Counting System Index Generation Service
Basic Strategy is the best way to play blackjack hands dealt from a full shoe. As cards are removed from a full shoe it's possible that the optimal playing strategy for a given player hand may vary from basic strategy. Maybe a hand whose basic strategy is hit may be more optimally played by standing or a pair whose basic strategy is to split is more optimally played by not splitting. These are just possible arbitrary examples. All of theses strategy variations are easily determined using a combinatorial analysis program such as cdca by simply removing dealt cards from a full shoe before computing to determine strategy. The problem is that it's not possible to organize this arbitrary input of removed cards to be usable without the use of a computer program. In order to somewhat deal with these removals a blackjack player can resort to a card counting system, which organizes the blackjack shoe ranks (A,2,3,4,5,6,7,8,9,T) into 2 or more groups of cards. The effect of cards removed from these groups can then be more simply determined.
Using Combinatorial Analysis to Compute Standard True Count Indexes
At the beginning of a blackjack shoe there is just one deck composition to consider (full shoe.) As cards are dealt the number of compositions to consider increases. Relative to a counting system for each number of cards removed from a full shoe there are a number of possible running counts. Each of these running counts has a probability of occurrence. True count is defined as running count per deck. True count can always be computed from running count as long as pen (measured by number of remaining cards or less accurately remaining decks) is known. The statistically most efficient true counts to reference for blackjack play to apply to all number of cards remaining are the ones computed from running counts at mid-shoe. When true count indexes are computed by combinatorial analysis at mid-shoe I have found that they are in agreement with those published by reputable sources that were obtained by simulation.
I have modified my combinatorial analyzer to remove cards as measured by most any counting system and compute according to that input. This serves the purpose of obtaining a reasonable estimate of the actual values as measured by the counting system being used. I can then determine the best way to play a blackjack hand according to a counting system's running count for any possible number of cards remaining to be dealt from this data. This is the best way to play according to a given counting system, but simplifying to true count as above is much more usable.
I can generate true count indexes for any counting system, known or unknown. I can also generate running count indexes for any viable penetration (number of cards remaining to be dealt.) True count is computed from running count at a given penetration. All I need are the count's tags.
I will need the following rules information:
1. number of decks
2. dealer stands or hits soft 17 (affects only up cards A-6, A and 6 more so)
3. odds paid to player blackjack (usually 1.5 to 1)
4. does player lose only original bet to dealer bj (full peek) or ENHC (no peek)
5. can player double on any number of cards (DAN or DO2)
6. can player double after splitting versus up cards 2-T (DAS or NDAS)
7. can player hit split aces (instead of 1 card) (hsa)
8. if hsa, can player double after resplitting aces
9. does player automatically win with 5 cards or 6 cards (5-card charlie, 6-card charlie)
I can generate either total dependent indexes, which only compensate for dealer up card or composition dependent indexes, which compensate for both player hand and dealer up card. If composition dependent I will need the exact player hand for which to generate indexes. If total dependent I will need the hand total and if hand total is less than 12 whether it is hard, soft, or a pair. (Normally a pair or a 2-card soft hand would be processed as composition dependent.)
Insurance indexes can be generated with or without regard for player hand composition and are listed by running count along with the true count conversion of rc to tc for all deck penetrations. Only the penetrations for which there is a change in running count are listed. The statistically best index will be at or near mid-shoe. This is an example of how true count is not constant relative to penetration.
I am looking for some compensation for the time I have spent analyzing blackjack over the years. The price for a generation is $5.00 and includes indexes for all up cards and all strategies (ES, LS, split if applicable, Double, Stand) for a given player hand input, listed in the order of precedence.
To order email requests to kc@bjstrat.net. Payment is made via PayPal. Once received I will email the generated indexes in a text file attachment.
Sample indexes: 2 decks, s17, bj pays 1.5:1, full peek, DO2, NDAS, one card to split aces (not hsa,) no auto winner
Player hand composition: 8-8
Output of true count indexes
Count tags {1,-1,-1,-1,-1,-1,0,0,0,1} S17, D2, OBO, noHSA, NDAS
Composition dependent indices for hand, rules, number of decks, and pen
Player hand composition: 0, 0, 0, 0, 0, 0, 0, 2, 0, 0: Hard 16, 2 cards
Decks: 2 (possible input for cards remaining: 1 to 104)
Cards remaining before up card = 53
No subgroups are defined
2 3 4 5 6 7 8 9 T A
ES - - - - - - - >=8 >=-1 >=-15
LS - - - - - - - >=8 >=2 >=36
Pair p p p p p p p p<17 p<8 p>=-13
h>=38 s>=37
Double - - - - - - - - - -
Stand >=-8 >=-10 >=-11 >=-13 >=-11 >=15 >=13 >=5 >=0 >=8
<38
Validity of Indexes
True count indexes generated at mid-shoe (mostly for HiLo) are very much in compliance with those I have seen published by other reliable sources. Index generation for any set of count tags uses this same methodology.
Side Counts
I can also generate indices to compensate for any number of side counts relative to the main count system.
