I have copied below a post I have made to my blog addressing generalized effects of removal methodology. One implication it has is that single-card EOR tags are provided with mathematical context that should influence how robust and how precise they may be considered to be. There are other implications as well.
O.M.
----------------
I have been considering for several weeks now the caveats underlying the computation and use of effects of removal (EOR). Implications of their being something other than uniquely determined and razor-sharp in their precision would include but not be limited to a serious devaluation of the potency of “betting correlation,” which would duplicate and perhaps worsen the false precision reflected in overconfident EOR figures (although assessment of popular, and unpopular, counts obviously could still be done with reference to sets of linearized card tags at various levels within the general model, just not with the accuracy sought by such correlative measures). Linearization theory does provide justification for the approximation constituted by determining and using single-card tags (meaning those computed by removing a single card, i.e. and not tags applying to one card or rank only), noting the gain achieved versus not recalling or making use of the recall of previously-viewed cards at all. However, the questions raised by considering the general context of linearization methodology do not go away if waved off, no more than do other issues regarding the mathematical assessments of various blackjack systems. Players who make use of figures derived from piecemeal methodology may find themselves led to misleading or even erroneous conclusions within a finite amount of time. It is even the case that certain posts on the Internet profess a lack of concern for “theory” while simultaneously arguing points that would have to be grounded in, or at least informed by, “theory” to be salient.
One issue where “ideal” tags are concerned is how to determine the values to use for the “ideal” tag when different levels of interaction produce different linearized values. To cling to a set of “ideal” tags each of which is specified to four decimal places is ascribing too much accuracy to the extrapolation of the tag values to all situations. Correlative measures relying on such overspecified tag values are an implicit attempt to reconcile simplified tag values, such as those used in popular blackjack systems, to an “ideal” that is not meaningful, one that is not “there” at all. To illustrate this point, I took an interest in how much effect considering the removal of two cards of the same rank would have in conjunction with considering the removal of only one. The desire for tags that were “accurate” to two decimal places might still be realized in practice. If it were not, the knowledge produced would at least give an idea of how much and in what way the single-card “ideal” tags needed to be modified and in what context “accuracy” could still be discussed: it seemed inconsistent to attribute overwhelming power and scope of applicability to EOR and related methodological concerns if the robustness and indeed the actual values of the tags to be used were not a concern.
I accessed the web page at http://www.bewersdorff-online.de/black-jack/ for the following calculations. Jorg Bewersdorff is a Ph.D. with a couple of books out (see his web site for details). He has taken the time to put together a page that produces a blackjack playing strategy for any remainder composition. The page supports any number of decks up to 24 and possibly more than that (24 is the figure I got from clicking the relevant button until getting bored). The page supports S17 only and does not allow hard doubles for player totals below nine nor soft doubles except on A9 and A8. Nonetheless, this source is sufficient to illustrate the mathematical concepts related to linearization methodology that I wish to address, as it also provides the player edge for the deck composition in question, if only for a set of rules that is not common (except perhaps in Europe).
I noted the neutral deck edge for the game when resplits were allowed but not doubling after splits. (It was -0.664.) I then computed the player edges after removal of one card of each rank in turn. Then I computed player edges when two cards of each rank were removed in turn. Finally, I compared the figures to determine the one-card EOR, the two-card cumulative EOR, and the two-card marginal EOR.
Here is the data for the ranks in order from Two to Ace:
1-card 2-card cumulative 2-card marginal
0.383 0.770 0.387
0.462 0.939 0.477
0.597 1.200 0.603
0.768 1.576 0.808
0.451 0.955 0.504
0.289 0.661 0.372
-0.018 -0.003 0.015
-0.187 -0.346 -0.159
-0.467 -0.919 -0.452
-0.631 -1.278 -0.647
Right away, it is apparent that card tags that are “accurate” to the second decimal place are misleading. Only the Four holds its “ideal” tag to two decimal places when the one-card and the two-card marginal values are compared. Ignoring the cumulative effect of two cards of the same rank is already a mistake unless you want your “ideal” tags to be completely arbitrary. And we haven’t even investigated other second-order interaction effects or the effects of more than two cards of the same rank being removed. This does not argue that linearization is hopeless (even single-tag linearization is effective in sims), only that confining “ideal” tag computations to single-tag effects constitutes false precision and that “betting correlation” is compromised by this realization. In particular, multilevel counts designed specifically to take advantage of single-tag peculiarities take another blow to their credibility in proportion to the number of levels they implement in their naïve pursuit of unity with the single-card tags (I have listed other such credibility issues elsewhere).
