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Thread: Unplayable splitting strategies

  1. #14


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    Quote Originally Posted by k_c View Post
    I made another attempt at this because I didn't vary some of the data by up card as it should have been. I also tried to address a starting shoe comp of {0,0,0,0,0,11,0,5,0,0} where 4 hands of split 6s are ready to be played as I originally intended. I hope this is an improvement.

    ** prob 6 = 7/12, prob 8 = 5/12 **
    ** prob of first card of 6 = 1 **

    Code:
    ** hand 1 **
    EVPair_ppp versus 6: Stand EV = -.5556, Double EV = .1667, Hit EV = .0833
    EVPair_ppp versus 8: Stand EV = .3333, Double EV = 1.000, Hit EV = .500
    
    Compute EVx_pp(strat)
    ----------------------------------Up card = 6--------------------------------
    Hand     Prob          Stand          Double         Hit            Best
    6-6      7/12*6/11     -.5556         .1667          .08333         .1667
    6-8      5/12*7/11     -.7333         -.2667         -.1333         -.1333
    EVx_pp                 -.6364         -.0303         -.015155       .0303
    
    ----------------------------------Up card = 8--------------------------------
    Hand     Prob          Stand          Double         Hit            Best
    6-6      7/12*5/11     .3333          1.000          .5000          1.000
    6-8      5/12*4/11     .0667          .2167          .1083          .2167
    EVx_pp                 .23634         .7152          .35756         .7152
    
    
    Compute EVn_pp(strat) = (EVx_pp(strat) - 7/12*EVPair_ppp(strat)) / (1 - 7/12)
    ----------------------------------Up card = 6--------------------------------
    EVn_pp(stand) = (-.6364 - 7/12*(-.5556)) / (1 - 7/12) = -.74952
    EVn_pp(double) = (-.0303 - 7/12*(.1667)) / (1 - 7/12) = -.3061
    EVn_pp(hit) = (-.015155 - 7/12*(.0833)) / (1 - 7/12) = -.151992 (best)
    
    ----------------------------------Up card = 8--------------------------------
    EVn_pp(stand) = (.23634 - 7/12*(.3333)) / (1 - 7/12) = .100596
    EVn_pp(double) = (.7152 - 7/12*(1.000)) / (1 - 7/12) = .31648 (best)
    EVn_pp(hit) = (.35756 - 7/12*(.5000) / (1 - 7/12) = .158144
    
    Hand 1    Up Card     EV         Strategy
    6-6       6           .1667      Double
              8           1.000      Double
    
    6-8       6           -.151992   Hit
              8           .31648     Double
    
    ** Hands 2,3,4 **
    Hand 2 possible additional removals: pp, pn*2, nn
    Hand 3 possible additional removals: ppp, ppn*3, pnn*3, nnn
    Hand 4 possible additional removals: pppp, pppn*4, ppnn*6, pnnn*4, nnnn
    k_c
    I'm having trouble interpreting this. Before digging deeper, it's still unclear to me what the intended answer to my original question is? That is, having split and resplit 6s (specifically vs. dealer 8) from the given shoe, to four now-all-incomplete hands, and subsequently drawing an 8 to the first split hand (so that seven 6s and three 8s remain face-down in the shoe), how do we read the table of information above to determine what we should do at that point (stand or double down)? Which is it?

    And what should the strategy be for the subsequent second, third, and fourth split hands as well, to complete the round? Once we know that, we can compute the overall EV for the original 6-6 vs. 8 split (since presumably we aren't trying to change the split-otherwise-double strategy for the other split situations) ... where the objective is to specify such a strategy that realizes the overall EV for the original 6-6 vs. 8 split of 1548/715=2.165, that CDP strategy suggests is achievable.

