(Note: Unless you are interested in how to calculate the playing efficiency of multiparameter systems, this post should be avoided. It is written in response to a query at AP.com. Since I do not wish to post there, or be associated with that site in any way, I have posted my reply here.)
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INTRODUCTION
Calculating BC and IC is relatively straightforward. Calculating PE using Griffin's method is somewhat more involved. For single parameter systems, there is an easy shortcut - a simple formula used by Snyder. (The formula can be found in chapter 3 of Cant's _Blackjack Therapy_, accessible at BJ Review Net.) For multiparameter systems, calculating PE without the aid of a computer is time consuming, though doable. Allow maybe 2-3 hours with pen, paper and pocket calculator. The good news is that you have all the information you need within the covers of Griffin's book, though it is scattered to all parts.
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PRELIMINARIES: THE CORRELATION COEFFICIENT
To calculate BC, IC and PE, you first need to know how to calculate the various correlation coefficients for your count system. Some simple examples will illustrate the procedure. Unless otherwise specified, all worked examples will assume use of Revere's level 1 primary count (0 1 1 1 1 1 0 0 -1 -1) and, depending on the circumstances, (A) and/or (78) side counts.
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- Definition -
The correlation coefficient is calculated by taking the inner product of the count system's tags and the EORs and dividing by the product of the sum of squares of the system and the sum of squares of the EORs. That is,
CC = IP/[(SSc.SSr)^1/2]
where IP is the inner product of the count system's tags and the EORs, SSc is the sum of squares of the count system's tags and SSr is the sum of squares of the effects of removal.
-----
- Single-Parameter Balanced Count Systems -
Let's calculate the betting correlation of Revere's level 1 system, R1, in Griffin's benchmark game (described p.11). We have:
R1: (0 1 1 1 1 1 0 0 -1 -1)
EORs: (-.61 .38 .44 .55 .69 .46 .28 .00 -.18 -.51)
Here IP = 0(-.61) + 1(.38) + 1(.44) + 1(.55) + 1(.69) + 1(.46) + 0(.28) + 0(.00) + -1(-.18) + 4(-1)(-.51) = 4.74
SSc = 0^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 0^2 + 0^2 + (-1)^2 + 4(-1)^2 = 10
SSr = -.61^2 + .38^2 + .44^2 + .55^2 + .69^2 + .46^2 + .28^2 + .00^2 + -.18^2 + 4(-.51^2) = 2.84
So BC = 4.74/[(10 x 2.84)^1/2] = .8894
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- Single-Parameter Unbalanced Count Systems -
If the count system is unbalanced, it is first necessary to convert it to its balanced equivalent before calculating the correlation coefficient. This is done by adding -u/52 to each tag, where u is the system's 'unbalance'. For wagering decisions you might use an ace side count together with Revere's level 1 count:
R1 - (A) = (-1 1 1 1 1 1 0 0 -1 -1)
which is unbalanced. The unbalance, u, is -4 (since if you count through a deck using IRC = 0, your running count will end up at -4). Thus to get the balanced equivalent, call it x, add -u/52 = 4/52 = 1/13 = .077 to each tag:
x = (-.923 1.077 1.077 1.077 1.077 1.077 .077 .077 -.923 -.923)
Notice x is balanced (all the tags add up to zero). The sum of squares of x's tags is 10.923.
Here IP = (-.923)[-.61 + -.18 + 4(-.51)] + (1.077)(.38 + .44 + .55 + .69 + .46) + (.077)(.28 + .00) = 5.35
So BC = 5.35/[(10.923 x 2.84)^1/2] = .9601
To take another example, for insurance you might use:
R1 + (A) + (78) = (1 1 1 1 1 1 1 1 -1 -1)
The unbalance is now 12, so we need to add -12/52 = -3/13 = -.231 to each tag to get the balanced equivalent, y:
y = (.769 .769 .769 .769 .769 .769 .769 .769 -1.231 -1.231)
The sum of squares of y's tags is 12.308. The EORs for insurance are given by Griffin on p.71:
EORs: (1.81 1.81 1.81 1.81 1.81 1.81 1.81 1.81 1.81 -4.07)
The sum of squares of the insurance EORs is 95.7. The inner product of y's tags and the insurance EORs is 28.95. So the insurance correlation is:
IC = 28.95/[(95.7 x 12.308)^1/2] = .8435
The correlation coefficient can also be calculated for the play of any strategy decision, using the EORs in Griffin's tables on pp.74-85. For example, for 14 v T you might use:
R1 + 2(78) = (0 1 1 1 1 1 2 2 -1 -1)
By converting this to its balanced equivalent, the correlation coefficient for 14 v T can be calculated in the usual way. As will soon be explained, the correlation coefficients, calculated separately for all strategy decisions, form the basis of the calculation of PE.
