Unplayable splitting strategies

[k_c]

We might be just differently indexing the number of pair cards removed, but as described in the post at the start of this thread (actually my 2nd post), the place where CDP strategy differs from CDZ- is with 4 pair cards observed to have been removed, not 2. This is sort of the point-- there *can eventually* be 4 pair cards observed, but not at the point that we have to make a strategy decision for that first 6-8 hand.



I might misunderstand you here, but note that CDP strategy, and the "component" expected values, agree with CDZ- in all of those other possible drawing sequences of splitting/resplitting; the only place where things differ is in the PPxxxx outcome (and the "problem" situation occurs when trying to play out that first "x" in that outcome).

If there were anything we might call a "solution" here, IMO it's nothing really earth-shaking-- just don't use CDP or CDPN as a measure of realizable expected return. CDZ- works... and CDP1 does as well, since it doesn't suffer from this problem of asking the player to *change* strategy decisions as a function of information that the player doesn't yet have. (In the case of CDP1, the only additional information that can affect a player's decision is *whether* he has split the original hand or not.)

E

I'm trying to mimic CDZ- from a full shoe. Don't you determine CDZ- from a full shoe by simply recording the strategies for each hand as dealt from the top of a full shoe? That's how I do it and I save the results in order to use this as a CD basic strategy option to apply to less than full shoe. I would think CDZ- given 2 cards have been removed would be done the same except that these same 2 additional cards have been removed. For the example full shoe is {0,0,0,0,0,11,0,5,0,0}.

Anyway my viewpoint is that for drawing sequence ppxxxx CDP strategy is clear. When an n is involved it seems more complicated.
For example pnnn and npnn yield the same 3 hands (pn, pn, pn) but strategy of the first n in each case would need to be determined with a differing number of pRemoved when played in sequence in a blackjack game.

I guess it's just a different approach to figure out that a problem exists.

k_c

[ericfarmer]

I'm trying to mimic CDZ- from a full shoe. Don't you determine CDZ- from a full shoe by simply recording the strategies for each hand as dealt from the top of a full shoe? That's how I do it and I save the results in order to use this as a CD basic strategy option to apply to less than full shoe. I would think CDZ- given 2 cards have been removed would be done the same except that these same 2 additional cards have been removed. For the example full shoe is {0,0,0,0,0,11,0,5,0,0}.

Yep, this is correct... sort of. For that full shoe with 11 6s and 5 8s, the strategy for 6-8 vs. 8 is indeed to double down. And if we remove two 6s and recompute, the strategy is still to double down, and so on... but when we remove *four* additional 6s from the shoe, and recompute strategy, that we *stand* on 6-8 vs. 8 instead of double down, and that's the strategy departure that matters here. Now, there are-- or will be later in the resolution of a PPxxxx hand-- possible situations where we may observe that many 6s. But as you can verify, when we actually collect the individual conditional expected returns into one overall CDP EV (or CDPN, for that matter), the result is not (generally) achievable.

You don't have to take my word for it-- I went searching for examples of this in shoes where things are easy to compute and verify "by hand." For example, in this split case of 6-6 vs. 8, there are only two possible useful strategies to consider, standing or doubling (hitting will never be optimal, since if it's desirable, then doubling is more so).

Anyway my viewpoint is that for drawing sequence ppxxxx CDP strategy is clear.

How is it clear? If it were clear, then someone would be able to answer my rhetorical/didactic question that I've been asking, namely, what's the "clear" strategy if that first "x" in PPxxxx is an 8, i.e., 6-8 vs. 8, with those three other 6s waiting to be fleshed out? Where remember, the goal is to realize the CDP value that each of our CAs reports of 1548/715 (or at least, yours *could* if you also implemented it).

E

[k_c]

Yep, this is correct... sort of. For that full shoe with 11 6s and 5 8s, the strategy for 6-8 vs. 8 is indeed to double down. And if we remove two 6s and recompute, the strategy is still to double down, and so on... but when we remove *four* additional 6s from the shoe, and recompute strategy, that we *stand* on 6-8 vs. 8 instead of double down, and that's the strategy departure that matters here. Now, there are-- or will be later in the resolution of a PPxxxx hand-- possible situations where we may observe that many 6s. But as you can verify, when we actually collect the individual conditional expected returns into one overall CDP EV (or CDPN, for that matter), the result is not (generally) achievable.