    Of course, if the intent of the above response is to describe how to compute a strategy that eventually realizes a different overall EV for the split, i.e. any value other than 1548/715=2.165, then, well, okay... but then I'm not sure what the point is. We can describe any number of other strategies-- split-and-resplit-then-mimic-the-dealer has a particular EV that we could easily compute; truly optimal perfect play a la ICountNTrack has a particular EV that we (or rather ICountNTrack) could also compute, etc. But the whole point here is to demonstrate that the *particular* strategies, namely those referred to by CDP and CDPN, require making strategy-varying decisions that depend on information that the player doesn't have when he needs it.

    E

  2. #15


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    Quote Originally Posted by ericfarmer View Post
    I'm having trouble interpreting this. Before digging deeper, it's still unclear to me what the intended answer to my original question is? That is, having split and resplit 6s (specifically vs. dealer 8) from the given shoe, to four now-all-incomplete hands, and subsequently drawing an 8 to the first split hand (so that seven 6s and three 8s remain face-down in the shoe), how do we read the table of information above to determine what we should do at that point (stand or double down)? Which is it?

    And what should the strategy be for the subsequent second, third, and fourth split hands as well, to complete the round? Once we know that, we can compute the overall EV for the original 6-6 vs. 8 split (since presumably we aren't trying to change the split-otherwise-double strategy for the other split situations) ... where the objective is to specify such a strategy that realizes the overall EV for the original 6-6 vs. 8 split of 1548/715=2.165, that CDP strategy suggests is achievable.

    Of course, if the intent of the above response is to describe how to compute a strategy that eventually realizes a different overall EV for the split, i.e. any value other than 1548/715=2.165, then, well, okay... but then I'm not sure what the point is. We can describe any number of other strategies-- split-and-resplit-then-mimic-the-dealer has a particular EV that we could easily compute; truly optimal perfect play a la ICountNTrack has a particular EV that we (or rather ICountNTrack) could also compute, etc. But the whole point here is to demonstrate that the *particular* strategies, namely those referred to by CDP and CDPN, require making strategy-varying decisions that depend on information that the player doesn't have when he needs it.

    E

    I think I had some code to compute splits based upon whether the first card drawn to each hand is either a p or an n. I can't find it and I think I deleted it. I remember that I didn't like the method because it did not necessarily yield more optimal results than a lesser information computation.

    The point was to try to get started in revisiting the problem by computing the first hand, which would be the first problem an actual player would face. Hands 2, 3, and 4 would be increasingly complex.

    In my view this approach is an attempt at pseudo-optimality. Hand 1 is relatively simple. Hand 2 depends what is drawn to hand 1, hand 3 to what is drawn to hands 1 and 2, hand 4 to what is drawn to hands 1, 2, and 3. However, there is no guarantee that only 1 card is drawn to any given hand using best strategy so this may not really be more optimal and I guess that's what I didn't like about it.

    I was just trying to outline something that might eventually answer your question if all computational complexities are solved.

    k_c

  3. #16


    2 out of 2 members found this post helpful. Did you find this post helpful? Yes | No
    I have to laugh. Eric's "require making strategy-varying decisions that depend on information that the player doesn't have when he needs it" reminds me of Godel's incompleteness theorem. Then, k_c's "I can't find it and I think I deleted it" is definitely Fermat's "I have discovered a truly remarkable proof of this theorem which this margin is too small to contain."

    Don

  4. #17


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    Quote Originally Posted by DSchles View Post
    I have to laugh. Eric's "require making strategy-varying decisions that depend on information that the player doesn't have when he needs it" reminds me of Godel's incompleteness theorem. Then, k_c's "I can't find it and I think I deleted it" is definitely Fermat's "I have discovered a truly remarkable proof of this theorem which this margin is too small to contain."

    Don
    I found my code. It was hiding in a gui version of my CA that I changed with a goal of computing splits more optimally than CDP1. I usually make any changes in a console version first, but in this case I didn't.