For more information on the conversion of unbalanced counts to their balanced equivalents do a search on Harris' unbalanced true-count theorem.
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- Multiparameter Systems Using Traditional Adjustments -
In the traditional method outlined by Griffin, the player adjusts the primary count according to the richness/poorness of the side-counted denomination/s. If you plan to use this method, you need to convert your multiparameter system into a single-parameter 'effects of removal' count. Griffin explains how to do this on pp.244-5. He also provides a simple-to-follow numerical example, so there is no need to repeat the explanation here. Once you have converted your multiparameter system appropriately, you simply take the correlation coefficient between the normal EORs and the 'tags' of the 'effects of removal' count.
-----
- Summary of Preliminaries -
The discussion so far has explained how to calculate correlation coefficients. This is all that is needed to calculate a system's BC and IC. However, for PE, what has been explained so far is merely a prerequisite. Once you have correlation coefficients for every hand - actually, by convention, the 71 decisions 10-16 v 2-A plus insurance (see Griffin, p.45) - you use these to determine PE.
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CALCULATING PLAYING EFFICIENCY
Armed with correlation coefficients for every strategy decision, you are now ready to calculate PE. The method is explained, with examples, by Griffin in chapter 6, pp.88-90. To carry out the calculations, you will need to make constant reference to:
1) The UNLLI table (p.87).
2) Approximations of the probabilities of occurrence for each hand (endnote F, p.39).
-----
- The Basic Approach -
The method requires two sets of calculations to be made. Firstly, you need to calculate the gains (over basic strategy) that are attainable from 'perfect play' (such as could be carried out by a computer that kept track of all denominations separately and made optimal decisions based on the precise composition of the cards remaining). Secondly, you must calculate the gains from 'actual play' - ie, those attainable through 'best use' of your particular count system ('best use' in the sense that all indices and multiparameter adjustments, if applicable, are correctly determined and accurately employed). Because count systems devised for human use are necessarily approximate in nature, the gains from actual play will obviously be less than the gains from perfect play. PE is calculated simply as (Gains from Actual Play)/(Gains from Perfect Play); ie, it is an estimate of how much of the potential theoretical gain your count system can capture if employed flawlessly.
- The Specifics -
To calculate gains from perfect play, the steps are as follows:
1) Calculate b by taking the square root of [ss.(N-n)/13.(N-1).n)] and multiplying this amount by 51. Here ss is the sum of squares of the EORs for the particular hand under consideration (provided in column 12 of the EOR charts), N is the number of cards initially in the pack, and n is the number of cards remaining to be dealt (assumed to be 20).
2) Calculate z = m/b, where m is the full deck favourability of hitting (12-16) or doubling (10-11) and is provided in column 11 of the EOR charts.
3) Look up the UNLLI chart to find the corresponding number for z.
4) Multiply the number found in the UNLLI chart by b.
Steps 1 to 4 give you the conditional gain from perfect play.
5) Weight the conditional gain by the probability of the relevant hand occurring. For example, Griffin estimates the probability of being dealt 15 v T as (165/1326)(188/663), so you would multiply the conditional gain for 15 v T by this probability of occurrence. This will give you the weighted gain for this particular hand.
Repeat these 5 steps for all 71 decisions to get all the weighted gains. You then simply add them all together to estimate the gain from perfect play of these 71 decisions.
To calculate the gains from actual play, do exactly the same 5 steps as above, except that instead of using b in the calculations, you use b', where b' is simply b multiplied by the CC of your count system for the hand in question.