You don't have to take my word for it-- I went searching for examples of this in shoes where things are easy to compute and verify "by hand." For example, in this split case of 6-6 vs. 8, there are only two possible useful strategies to consider, standing or doubling (hitting will never be optimal, since if it's desirable, then doubling is more so).



How is it clear? If it were clear, then someone would be able to answer my rhetorical/didactic question that I've been asking, namely, what's the "clear" strategy if that first "x" in PPxxxx is an 8, i.e., 6-8 vs. 8, with those three other 6s waiting to be fleshed out? Where remember, the goal is to realize the CDP value that each of our CAs reports of 1548/715 (or at least, yours *could* if you also implemented it).

E

I'm not sure if I'm right or wrong but what I think is there are 3 separate CDZ- strategies that apply to CDP for 3 allowed splits:
1. Strategy for full shoe (starting composition) - applies to a decision where no additional pair cards have been removed
2. Strategy with 1 pair card removed from starting composition - applies to a decision where 1 additional pair card has been removed
3. Strategy with 2 pair cards removed from starting composition - applies to a decision where 2 additional pair cards have been removed

For drawing sequence ppxxxx there are always 2 additional (pertinent) pair cards removed for each decision so wouldn't strategy always be defined? I guess what I'm saying is that there is a strategy for each px that is fixed. What I'm not saying is what it computes to. I believe you when you say there is no workable fixed strategy that computes to the expected CDP value assuming there are no other CDZ- variations in the other drawing sequences.

k_c

[k_c]

Can the problem be simplified by looking first at the case of 1 allowed split?

Shoe composition: {0,0,0,0,0,11,0,5,0,0}
Split EV for 6-6:
Up card     CDZ- (Farmer console program)      CDZ+ (my console program - computes optimal SPL1)     CDP1 (from my gui program rounded off)
   6        -1.864801865%                      0.0217560217560218                                    -1.399%
   8        103.496503497%                     1.04055944055944                                      103.5%

Is it in general possible to get a partially optimal split EV where EV(CDZ-) <= EVPN(CDZ-) <= EV(CDZ+) for a single split with simplified removals of just p's and n's?

k_c

[ericfarmer]

Can the problem be simplified by looking first at the case of 1 allowed split?

Shoe composition: {0,0,0,0,0,11,0,5,0,0}
Split EV for 6-6:
Up card     CDZ- (Farmer console program)      CDZ+ (my console program - computes optimal SPL1)     CDP1 (from my gui program rounded off)
   6        -1.864801865%                      0.0217560217560218                                    -1.399%
   8        103.496503497%                     1.04055944055944                                      103.5%

Is it in general possible to get a partially optimal split EV where EV(CDZ-) <= EVPN(CDZ-) <= EV(CDZ+) for a single split with simplified removals of just p's and n's?

k_c

This is interesting; can you describe what is meant by "CDZ+", i.e. the optimal SPL1 is subject to what constraints? (I can't find reference to CDZ+ anywhere, other than earlier in this thread where it isn't clearly defined.) What is the hand(s) where strategy differs from CDZ- to yield that increase in EV from 740/715 to 744/715 (where 715 is the number of possible shuffled arrangements of the remainder of the shoe)?

Thanks,
E

[k_c]

This is interesting; can you describe what is meant by "CDZ+", i.e. the optimal SPL1 is subject to what constraints? (I can't find reference to CDZ+ anywhere, other than earlier in this thread where it isn't clearly defined.) What is the hand(s) where strategy differs from CDZ- to yield that increase in EV from 740/715 to 744/715 (where 715 is the number of possible shuffled arrangements of the remainder of the shoe)?

Thanks,
E

CDZ+ means making each post split decision considering all removals at the time of the decision.

For SPL1 there are 2 hands, each with its own EV. hand1EV is simply the optimal EV of drawing to the first pair card allowing for one pair
card removed. This is what is done to compute CDP1. For CDP1, EVx = optimal EV of first split hand and EV(CDP1) for SPL1 = 2*EVx where
strategy for hand2 is the same as strategy for hand1. For CDZ+, hand2EV and strategy for one allowed split is variable depending upon what
has been drawn to hand1 as well as what is currently drawn to hand2.