    There was a thread on blackjackinfo.com several years ago that discussed this type of problem. I composed this but I never posted it. Instead I decided to remove this type of approach from my CA. http://www.blackjackinfo.com/bb/show...t=22710&page=9

    Code:
    I wrote an algorithm with the goal of computing split EVs that are a little better at approximating optimal split EVs. In computing splits there are 3 types 
    
    of hands that come from drawing to a one or more single pair cards. When splits remain there are 2 types - N when a non-pair card is drawn and P when a pair 
    
    card is drawn. A special case exists when exactly 1 pair card remains since no matter how many splits remain if an N card is drawn then that particular 
    
    drawing sequence is finished. When no splits remain there is 1 type of hand - x hands. A card of any rank, including a pair card, can be drawn on an x hand 
    
    since no more splitting can be done.
    
    I think maybe the reason CDP is not necessarily more optimal than CDP1 even though it considers more information has something to do with the way it 
    
    considers N hands on the condition that a non-pair card has already been removed. In that case the EV is figured as the conditional probability involving 
    
    differing numbers of pair cards removed. The EV is right but each non-pair card needs to be specifically removed to be included in computing subsequent 
    
    playing strategy. So my algorithm does this. Up to 2 specific non-pair cards can be removed.
    
    For x hands when a fixed strategy is used it turns out the EV is the same for each x hand even though different drawing sequences are possible. I guess the 
    
    reason is because the chance of each hand being played versus each possible dealer composition turns out to be the same. For SPL1 there are 2 x hands. If the 
    
    first hand is played optimally without any consideration of the second then the EV for the first hand is simply the optimal EV for burning one pair card and 
    
    drawing to one pair card. If the second hand is played with the same strategy as the first then the EV of the second hand is the same EV as well and SPL1 EV 
    
    = 2*(hand 1 EV). In order to consider more information for x hands my algorithm specifically considers the first card drawn to the first hand in computing a 
    
    strategy for the second hand. For SPL2 there are up to 3 x hands and for SPL3 there are up to 4 x hands. What my algorithm does is to specifically remove all 
    
    combinations of 0, 1, 2, or 3 ranks progressively depending upon the current number of x hands being computed.
    
    Broadly speaking the algorithm removes the rank of the first card drawn to each hand so it can be considered for subsequent hands. It doesn't consider each 
    
    and every card drawn though. Considering more than the first card drawn to each pair card would require considering separate compositions for each up card 
    
    since strategy for each up card is potentially different. It assumes EV for the current calculation is the computed EV for whatever is presently known so no 
    
    possible covariability between split hands is considered.
    
    The algorithm is not fast. It seems to get reasonable values but I don't know if it correctly does what in theory I'm trying to do.
    I named the algorithm CDPnnxxx since it's supposed to specifically remove up to 2 N cards and/or 3 x cards.
    
    Attached sample is hand of 2-2, single deck, s17, DAS.
    
    2-2, single deck, s17, DAS
    It was at this point I decided there seemed to be a deficiency in this type of approach, which Eric has pointed out - EV is computable but at some point strategy may be undeterminable because sooner or later it may be required to move to the next hand before the current hand is completed. Also EV is not necessarily more optimal than a lesser information strategy, although it may be more likely.

    k_c
    Last edited by k_c; 09-15-2021 at 03:46 PM.

  5. #18


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    Quote Originally Posted by k_c View Post
    I found my code. It was hiding in a gui version of my CA that I changed with a goal of computing splits more optimally than CDP1. I usually make any changes in a console version first, but in this case I didn't.

    There was a thread on blackjackinfo.com several years ago that discussed this type of problem. I composed this but I never posted it. Instead I decided to remove this type of approach from my CA. http://www.blackjackinfo.com/bb/show...t=22710&page=9

    Code:
    I wrote an algorithm with the goal of computing split EVs that are a little better at approximating optimal split EVs. In computing splits there are 3 types 
    
    of hands that come from drawing to a one or more single pair cards. When splits remain there are 2 types - N when a non-pair card is drawn and P when a pair 
    
    card is drawn. A special case exists when exactly 1 pair card remains since no matter how many splits remain if an N card is drawn then that particular 
    
    drawing sequence is finished. When no splits remain there is 1 type of hand - x hands. A card of any rank, including a pair card, can be drawn on an x hand 
    
    since no more splitting can be done.
    