The Seven is the rank of most interest, if only because the absolute difference of its tags is the largest of any rank. This should be unsurprising news to single-deck specialists, who have been taught since birth to stand on (7,7) vs. dealer T even if they observe no other compositional modifications to total-dependent basic strategy. The Eight tag value changes signs, it should be noted, and other tags are volatile enough that their effect even on the “betting correlation” of low-level counts would be of interest.
Opponents may argue that a man with one watch always knows what time it is while a man with two watches is never sure. Regardless, the “confusion” that results from considering all of this data (and perhaps additional data in the form of effects of removal of more than two cards of the same rank and interaction effects among the ranks) instead of the single-card tags alone reflects the ambiguity and variance inherent in the subject, and turning a blind eye to it isn’t a solution as much as it is a refusal to consider the full scope of what linearization methodology would tell us at each level. To consider single-card tags only, while incrementally helpful, is not “linearization” but a subset of “linearization.” To treat the word as if it had an a priori definition is misinformed enough, but assigning it an a priori definition that disavows the possibility of the generalized approach informing the specific one that is retained is openly inconsistent.
Parallels in various quantitative fields exist and may be helpful in illustrating the distinction, but it is perhaps sufficient to realize that in mathematics, there are no “magic formulas,” ones that just work for the user without any understanding of the context in which they are intended to be applied. The semiprofessional blackjack player, while perhaps justified in his unwillingness to abandon his single-tag approach, might consider developing the background to aid in critical thinking regarding the subject. Numerous sources in the blackjack world have cautioned against becoming overly excited about multilevel counts, but uncritically accepting such dictates without an understanding of the context in which they were made is as dangerous as uncritically accepting the systems they seek to debunk. The next time, for example, a system seller offers a Level Four count with an astronomical “betting correlation,” perhaps the semipro will understand enough about the mathematical context of such a claim that he will be able to put it in perspective by considering fully both the advantages and the disadvantages such a system might have. This is an improvement on having to decide solely on the basis of argument from authority, which would be especially hopeless here since the player has already declined to garner any rigorous information upon which to base a decision as to which sources are “authoritative” and in what respect. “What difference does it make?” is a question that the individual must ultimately and inescapably address himself, and the kind of “difference” must be understood first, otherwise the magnitude of “difference” will be a meaningless quantity.
O.M.
----------------
I have been considering for several weeks now the caveats underlying the computation and use of effects of removal (EOR). Implications of their being something other than uniquely determined and razor-sharp in their precision would include but not be limited to a serious devaluation of the potency of “betting correlation,” which would duplicate and perhaps worsen the false precision reflected in overconfident EOR figures (although assessment of popular, and unpopular, counts obviously could still be done with reference to sets of linearized card tags at various levels within the general model, just not with the accuracy sought by such correlative measures). Linearization theory does provide justification for the approximation constituted by determining and using single-card tags (meaning those computed by removing a single card, i.e. and not tags applying to one card or rank only), noting the gain achieved versus not recalling or making use of the recall of previously-viewed cards at all. However, the questions raised by considering the general context of linearization methodology do not go away if waved off, no more than do other issues regarding the mathematical assessments of various blackjack systems. Players who make use of figures derived from piecemeal methodology may find themselves led to misleading or even erroneous conclusions within a finite amount of time. It is even the case that certain posts on the Internet profess a lack of concern for “theory” while simultaneously arguing points that would have to be grounded in, or at least informed by, “theory” to be salient.
One issue where “ideal” tags are concerned is how to determine the values to use for the “ideal” tag when different levels of interaction produce different linearized values. To cling to a set of “ideal” tags each of which is specified to four decimal places is ascribing too much accuracy to the extrapolation of the tag values to all situations. Correlative measures relying on such overspecified tag values are an implicit attempt to reconcile simplified tag values, such as those used in popular blackjack systems, to an “ideal” that is not meaningful, one that is not “there” at all. To illustrate this point, I took an interest in how much effect considering the removal of two cards of the same rank would have in conjunction with considering the removal of only one. The desire for tags that were “accurate” to two decimal places might still be realized in practice. If it were not, the knowledge produced would at least give an idea of how much and in what way the single-card “ideal” tags needed to be modified and in what context “accuracy” could still be discussed: it seemed inconsistent to attribute overwhelming power and scope of applicability to EOR and related methodological concerns if the robustness and indeed the actual values of the tags to be used were not a concern.
