09739
TD Basic	.0627	.0713	.0911	.1139	.0740	.0445	-.0016	-.0308	-.0816	-.0981
Generic Basic	.0623	.0707	.0906	.1135	.0737	.0444	-.0012	-.0306	-.0814	-.0986

Normalized Single Deck EORs (in %)
2	  3	  4	  5	  6	  7	  8	  9	  T	  A
CD Basic 	.3751	.4276	.5402	.6854	.4064	.2799	.0081	-.1654	-.4932	-.5842
TD Basic	.3794	.4299	.5264	.6634	.4466	.2830	.0224	-.1582	-.5025	-.5831
Generic Basic	.3635	.4037	.5033	.6331	.4243	.2692	.0191	-.1526	-.4693	-.5860

Normalized Six Deck EORs (in %)
2	  3	  4	  5	  6	  7	  8	  9	  T	  A
CD Basic 	.3776	.4312	.5612	.7067	.4154	.2679  -.0141	-.1962	-.4918	-.5825
TD Basic	.3750	.4264	.5264	.6812	.4426	.2661  -.0096	-.1842	-.4880	-.5867
Generic Basic	.3726	.4228	.5419	.6788	.4408	.2655  -.0072	-.1830	-.4868	-.5897
```

The normalized eors for 1 and 6 decks are reasonably similar. Since we have discarded the optimum count approach as being impractical it does not seem unreasonable to apply whatever the end method turns out to be to any number of decks.

## Putting a Counting System Together

Review -

1. At the start of this article it was stated that the only perfect way to count cards in blackjack would be to keep track of how many of each and every rank remains to be dealt and to use that information to bet and play but that this is an unrealistic way to go about counting cards in blackjack.
2. Following that it was shown that the effect of removing one of each rank on player's advantage/disadvantage using a given strategy, in this case basic strategy, can be exactly calculated.
3. Finally it was shown how go about comparing the effect of removing one of a rank from a differing number of decks.

Using above information - Up until this point we have managed to obtain unequivocal computed data. However from here on we can't rely on exact calculations since player cannot be realistucally expected to know every card played and on top of that to be able to use this information. Instead we note the following relative to using basic strategy:

• Removal of a single 2, 3, 4, 5, 6, or 7 increases player's advantage
• Removal of a single 8 results in close to no change in player advantage/disadvantage
• Removal of a single 9 results in a moderate decrease in player advantage
• Removal of a single T results in a definite decrease in player advantage
• Removal of a single A results in a definite decrease in player advantage

The final step in putting together a counting system is to look at the eor data and group ranks with similar attributes. There are numerous counting systems. Each groups cards differently according to whatever theory is deemed appropriate relative to the eors. However there is no perfect system. Employing one results in an approximation of what the actual advantage/disadvantage may be. Also grouping ranks in order to improve a system's ability to better approximate correct playing strategy is something else that might be considered.

## Hi-Lo Counting System as Example

The purpose of this article was to describe what considerations there are in arriving at a usable blackjack counting system. It would be remiss to not include an example so we'll end by listing how the classic HiLo system groups cards.

• Group one consists of (2, 3, 4, 5, 6) - Removal of these cards tends to increase EV
• Group two consists of (7, 8, 9) - Removal of these cards tends to keep EV relatively unchanged
• Group three consists of (T, A) - Removal of these cards tends to decrease EV

The actual process of evaluating and using a counting system will be left as a separate subject.

## Conclusion

Card counting is an imprecise method of determining betting and or playing strategies in the game of blackjack. It is based upon known expectations relative to a full shoe. As cards are dealt there will possibly be shoe compositions where it is relatively accurate as well some where it is not so accurate. However, it is the only practical way to evaluate expectation on a card by card basis as is done with unequivocal accuracy using the optimum method.