    I think maybe the reason CDP is not necessarily more optimal than CDP1 even though it considers more information has something to do with the way it 
    
    considers N hands on the condition that a non-pair card has already been removed. In that case the EV is figured as the conditional probability involving 
    
    differing numbers of pair cards removed. The EV is right but each non-pair card needs to be specifically removed to be included in computing subsequent 
    
    playing strategy. So my algorithm does this. Up to 2 specific non-pair cards can be removed.
    
    For x hands when a fixed strategy is used it turns out the EV is the same for each x hand even though different drawing sequences are possible. I guess the 
    
    reason is because the chance of each hand being played versus each possible dealer composition turns out to be the same. For SPL1 there are 2 x hands. If the 
    
    first hand is played optimally without any consideration of the second then the EV for the first hand is simply the optimal EV for burning one pair card and 
    
    drawing to one pair card. If the second hand is played with the same strategy as the first then the EV of the second hand is the same EV as well and SPL1 EV 
    
    = 2*(hand 1 EV). In order to consider more information for x hands my algorithm specifically considers the first card drawn to the first hand in computing a 
    
    strategy for the second hand. For SPL2 there are up to 3 x hands and for SPL3 there are up to 4 x hands. What my algorithm does is to specifically remove all 
    
    combinations of 0, 1, 2, or 3 ranks progressively depending upon the current number of x hands being computed.
    
    Broadly speaking the algorithm removes the rank of the first card drawn to each hand so it can be considered for subsequent hands. It doesn't consider each 
    
    and every card drawn though. Considering more than the first card drawn to each pair card would require considering separate compositions for each up card 
    
    since strategy for each up card is potentially different. It assumes EV for the current calculation is the computed EV for whatever is presently known so no 
    
    possible covariability between split hands is considered.
    
    The algorithm is not fast. It seems to get reasonable values but I don't know if it correctly does what in theory I'm trying to do.
    I named the algorithm CDPnnxxx since it's supposed to specifically remove up to 2 N cards and/or 3 x cards.
    
    Attached sample is hand of 2-2, single deck, s17, DAS.
    
    2-2, single deck, s17, DAS
    It was at this point I decided there seemed to be a deficiency in this type of approach, which Eric has pointed out - EV is computable but at some point strategy may be undeterminable because sooner or later it may be required to move to the next hand before the current hand is completed. Also EV is not necessarily more optimal than a lesser information strategy, although it may be more likely.

    k_c
    I think there is a misunderstanding in this thread. Eric is simply stating that CDP splitting strategy while it can me clearly computed, cannot be achieved at the table because it requires information not available to the player as the split hands being resolved. So again it's not a computational issue, it's an intrinsic problem with the way split hands out are played out at a blackjack table.
    Chance favors the prepared mind

  6. #19


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    Quote Originally Posted by iCountNTrack View Post
    I think there is a misunderstanding in this thread. Eric is simply stating that CDP splitting strategy while it can me clearly computed, cannot be achieved at the table because it requires information not available to the player as the split hands being resolved. So again it's not a computational issue, it's an intrinsic problem with the way split hands out are played out at a blackjack table.
    That's what I'm trying to say while still leaving open the possibility that MGP's conditional probability approach could somehow overcome this problem.

    It has been awhile but I looked into some approaches but eventually determined it wasn't worth the effort so I decided not to include support for either CDP or CDPN in my CA. I just didn't care for the seemingly inherent anomalies.

    k_c

  7. #20


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    Code:
    Rounds         
    SPL1		MGP			Eric
    xx		2*EV(x)			2*EV(x) 
    
    
    SPL2		MGP			Eric 
    Pxxx		3*EV(x-P)		3*EV(x-P)
    NN		EV(N) + EV(N-N)		2*EV(N-N) 
    NPxx		EV(N) + 2*EV(x-PN)	EV(N-P) + 2*EV(x-PN)


    Everyone's talking in circles and their own language lol. Eric, let's see if I can try and restate your question:

    Is it possible to play a more optimal strategy after during a split round given the knowledge of the cards as they are being played?
    Is that what you're asking?

    iCountNTrack sounds like he can have it play perfect strategy based on every card played, so yes, if you want perfect play.