Once you have the gains from actual and perfect play, take their quotient to obtain your estimate of your count system's PE.
----- ----- -----
INTRODUCTION
Calculating BC and IC is relatively straightforward. Calculating PE using Griffin's method is somewhat more involved. For single parameter systems, there is an easy shortcut - a simple formula used by Snyder. (The formula can be found in chapter 3 of Cant's _Blackjack Therapy_, accessible at BJ Review Net.) For multiparameter systems, calculating PE without the aid of a computer is time consuming, though doable. Allow maybe 2-3 hours with pen, paper and pocket calculator. The good news is that you have all the information you need within the covers of Griffin's book, though it is scattered to all parts.
----- ----- -----
PRELIMINARIES: THE CORRELATION COEFFICIENT
To calculate BC, IC and PE, you first need to know how to calculate the various correlation coefficients for your count system. Some simple examples will illustrate the procedure. Unless otherwise specified, all worked examples will assume use of Revere's level 1 primary count (0 1 1 1 1 1 0 0 -1 -1) and, depending on the circumstances, (A) and/or (78) side counts.
-----
- Definition -
The correlation coefficient is calculated by taking the inner product of the count system's tags and the EORs and dividing by the product of the sum of squares of the system and the sum of squares of the EORs. That is,
CC = IP/[(SSc.SSr)^1/2]
where IP is the inner product of the count system's tags and the EORs, SSc is the sum of squares of the count system's tags and SSr is the sum of squares of the effects of removal.
-----
- Single-Parameter Balanced Count Systems -
Let's calculate the betting correlation of Revere's level 1 system, R1, in Griffin's benchmark game (described p.11). We have:
R1: (0 1 1 1 1 1 0 0 -1 -1)
EORs: (-.61 .38 .44 .55 .69 .46 .28 .00 -.18 -.51)
Here IP = 0(-.61) + 1(.38) + 1(.44) + 1(.55) + 1(.69) + 1(.46) + 0(.28) + 0(.00) + -1(-.18) + 4(-1)(-.51) = 4.74
SSc = 0^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 0^2 + 0^2 + (-1)^2 + 4(-1)^2 = 10
SSr = -.61^2 + .38^2 + .44^2 + .55^2 + .69^2 + .46^2 + .28^2 + .00^2 + -.18^2 + 4(-.51^2) = 2.84
So BC = 4.74/[(10 x 2.84)^1/2] = .8894
-----
- Single-Parameter Unbalanced Count Systems -
If the count system is unbalanced, it is first necessary to convert it to its balanced equivalent before calculating the correlation coefficient. This is done by adding -u/52 to each tag, where u is the system's 'unbalance'. For wagering decisions you might use an ace side count together with Revere's level 1 count:
R1 - (A) = (-1 1 1 1 1 1 0 0 -1 -1)
which is unbalanced. The unbalance, u, is -4 (since if you count through a deck using IRC = 0, your running count will end up at -4). Thus to get the balanced equivalent, call it x, add -u/52 = 4/52 = 1/13 = .077 to each tag:
x = (-.923 1.077 1.077 1.077 1.077 1.077 .077 .077 -.923 -.923)
Notice x is balanced (all the tags add up to zero). The sum of squares of x's tags is 10.923.
Here IP = (-.923)[-.61 + -.18 + 4(-.51)] + (1.077)(.38 + .44 + .55 + .69 + .46) + (.077)(.28 + .00) = 5.35
So BC = 5.35/[(10.923 x 2.84)^1/2] = .9601
To take another example, for insurance you might use:
R1 + (A) + (78) = (1 1 1 1 1 1 1 1 -1 -1)
The unbalance is now 12, so we need to add -12/52 = -3/13 = -.231 to each tag to get the balanced equivalent, y:
y = (.769 .769 .769 .769 .769 .769 .769 .769 -1.231 -1.231)
The sum of squares of y's tags is 12.308. The EORs for insurance are given by Griffin on p.71:
EORs: (1.81 1.81 1.81 1.81 1.81 1.81 1.81 1.81 1.81 -4.07)
The sum of squares of the insurance EORs is 95.7. The inner product of y's tags and the insurance EORs is 28.95. So the insurance correlation is:
IC = 28.95/[(95.7 x 12.308)^1/2] = .8435
The correlation coefficient can also be calculated for the play of any strategy decision, using the EORs in Griffin's tables on pp.74-85. For example, for 14 v T you might use:
R1 + 2(78) = (0 1 1 1 1 1 2 2 -1 -1)
By converting this to its balanced equivalent, the correlation coefficient for 14 v T can be calculated in the usual way. As will soon be explained, the correlation coefficients, calculated separately for all strategy decisions, form the basis of the calculation of PE.
