"Fast" optimal split EVs

[ericfarmer]

My EVs match in all cases for the twelve states you listed.

Sincerely,
Cac

I think that's expected-- those last dozen at the end of the list are all terminal states, so that the only way we might differ is if (1) we compute dealer outcome probabilities differently, which is nearly certainly not the case; or (2) you're deciding to hit one of those hands instead of stand, also unlikely given the number of cards consumed.

What I meant to suggest was to *continue* working backward/upward, starting with but continuing beyond those last dozen states, until finding the latest in the list where we differ. That will be a more convenient round for us to focus on-- we *know* we differ in the *first* round's EV, since that's the "root" of the tree of possible playing decisions, but to work out *why* we differ requires exploring that entire subtree. If we instead find the "deepest" state where we differ, then the subtree that we have to work out (probably by hand) to identify differences will be as *shallow* as possible.

[Gramazeka]

ericfarmer,
Have you forgotten about me?

[ericfarmer]

ericfarmer,
Have you forgotten about me?

I haven't forgotten-- it wasn't clear to me that it was useful to respond. You have yet to answer my question, nor Cacarulo's. (Admittedly, you did reply to them, but your reply suggested that your software was incapable of handling even *single deck* analysis, despite your claim that it has "a wider range of options.")

So, I'll repeat my question here for convenience: what is the maximum achievable expected return for splitting 2-2 versus dealer 2 in 4D, H17, DOA, DAS, SPL2?

Granted, you did reply, with a screenshot from your program, with GUI labels in Russian. But this screenshot does not provide an answer to the question-- such an answer will be in the form of a single number, namely, the expected value E(X) of the random variable X indicating the overall outcome of a round after dealing 2-2 vs. 2 to a player who subsequently plays strategy maximizing that E(X).

This expected value E(X) is one number. Not a screenshot. Not a screenshot with labels in Russian. And certainly not a screenshot with not one number, but eighteen numbers.

I can't read the labels in your screenshot, so I'm guessing, but the number of values displayed, along with the consecutive range of corresponding integer values in the first column labeled "S", together very strongly suggest that those integers are true counts. If this is not the case, then I welcome a more detailed explanation of what we're looking at, accompanied by the (single-value) answer to the original question. If, on the other hand, these *are* true counts as I suspect, then that would confirm that you're simply misunderstanding the subject under discussion in this thread.

[Gramazeka]

So, I'll repeat my question here for convenience: what is the maximum achievable expected return for splitting 2-2 versus dealer 2 in 4D, H17, DOA, DAS, SPL2?

Granted, you did reply, with a screenshot from your program, with GUI labels in Russian. But this screenshot does not provide an answer to the question-- such an answer will be in the form of a single number, namely, the expected value E(X) of the random variable X indicating the overall outcome of a round after dealing 2-2 vs. 2 to a player who subsequently plays strategy maximizing that E(X).

I answered. EV = -0.0805.

4D, H17, DOA, DAS, SPL2.

Calculation speed is 20 seconds.

SPL 3 = -0.0798
SPL 1 = -0.0851

[DSchles]

You clearly don't mean that answer. Check the parameters again.

Don

[Gramazeka]

Yes , I'll correct Don )) I'm glad that you understand me...

[Gramazeka]

-1136 ? correct Don ? We have shelling going on, I'm thinking about it... Sorry...

[DSchles]

Yes, that disadvantage should be lower, around -.0755. As for the shelling, can't imagine what you've been through. Praying for peace.

Don

[Gramazeka]

I answered above. I can attach a screenshot.

[DSchles]

Sorry. I never saw your amended answer. But it still seems a little off. See BJA3, p. 438.

Don

[k_c]

My fixed strategy CA computes split EVs efficiently by adapting to removal of generic non-pair cards in a a shoe composition, represented by np and to (generic) pair cards represented by p. I decided to see if I could implement the same methodology for optimal splits.