    What if you want something in-between perfect play and a CDZ- (composition dependent playing pre and post-split hands the same way but ignoring post-split hand evs) strategy?

    That's what I call a CDPN (composition dependent including knowledge of whether a paircard or non-paircard has been dealt) strategy.

    Let's look at SPL1. When looking at any-post-split hand, note that a paircard has been removed from the deck. So when we say EV(x) and are let's say looking at 8, 8 vs 5. Now let's say a 2 is dealt to the first 8. The EV for this hand is 8,2 vs 5 with an 8 removed from the deck.

    With a CDPN knowledge for SPL1, you would:

    1) Remove 1 paircard
    2) Figure out the EVs for the possible hands/strategies against that upcard removing one paircard
    3) Determine for the post-split hand the best composition dependent strategy for each hand
    4) Play out both hands the same composition dependent way

    NOTE: You are using the knowledge that this is a post-split hand and possibly changing your strategy based on that.

    If you are doing a full CD+ (plus simply means composition dependent PLUS composition dependent post-split) strategy as iCountNTrack does you would:

    1) Remove one paircard
    2) Play out the first hand optimally
    3) When that hand plays out, play out the second hand optimally based on the current deck composition

    Also note that if you are trying to find a TDPN (Total dependent strategy that takes into account pair and non-pair card post-split hands) strategy (which was another question - i.e. does basic (TD - total dependent) strategy change if you take into account post-split hands), you would do the following:

    1) Remove 1 paircard
    2) Figure out the EVs for the possible hands/strategies against that upcard removing one paircard
    3) Weight the outcomes of each hand in the usual TD calculation BUT include the EV(x) calcs given their probability of occuring
    4) Play out both hands the same way as the calculated net TD strategy.

    Now, let's look at SPL2. Here there are 3 possible hand types.

    A) Pxxx: This is calculated the exact same way as SPL1. Again note that all 3 hands are played the same way and all 3 take into account that 2 paircards are removed before playing out a hand optimally.

    B) NN: Let's play this hand out.

    1) First you get a non-paircard and play out your hand. So you play this hand after removing 1 paircard. You have no idea what the next card will be but that's ok. You know you didn't get an 8.
    2) Figure out the optimal play for this hand and play it. Note that I'm pretty sure this will be the same as SPL1 (x).

    3) Now you get another non-pair card. Here you use the (N-N) calculations for the strategy calculations. You know you didn't get a third paircard on the first paircard.
    4) Play out that hand optimally for (N-N)

    C) NPxx

    1)
    First you get a non-paircard and play out your hand. So you play this hand after removing 1 paircard. You have no idea what the next card will be but that's ok. You know you didn't get an 8.
    2) Figure out the optimal play for this hand and play it.

    3) Now you get a paircard P.

    4) Now you calculate the optimal strategy given that an N and P were removed (x-PN)
    5) Play both remaining hands the same way using the knowledge of the current shoe state.

    NOTE: If you want to use the above for a TDPN strategy then you need to take the weighted probabilities of each possible round and hand in aggregate based on the total, given that the proper move is to split for the pair and add those in to the regular TD strategy calcs.

    NOTE: For a CDZ+ (composition dependent playing post-split and pre-split hands the same but including the EVs from post-split hands), you need to take the weighted probabilities of each possible round and hand in aggregate based on the composition of the hand, given that the proper move is to split for the pair and add those in to the regular CD strategy calcs.