Suppose our main count is HiLo whose tags for its 3 groups are: {2,3,4,5,6} -1; {7,8,9} 0; {T,A} +1. (I define the sign of the tags to be relative to what remains in the shoe as opposed to what is removed but however the sign of the tags is defined results in the same indices.)
I allow for each of the 3 main groups to contain subgroups to side count which are comprised of any combination of cards within that group. Therefore the maximum number of subgroups is 10 (1 for each rank,) which would bring us back to the original problem of determining a strategy without having to consult a combinatorial analysis computer program. I believe it's best to limit subgroups to no more than 1 to be as simple as possible, although I am able to generate data for multiple subgroups. Any cards from a defined subgroup are directly removed so an index can be computed for each possible input of subgroup cards. The problem of inputting this way is that a simple usable correlation between the variation of the subgroup(s) and the main count should be determined. Hopefully this correlation always turns out to be linear or close to linear. However, it seems this isn't necessarily the case. For this reason I am not a fan of this way of side counting, but I can generate data without worrying about correlation. It would then be up to you to determine a correlating use of the side count(s).
Sample indexes: 2 decks, s17, bj pays 1.5:1, full peek, DO2, NDAS, one card to split aces (not hsa,) no auto winner
Side count 7s: 1 removed, 7 remain
Player hand composition: 8-8
Output of true count indexes
Count tags {1,-1,-1,-1,-1,-1,0,0,0,1} S17, D2, OBO, noHSA, NDAS
Composition dependent indices for hand, rules, number of decks, and pen
Player hand composition: 0, 0, 0, 0, 0, 0, 0, 2, 0, 0: Hard 16, 2 cards
Decks: 2 (possible input for cards remaining: 1 to 104)
Cards remaining before up card = 53
Subgroup removals: {7}1
2 3 4 5 6 7 8 9 T A
ES - - - - - - - >=12 >=2 >=-16
LS - - - - - - - >=12 >=5 >=35
Pair p p p p p p p p<17 p<17 p<-31
h>=36 p>=-18
Double - - - - - - - - - -
Stand >=-9 >=-11 >=-12 >=-14 >=-13 >=6 >=5 >=-3 >=3 >=9
<36
There is another method of side counting which I have recently been more enabled to analyze that I will try to explain by example.
I can tag all 10 blackjack ranks {A,2,3,4,5,6,7,8,9,T} with the same tag. When I do this and generate indexes either total or composition dependent basic strategy is output because a counting system requires 2 or more tag groups in order to measure variation. However, I am able to side count any combination of cards within this 1 main group as a subgroup. So I am going to define subgroup {2,3,4,5,6} and display results for generating indexes for a player hand of hard 16 for 209 cards remaining, dealt from 8 decks before up card is dealt. In all cases definite playing strategies are output just like with basic strategy. The strategies may (or may not) vary from basic strategy depending upon input.
Count tags {0,0,0,0,0,0,0,0,0,0} S17, D2, OBO, noHSA, NDAS
Total dependent indices for hand total, rules, number of decks, and pen
Hand total: 16 Hand Type: h
Decks: 8 (possible input for cards remaining: 1 to 416)
Cards remaining before up card = 209 (Note: 4 decks + 1 card)
Subgroup removals: {2,3,4,5,6}80 (Note: 80 remain = 20 per deck)
2 3 4 5 6 7 8 9 T A
ES - - - - - - - r r r
LS - - - - - - - r r r
Pair - - - - - - - - - -
Double - - - - - - - - - -
Stand s s s s s h h h h h
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Count tags {0,0,0,0,0,0,0,0,0,0} S17, D2, OBO, noHSA, NDAS
Total dependent indices for hand total, rules, number of decks, and pen
Hand total: 16 Hand Type: h
Decks: 8 (possible input for cards remaining: 1 to 416)
Cards remaining before up card = 209 (Note: 4 decks + 1 card)
Subgroup removals: {2,3,4,5,6}92 (Note: 68 remain = 17 per deck)
2 3 4 5 6 7 8 9 T A
ES - - - - - - r r r r
LS - - - - - - r r r r
Pair - - - - - - - - - -
Double - - - - - - - - - -
Stand s s s s s s s s s h
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Present Restrictions
Not all counting system indexes can be computed with the same degree of efficiency. I can compute less complex systems with enough efficiency to get indexes in a reasonable amount of time, even on my present computer which is not so efficent by today's standards. I use the counting system known as Wong Halves as an example of the most complex system that will be encountered. Presently computing more than 4 decks for Wong Halves is too slow. However, I have successfully created a prototype which can record all of the necessary data needed to compute all running counts for mid-shoe penetration for 8 decks in a little bit over a minute. Mid-shoe computation is the most time consuming. There are over a half million Wong Halves subsets for many of the running counts at mid-shoe. Presently I have to individually compute all of the probabilities associated with each possible running count which is way too time consuming. The protoype allows for computation of all possible running counts at once which eliminates many duplicate calculations and should be much more efficient. However, I have not yet implemented it.