For more information on the conversion of unbalanced counts to their balanced equivalents do a search on Harris' unbalanced true-count theorem.
-----
- Multiparameter Systems Using Traditional Adjustments -
In the traditional method outlined by Griffin, the player adjusts the primary count according to the richness/poorness of the side-counted denomination/s. If you plan to use this method, you need to convert your multiparameter system into a single-parameter 'effects of removal' count. Griffin explains how to do this on pp.244-5. He also provides a simple-to-follow numerical example, so there is no need to repeat the explanation here. Once you have converted your multiparameter system appropriately, you simply take the correlation coefficient between the normal EORs and the 'tags' of the 'effects of removal' count.
-----
- Summary of Preliminaries -
The discussion so far has explained how to calculate correlation coefficients. This is all that is needed to calculate a system's BC and IC. However, for PE, what has been explained so far is merely a prerequisite. Once you have correlation coefficients for every hand - actually, by convention, the 71 decisions 10-16 v 2-A plus insurance (see Griffin, p.45) - you use these to determine PE.
----- ----- -----
CALCULATING PLAYING EFFICIENCY
Armed with correlation coefficients for every strategy decision, you are now ready to calculate PE. The method is explained, with examples, by Griffin in chapter 6, pp.88-90. To carry out the calculations, you will need to make constant reference to:
1) The UNLLI table (p.87).
2) Approximations of the probabilities of occurrence for each hand (endnote F, p.39).
-----
- The Basic Approach -
The method requires two sets of calculations to be made. Firstly, you need to calculate the gains (over basic strategy) that are attainable from 'perfect play' (such as could be carried out by a computer that kept track of all denominations separately and made optimal decisions based on the precise composition of the cards remaining). Secondly, you must calculate the gains from 'actual play' - ie, those attainable through 'best use' of your particular count system ('best use' in the sense that all indices and multiparameter adjustments, if applicable, are correctly determined and accurately employed). Because count systems devised for human use are necessarily approximate in nature, the gains from actual play will obviously be less than the gains from perfect play. PE is calculated simply as (Gains from Actual Play)/(Gains from Perfect Play); ie, it is an estimate of how much of the potential theoretical gain your count system can capture if employed flawlessly.
- The Specifics -
To calculate gains from perfect play, the steps are as follows:
1) Calculate b by taking the square root of [ss.(N-n)/13.(N-1).n)] and multiplying this amount by 51. Here ss is the sum of squares of the EORs for the particular hand under consideration (provided in column 12 of the EOR charts), N is the number of cards initially in the pack, and n is the number of cards remaining to be dealt (assumed to be 20).
2) Calculate z = m/b, where m is the full deck favourability of hitting (12-16) or doubling (10-11) and is provided in column 11 of the EOR charts.
3) Look up the UNLLI chart to find the corresponding number for z.
4) Multiply the number found in the UNLLI chart by b.
Steps 1 to 4 give you the conditional gain from perfect play.
5) Weight the conditional gain by the probability of the relevant hand occurring. For example, Griffin estimates the probability of being dealt 15 v T as (165/1326)(188/663), so you would multiply the conditional gain for 15 v T by this probability of occurrence. This will give you the weighted gain for this particular hand.
Repeat these 5 steps for all 71 decisions to get all the weighted gains. You then simply add them all together to estimate the gain from perfect play of these 71 decisions.
To calculate the gains from actual play, do exactly the same 5 steps as above, except that instead of using b in the calculations, you use b', where b' is simply b multiplied by the CC of your count system for the hand in question.
Once you have the gains from actual and perfect play, take their quotient to obtain your estimate of your count system's PE.