    Anyways, my point is that you can indeed use knowledge from split hands to calculate a more optimal strategy as the hand is being played out, which I thought was the original question Eric posed. The column labelled Eric won't work for the reasons that led to the question, they assume that the N and P cards are all played out first which they are not. That's why it fascinated me that those calculations worked at all when I knew my calculations were already correct.

    Does this help clarify this discussion at all?
    Last edited by MGP; 09-27-2021 at 08:59 PM.

  8. #21


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    Quote Originally Posted by MGP View Post
    Code:
    Rounds         
    SPL1        MGP            Eric
    xx        2*EV(x)            2*EV(x) 
    
    
    SPL2        MGP            Eric 
    Pxxx        3*EV(x-P)        3*EV(x-P)
    NN        EV(N) + EV(N-N)        2*EV(N-N) 
    NPxx        EV(N) + 2*EV(x-PN)    EV(N-P) + 2*EV(x-PN)


    Everyone's talking in circles and their own language lol. Eric, let's see if I can try and restate your question:



    Is that what you're asking?

    iCountNTrack sounds like he can have it play perfect strategy based on every card played, so yes, if you want perfect play.

    What if you want something in-between perfect play and a CDZ- (composition dependent playing pre and post-split hands the same way but ignoring post-split hand evs) strategy?

    That's what I call a CDPN (composition dependent including knowledge of whether a paircard or non-paircard has been dealt) strategy.

    Let's look at SPL1. When looking at any-post-split hand, note that a paircard has been removed from the deck. So when we say EV(x) and are let's say looking at 8, 8 vs 5. Now let's say a 2 is dealt to the first 8. The EV for this hand is 8,2 vs 5 with an 8 removed from the deck.

    With a CDPN knowledge for SPL1, you would:

    1) Remove 1 paircard
    2) Figure out the EVs for the possible hands/strategies against that upcard removing one paircard
    3) Determine for the post-split hand the best composition dependent strategy for each hand
    4) Play out both hands the same composition dependent way

    NOTE: You are using the knowledge that this is a post-split hand and possibly changing your strategy based on that.

    If you are doing a full CD+ (plus simply means composition dependent PLUS composition dependent post-split) strategy as iCountNTrack does you would:

    1) Remove one paircard
    2) Play out the first hand optimally
    3) When that hand plays out, play out the second hand optimally based on the current deck composition

    Also note that if you are trying to find a TDPN (Total dependent strategy that takes into account pair and non-pair card post-split hands) strategy (which was another question - i.e. does basic (TD - total dependent) strategy change if you take into account post-split hands), you would do the following:

    1) Remove 1 paircard
    2) Figure out the EVs for the possible hands/strategies against that upcard removing one paircard
    3) Weight the outcomes of each hand in the usual TD calculation BUT include the EV(x) calcs given their probability of occuring
    4) Play out both hands the same way as the calculated net TD strategy.

    Now, let's look at SPL2. Here there are 3 possible hand types.

    A) Pxxx: This is calculated the exact same way as SPL1. Again note that all 3 hands are played the same way and all 3 take into account that 2 paircards are removed before playing out a hand optimally.

    B) NN: Let's play this hand out.

    1) First you get a non-paircard and play out your hand. So you play this hand after removing 1 paircard. You have no idea what the next card will be but that's ok. You know you didn't get an 8.
    2) Figure out the optimal play for this hand and play it. Note that I'm pretty sure this will be the same as SPL1 (x).

    3) Now you get another non-pair card. Here you use the (N-N) calculations for the strategy calculations. You know you didn't get a third paircard on the first paircard.
    4) Play out that hand optimally for (N-N)

    C) NPxx

    1)
    First you get a non-paircard and play out your hand. So you play this hand after removing 1 paircard. You have no idea what the next card will be but that's ok. You know you didn't get an 8.
    2) Figure out the optimal play for this hand and play it.

    3) Now you get a paircard P.

    4) Now you calculate the optimal strategy given that an N and P were removed (x-PN)
    5) Play both remaining hands the same way using the knowledge of the current shoe state.