I accessed the web page at http://www.bewersdorff-online.de/black-jack/ for the following calculations. Jorg Bewersdorff is a Ph.D. with a couple of books out (see his web site for details). He has taken the time to put together a page that produces a blackjack playing strategy for any remainder composition. The page supports any number of decks up to 24 and possibly more than that (24 is the figure I got from clicking the relevant button until getting bored). The page supports S17 only and does not allow hard doubles for player totals below nine nor soft doubles except on A9 and A8. Nonetheless, this source is sufficient to illustrate the mathematical concepts related to linearization methodology that I wish to address, as it also provides the player edge for the deck composition in question, if only for a set of rules that is not common (except perhaps in Europe).
I noted the neutral deck edge for the game when resplits were allowed but not doubling after splits. (It was -0.664.) I then computed the player edges after removal of one card of each rank in turn. Then I computed player edges when two cards of each rank were removed in turn. Finally, I compared the figures to determine the one-card EOR, the two-card cumulative EOR, and the two-card marginal EOR.
Here is the data for the ranks in order from Two to Ace:
1-card 2-card cumulative 2-card marginal
0.383 0.770 0.387
0.462 0.939 0.477
0.597 1.200 0.603
0.768 1.576 0.808
0.451 0.955 0.504
0.289 0.661 0.372
-0.018 -0.003 0.015
-0.187 -0.346 -0.159
-0.467 -0.919 -0.452
-0.631 -1.278 -0.647
Right away, it is apparent that card tags that are “accurate” to the second decimal place are misleading. Only the Four holds its “ideal” tag to two decimal places when the one-card and the two-card marginal values are compared. Ignoring the cumulative effect of two cards of the same rank is already a mistake unless you want your “ideal” tags to be completely arbitrary. And we haven’t even investigated other second-order interaction effects or the effects of more than two cards of the same rank being removed. This does not argue that linearization is hopeless (even single-tag linearization is effective in sims), only that confining “ideal” tag computations to single-tag effects constitutes false precision and that “betting correlation” is compromised by this realization. In particular, multilevel counts designed specifically to take advantage of single-tag peculiarities take another blow to their credibility in proportion to the number of levels they implement in their naïve pursuit of unity with the single-card tags (I have listed other such credibility issues elsewhere).
The Seven is the rank of most interest, if only because the absolute difference of its tags is the largest of any rank. This should be unsurprising news to single-deck specialists, who have been taught since birth to stand on (7,7) vs. dealer T even if they observe no other compositional modifications to total-dependent basic strategy. The Eight tag value changes signs, it should be noted, and other tags are volatile enough that their effect even on the “betting correlation” of low-level counts would be of interest.
Opponents may argue that a man with one watch always knows what time it is while a man with two watches is never sure. Regardless, the “confusion” that results from considering all of this data (and perhaps additional data in the form of effects of removal of more than two cards of the same rank and interaction effects among the ranks) instead of the single-card tags alone reflects the ambiguity and variance inherent in the subject, and turning a blind eye to it isn’t a solution as much as it is a refusal to consider the full scope of what linearization methodology would tell us at each level. To consider single-card tags only, while incrementally helpful, is not “linearization” but a subset of “linearization.” To treat the word as if it had an a priori definition is misinformed enough, but assigning it an a priori definition that disavows the possibility of the generalized approach informing the specific one that is retained is openly inconsistent.
Parallels in various quantitative fields exist and may be helpful in illustrating the distinction, but it is perhaps sufficient to realize that in mathematics, there are no “magic formulas,” ones that just work for the user without any understanding of the context in which they are intended to be applied. The semiprofessional blackjack player, while perhaps justified in his unwillingness to abandon his single-tag approach, might consider developing the background to aid in critical thinking regarding the subject. Numerous sources in the blackjack world have cautioned against becoming overly excited about multilevel counts, but uncritically accepting such dictates without an understanding of the context in which they were made is as dangerous as uncritically accepting the systems they seek to debunk. The next time, for example, a system seller offers a Level Four count with an astronomical “betting correlation,” perhaps the semipro will understand enough about the mathematical context of such a claim that he will be able to put it in perspective by considering fully both the advantages and the disadvantages such a system might have. This is an improvement on having to decide solely on the basis of argument from authority, which would be especially hopeless here since the player has already declined to garner any rigorous information upon which to base a decision as to which sources are “authoritative” and in what respect. “What difference does it make?” is a question that the individual must ultimately and inescapably address himself, and the kind of “difference” must be understood first, otherwise the magnitude of “difference” will be a meaningless quantity.