*Here are some things I do to compute hand2:
1. In the course of computing hand1 I use busted as well as unbusted hands. Busted hand1 hands have no effect on hand1EV and strategy but do
effect hand2.
2. For each hand1 composition I compute a parameter I call h2EVX. h2EVx is the optimal hand2EV
for that particular hand1 comp. To compute h2EVx only unbusted hands need to be considered because for SPL1
no more split hands follow. I do not save this set of hand2 hands, only the h2EVx parameter for this hand1 comp.
3. I have parameters I call h2EVx_stand, h2EVx_double, and h2EVx_hit which depend on hand1 strategy. h2EVx_stand = h2EVx because no cards are
drawn for the hand1 comp being referenced. hand2EV is computed while going through all hand1 hands using above values. If by some quirk of
fate strategy of all hand1 hands is stand then hand2EV would equal hand1EV.

I'm sure there's room for improvement.

- This is what I get for the sample shoe comp -

Shoe composition: {0,0,0,0,0,11,0,5,0,0}
Split EV for 6-6 (SPL1), S17, DAS:

** versus 6 **
hand1EV = -0.0069930069930069488
hand2EV = 0.028749028749028734
hand1EV + hand2EV = 0.0217560217560217852 = CDZ+(single split)

** versus 8 **
hand1EV = 0.51748251748251750
hand2EV = 0.52307692307692299
hand1EV + hand2EV = 1.04055944055944049 = CDZ+(single split)

k_c

[ericfarmer]

CDZ+ means making each post split decision considering all removals at the time of the decision.

For SPL1 there are 2 hands, each with its own EV. hand1EV is simply the optimal EV of drawing to the first pair card allowing for one pair
card removed. This is what is done to compute CDP1. For CDP1, EVx = optimal EV of first split hand and EV(CDP1) for SPL1 = 2*EVx where
strategy for hand2 is the same as strategy for hand1. For CDZ+, hand2EV and strategy for one allowed split is variable depending upon what
has been drawn to hand1 as well as what is currently drawn to hand2.

*Here are some things I do to compute hand2:
1. In the course of computing hand1 I use busted as well as unbusted hands. Busted hand1 hands have no effect on hand1EV and strategy but do
effect hand2.
2. For each hand1 composition I compute a parameter I call h2EVX. h2EVx is the optimal hand2EV
for that particular hand1 comp. To compute h2EVx only unbusted hands need to be considered because for SPL1
no more split hands follow. I do not save this set of hand2 hands, only the h2EVx parameter for this hand1 comp.
3. I have parameters I call h2EVx_stand, h2EVx_double, and h2EVx_hit which depend on hand1 strategy. h2EVx_stand = h2EVx because no cards are
drawn for the hand1 comp being referenced. hand2EV is computed while going through all hand1 hands using above values. If by some quirk of
fate strategy of all hand1 hands is stand then hand2EV would equal hand1EV.

I'm sure there's room for improvement.

- This is what I get for the sample shoe comp -

Shoe composition: {0,0,0,0,0,11,0,5,0,0}
Split EV for 6-6 (SPL1), S17, DAS:

** versus 6 **
hand1EV = -0.0069930069930069488
hand2EV = 0.028749028749028734
hand1EV + hand2EV = 0.0217560217560217852 = CDZ+(single split)

** versus 8 **
hand1EV = 0.51748251748251750
hand2EV = 0.52307692307692299
hand1EV + hand2EV = 1.04055944055944049 = CDZ+(single split)

k_c

Isn't this optimal in the sense of maximizing EV among *all* possible strategies? That is, the "Z" in CDZ+ is out of place; "zero memory" is pretty consistently agreed to mean a strategy that depends only on the cards in the current hand (vs. any additional cards or "state" of the round). Maybe ICountNTrack can confirm/weigh in on this small shoe example for comparison (since I know he has implemented optimal EV-maximization).