    NOTE: If you want to use the above for a TDPN strategy then you need to take the weighted probabilities of each possible round and hand in aggregate based on the total, given that the proper move is to split for the pair and add those in to the regular TD strategy calcs.

    NOTE: For a CDZ+ (composition dependent playing post-split and pre-split hands the same but including the EVs from post-split hands), you need to take the weighted probabilities of each possible round and hand in aggregate based on the composition of the hand, given that the proper move is to split for the pair and add those in to the regular CD strategy calcs.

    Anyways, my point is that you can indeed use knowledge from split hands to calculate a more optimal strategy as the hand is being played out, which I thought was the original question Eric posed. The column labelled Eric won't work for the reasons that led to the question, they assume that the N and P cards are all played out first which they are not. That's why it fascinated me that those calculations worked at all when I knew my calculations were already correct.

    Does this help clarify this discussion at all?
    Here is a potential problem when drawing an n (non-pair card):

    1. If it turns out your strategy is to stand then you can move to the next hand without problem
    2. If it turns out your strategy is to hit
    a. If you don't compute another draw then you have to go to the next hand before the current hand is completed
    b. You can continue to draw additional n's and p's until strategy is to stand but there is no way to know if hand is busted
    Maybe this is OK but it's hard to imagine
    (In the case of double only 1 additional card needs to be drawn)

    It seems that to be playable n's and p's must continue to be drawn until CDZ+ is actually realized, assuming this is possible.

    k_c

  9. #22


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    Quote Originally Posted by MGP View Post
    Everyone's talking in circles and their own language lol. Eric, let's see if I can try and restate your question:

    "Is it possible to play a more optimal strategy after during a split round given the knowledge of the cards as they are being played?"

    Is that what you're asking?
    No. I see now that I likely added to the confusion by phrasing as a question at all, since it was intended to be rhetorical. That is, to the question "should we stand or double down with 6-8 vs. 8, in the split situation we've described, if we want to achieve an overall EV of 1548/715 for the round, no more and no less, using CDP strategy as computed by *either* "Eric's" or "MGP's" method?", the answer is, "neither, because if we always stand in that situation, or if we always double down in that situation, we have no hope of achieving the computed EV of 1548/715."

    Note that I'm not "asking" anything here, I'm *claiming* that CDP-- *and CDPN*-- are broken as specifications of strategies, since they are unplayable, the situation described in this post being a concrete example. For some reason, the discussion keeps gravitating toward computational methods (what you label "MGP" and "Eric" in the tables of formulas you keep quoting)... it makes no difference which of these two computational approaches are used, the problem (with the *definition of the random variable*, which has a well-defined expected value no matter how we choose to compute it) remains.

    Quote Originally Posted by MGP View Post
    Now, let's look at SPL2. Here there are 3 possible hand types.

    A) Pxxx: This is calculated the exact same way as SPL1. Again note that all 3 hands are played the same way and all 3 take into account that 2 paircards are removed before playing out a hand optimally.
    This is exactly where the problem can arise, and it's still a problem for CDPN as well as CDP, which is what I seem to have not clearly communicated yet. This is where the problem arises in the example described in this post, albeit in the PPxxxx case for SPL3 instead of Pxxx for SPL2. The expected value that results from "all 3 hands are played the same way and all 3 take into account that 2 paircards are removed" cannot (in general) actually be achieved.

    I say "in general" because finding examples of this sort of thing has turned out to be hard to find. That is, consider the example I've described so far ((0,0,0,0,0,11,0,5,0,0), splitting 6s vs. 8). I had to search quite a bit to find that example, first looking for "nice" shoes with the appropriate differences in EV as we move from CDZ- to CDP[n]... but even after that reasonably automatable search, I still had to "re-evaluate" those same EVs by brute-force enumeration and playout of shoe arrangements, looking for situations where the *indicated* strategy variations *didn't* yield the corresponding EV.

    This is a detail that I'm not sure I've actually made explicit here yet. That is, it *is* possible to realize 1548/715 as an overall CDP EV for the split in this example... but to do so, not only do we need a dealer willing to deal all of our split hands out to two cards each before asking for a "non-split" strategy decision, but even then we still have to execute a strategy that looks surprisingly complicated. Coming back to the rhetorical question situation above, having split and resplit 6s to a maximum of 4 hands, and observing the first of those hands as 6-8, what should we do?

    We've already said that we can't *always* stand, and we can't *always* double down. It turns out that we have to distinguish *two* of the other three split hands (any two will do, but they have to be *fixed*), and stand if those hands both get fleshed out to 6-6, otherwise (if we draw an 8 to either of them) double down.

    But it's worse than that-- we have to execute the same "conditional stand/double" strategy for *each* of the other three split hands as well: for each, distinguish and fix two of the three "other slots," and stand only if both of those two other hands are 6-6.

    E

  10. #23


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    Quote Originally Posted by MGP View Post
    Honestly I wish I could answer that but it's programmed with Visual Studio 2010. I don't have any way of recompiling it to change it over to those hands since I use the ones in the Eric column after you showed that was the actual EV.

    I understand what you're saying about the PPxxxx case. All the x's have the same EV but may have a different strategy based on the effect of removal. What you're describing is a full CD strategy. Once the P's are drawn, the EV and strategy for the hands would be the same because no other information about the cards is known and thus the "x" designation. There is obviously no strategy for either pair card other than to split.

    Now, if you wanted to DECIDE to split or not after the first pair card, that's a whole other ball game and I wouldn't even know how to name that strategy.

    But I guess I misunderstood that's what you were going for. I am just explaining that if you use the effects of removal hands you'd be closer to a fully CD strategy based on every card and it would be the correct strategy based on removing N and P cards in a way that can actually be played in real time.
    Hello Eric
    This thread seems to be more about programming since a shoe left with only 6s and 8s has probably never happened in the history of blackjack. I am not a programmer but I find this thread interesting nonetheless, especially tha Don defined you somewhat as an “edge seeker”, something I can relate to.

    For my grain of salt, it’s clear that someone should always have a post-split strategy in mind before splitting any hand and it would be deck-composition dependent which may lead to different decisions after the first and second and splits. I guess programmers should lean that way.

    We both know that nobody will ever make a living splitting 6s but we have to make the best of this hand when it happens. For practical purposes, it seems like with 66v8, one would be stupid to split when there are extra tens in the deck. Who would like to get stiffed with two 16s against 8 on a high count and with more money on the table?

    On the other hand, when there are extra 789s, the clear cut decision is to hit 66, although again, at very rare times a double down could give the superior EV.

    With a decent amount of 4s and 5s left in the shoe and a fair count, the split/double sequence becomes more probable and more rewarding.

    However it becomes tricky at very low counts. Splitting once and ending up with at least one probable winning hand at 19, 20 or 21 with a couple of hits seems like a good choice. However, splitting more than once is pushing it.

    Do you have any extended data (deck composition dependent? you can share with us on splitting 6s (and resplitting) against various dealer cards?

    Thanks

  11. #24
    Random number herder Norm's Avatar
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    Quote Originally Posted by Secretariat View Post
    This thread seems to be more about programming since a shoe left with only 6s and 8s has probably never happened in the history of blackjack.
    Of course this is an extremely unlikely event. Problem is, there are an extremely large number of extremely unlikely events. Which is to say, extremely unusual events happen all the time. So, you need to worry about extremely unlikely events when looking at the whole.
    Last edited by Norm; 10-03-2021 at 06:13 AM.
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    This is exactly my approach, Norm. That's why one must go beyond the standard strategy deviation indices which are not the Holy Grail, but only educated approximations.

  13. #26
    Random number herder Norm's Avatar
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    That isn't close to what I was saying.
    "I don't think outside the box; I think of what I can do with the box." - Henri Matisse

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