I'm also not sure what EVPN(CDZ-) means; CDZ- refers to a well-defined strategy (maximize CD strategy temporarily prohibiting splits, then apply that strategy in evaluating candidate splits), in which case it has a definite expected value, no matter how we choose to implement an algorithm to compute it. Maybe you meant something like E(CDPN), or E(something-like-CDPN-that-we-haven't-named-yet-but-is-demonstrably-executable-at-the-table-unlike-CDP-or-CDPN)?

In that case, yes, there is certainly opportunity to define and evaluate playing strategies that depend on more information about the round than just the cards in the current hand (but constrained to information actually available at the time of each necessary playing decision). CDP1 is such a strategy. And one could certainly do better, with some yet-to-be-named-strategy S, so that E(CDZ-)<=E(CDP1)<=E(S)<=E(OPT, i.e. what you're calling CDZ+).

E

[k_c]

Isn't this optimal in the sense of maximizing EV among *all* possible strategies? That is, the "Z" in CDZ+ is out of place; "zero memory" is pretty consistently agreed to mean a strategy that depends only on the cards in the current hand (vs. any additional cards or "state" of the round). Maybe ICountNTrack can confirm/weigh in on this small shoe example for comparison (since I know he has implemented optimal EV-maximization).

I'm also not sure what EVPN(CDZ-) means; CDZ- refers to a well-defined strategy (maximize CD strategy temporarily prohibiting splits, then apply that strategy in evaluating candidate splits), in which case it has a definite expected value, no matter how we choose to implement an algorithm to compute it. Maybe you meant something like E(CDPN), or E(something-like-CDPN-that-we-haven't-named-yet-but-is-demonstrably-executable-at-the-table-unlike-CDP-or-CDPN)?

In that case, yes, there is certainly opportunity to define and evaluate playing strategies that depend on more information about the round than just the cards in the current hand (but constrained to information actually available at the time of each necessary playing decision). CDP1 is such a strategy. And one could certainly do better, with some yet-to-be-named-strategy S, so that E(CDZ-)<=E(CDP1)<=E(S)<=E(OPT, i.e. what you're calling CDZ+).

E

My intention was to hopefully compute optimal SPL1. I thought that CDZ- referred to optimal pre-split strategy so I thought CDZ+ would refer to optimal post-split strategy.

I took it that your original problem was that not only was CDP unplayable at the table but also that the EV computed by CA couldn't be accounted for. I don't know why but was wondering if SPL1 had the same problem except hopefully more simple. As long as strategy is fixed we know there are no problems for any number of splits.

k_c

[ericfarmer]

My intention was to hopefully compute optimal SPL1. I thought that CDZ- referred to optimal pre-split strategy so I thought CDZ+ would refer to optimal post-split strategy.

The earliest discussions of notation that I am familiar with were back in the bjmath.com days, where although the "Z" was consistently established to indicate "zero memory," meaning that strategy is only a function of the current hand, I never saw us settle on notation to refer to perfect EV-maximizing play. (There was discussion back in 2003 with Steve Jacobs proposing "CDB," where the "B" indicated knowledge of cards in all past "branches" of a round, which would effectively mean perfect play in the sense of maximizing EV for the round... but I neither observed nor used this notation myself at any time after that discussion. I've used "OPT" to refer to this in the past in my own internal scribblings; I suppose we haven't really settled on anything because nobody was able to actually compute it, other than ICountNTrack.)

I took it that your original problem was that not only was CDP unplayable at the table but also that the EV computed by CA couldn't be accounted for. I don't know why but was wondering if SPL1 had the same problem except hopefully more simple. As long as strategy is fixed we know there are no problems for any number of splits.

k_c

You are correct that CDP is unplayable at the table with realistic rules. However, if we hypothesize a dealer that fleshes out all split hands to two cards before accepting stand/hit/double decisions on any of them, then the expected return produced by the CDP calculation (in this example, 1548/715) *is* able to be realized/accounted for... it just takes an interestingly complicated playing strategy to realize it. I described this strategy in this earlier comment. (I don't want to give the impression that the strategy described there should be a priori intuitive or at all obvious-- it certainly wasn't to me, and it took quite a bit of experimentation with various strategies to yield an EV that agreed with that produced by CDP.)

At any rate, you're right that "as long as the strategy is fixed we know there are no problems for any number of splits." That is, SPL1 *doesn't* have this same problem, essentially because CDP is the same as CDP1 in that case, and CDP1 does *not* have this problem in general (for SPL3 or SPL1). That is, when we try to describe CDP1 strategy to a player that wants to use it, our laminated card only needs to indicate *whether* the current hand is a result of splitting a particular pair card P. It doesn't need to indicate the additional dependent information like *numbers* of observed pair (or non-pair) cards that are not generally available to the player. (Note also that the fact that CDP1 is "problem-free" has nothing to do with the fact that it is *optimal* in any particular sense; it's problem-free simply because of the limited information on which the strategy depends. For example, it would make sense to define a strategy that plays CDZ- pre-split, but plays mimic-the-dealer post-split. Such a strategy is both computable *and* playable.)

E

[k_c]

The earliest discussions of notation that I am familiar with were back in the bjmath.com days, where although the "Z" was consistently established to indicate "zero memory," meaning that strategy is only a function of the current hand, I never saw us settle on notation to refer to perfect EV-maximizing play. (There was discussion back in 2003 with Steve Jacobs proposing "CDB," where the "B" indicated knowledge of cards in all past "branches" of a round, which would effectively mean perfect play in the sense of maximizing EV for the round... but I neither observed nor used this notation myself at any time after that discussion. I've used "OPT" to refer to this in the past in my own internal scribblings; I suppose we haven't really settled on anything because nobody was able to actually compute it, other than ICountNTrack.)



You are correct that CDP is unplayable at the table with realistic rules. However, if we hypothesize a dealer that fleshes out all split hands to two cards before accepting stand/hit/double decisions on any of them, then the expected return produced by the CDP calculation (in this example, 1548/715) *is* able to be realized/accounted for... it just takes an interestingly complicated playing strategy to realize it. I described this strategy in this earlier comment. (I don't want to give the impression that the strategy described there should be a priori intuitive or at all obvious-- it certainly wasn't to me, and it took quite a bit of experimentation with various strategies to yield an EV that agreed with that produced by CDP.)

At any rate, you're right that "as long as the strategy is fixed we know there are no problems for any number of splits." That is, SPL1 *doesn't* have this same problem, essentially because CDP is the same as CDP1 in that case, and CDP1 does *not* have this problem in general (for SPL3 or SPL1). That is, when we try to describe CDP1 strategy to a player that wants to use it, our laminated card only needs to indicate *whether* the current hand is a result of splitting a particular pair card P. It doesn't need to indicate the additional dependent information like *numbers* of observed pair (or non-pair) cards that are not generally available to the player. (Note also that the fact that CDP1 is "problem-free" has nothing to do with the fact that it is *optimal* in any particular sense; it's problem-free simply because of the limited information on which the strategy depends. For example, it would make sense to define a strategy that plays CDZ- pre-split, but plays mimic-the-dealer post-split. Such a strategy is both computable *and* playable.)

E

I believe that for SPL1 computing hand 2 allowing for a single p or n drawn to hand 1 cannot improve (hand 1 EV + hand 2 EV) to anything more than a fixed strategy for both hand 1 and hand 2.

EV(p) = (EVPair_p + EVx_p) // draw a p card to hand 1
EV(non_p) = (EVn + EVx_n) // draw a non_p card to hand 1
EV(SPL1) = pP*EV(p) + (1-pP)*EV(non_p)

EV(SPL1) = pP*(EVPair_p + EVx_p) + (1-pP)*(EVn + EVx_n)

** EVn = (EVx - pP*EVPair_p) / (1-pP) **
** EVx_n = (EVx - pP*Evx_p) / (1-pP) **

EV(SPL1) = pP*EVPair_p + pP*EVx_p + (EVx - pP*EVx_p) + (EVx - pP*EVPair_p)
EV(SPL1) = 2*EVx

Compared to a fixed strategy where:
EVx_x = EVx
EV(SPL1) = EVx + EVx_x = 2*EVx

I think that in order to begin to see an increase to the fixed strategy EV for SPL1 considering only p's and n's at least 2 cards need to be allocated to hand 1 (pp,pn,np,nn) and considered in hand 2. However, this is also where the hand begins to be unplayable.

I can't prove that this is true but if it is couldn't the same principle apply to the x hands for multiple splits?

k_c