The fixed strategy algorithm requires the ability to compute EVx (= draw to a single pair card with 0 splits remaining) and EVPair (= draw to 2 unsplit pair cards) for shoe states of (2*allowedSplits-2) pair cards removed from initial shoe comp. Initial shoe comp is defined as shoe composition after 2 pair cards and dealer up card have been dealt. It requires removal of only p cards to compute split EVs.

if we let array pP[2*allowedSplits-2] = prob of drawing a pair card from each of relevant shoe states with (2*allowedSplits-2) pair cards removed,
underlying relationship for a fixed strategy is
EV(non-pair card) = (EVx[2*allowedSplits-2] - pP[2*allowedSplits-2]*EVPair[2*allowedSplits-2]) / (1 - pP[2*allowedSplits-2])
where EVPair[2*allowedSplits-2] refers to shoe state before matching p card is drawn
The same type of relationship applies also to optimal splits except EVx is replaced by optx as described below.

For fixed strategy drawing a matching p card to a single pair card results in one of 2 cases:
1, resplit occurs if more p cards & remaining splits are available
2. generic drawing sequence is concluded when no more p cards and/or splits remain
For optimal splits the value of an unsplit pair may apply when a matching p card is drawn to a single pair card. This is the only difference between fixed and optimal when a generic p card is drawn. Not addressing this difference results in output relative to splitting at every opportunity.

When a generic np card is drawn to a single p there is no difference since EV(non-pair card drawn) is the same whether the value of EVPair is for an unsplit or split pair. A generic np card is never algorithmically removed but is instead computed relative to not removing a p card so in the end only removal of p cards needs to be considered.

Difference in split EV comes from the handling of x hands (where no more splitting is allowed). Instead of drawing to a single pair card with a fixed strategy whose EV for each number of p removed is stored in EVx[pRem], we have optimal EV of drawing to (pRem + 2) p cards whose EV is stored in
optx[pRem].

                    number of hands     p removed     EV
Fixed strategy             n              pRem        n*EVx[pRem]
Optimal strategy           n              pRem        optx[pRem] where pRem = (n-2)

In order to compute SPL1 we need optimal EV of drawing to 2 p cards with 0 splits remaining (optx[0])
In order to compute SPL2 we need optimal EV of drawing to 2,3,&4 p cards with 0 splits remaining (optx[0] to optx[2])
In order to compute SPL3 we need optimal EV of drawing to 2,3,4,5,&6 p cards with 0 splits remaining (optx[0] to optx[4])
In order to compute SPLn we need optimal EV of drawing to 2,3,...,&2*n p cards with 0 splits remaining (optx[0] to optx[2*n-2])

Example: compute optimal SPL2, 6-6 v 6 dealt from shoe comp {0,0,0,0,0,11,0,5,0,0}

Pair cards present in shoe: 8   Non-pair cards present in shoe: 5
Number of allowed splits for which to display data (1 - 9): 2
Allowed splits: 2       Expected number of hands: 2.8717948718 = 112/39

P removed    optx(pRem + 2) multiplier    EVPair(pRem) multiplier

0            1.00000000000                -1.23076923077 = -16/13
1            0.82051282051 = 32/39        0.35897435897 = 14/39
2            -0.17948717949 = -7/39       0.00000000000

Press x to exit, i to reinput p & np, any other key to reinput num splits

optx[0] = -0.013986013986014 = -2/143 = EV(draw to 2 pair cards, 0 splits remain)     EVPair[0] = -1/132
optx[1] = -1/33 = EV(draw to 3 pair cards, 0 splits remain)                           EVPair[1] = 1/33
optx[2] = -2/33 = EV(draw to 4 pair cards, 0 splits remain)

EV(SPL2_forcedResplit) = 1*optx[0] - 16/13*EVPair[0] + 32/39*optx[1] + 14/39*EVPair[1] - 7/39*optx[2]
EV(SPL2_forcedResplit) = -2/143-32/39*1/33+7/39*2/33+16/13*1/132+14/39*1/33 = -0.0077700077700077700077700077700078

The algorithm gets the correct optimal SPL2 value of the simple example for splitting at every opportunity. I only have an algorithm to compute EV of drawing to 2 p cards with 0 splits remaining so I can't check values in general for more than 1 allowed split. For the simple example I am able to determine the needed values.

If it turns out that the algorithm works in general this would mean that computing optimal splits efficiently would only depend upon being able to compute optimal EV of drawing to a string of individual pair cards (with 0 splits remaining) efficiently as computation time of other needed values is negligible.

Regarding further optimization by declining to resplit at some point rather than resplitting at every opportunity: what happens here is that number of expected hands is variable as opposed to fixed as would be the case for forced resplitting. Suppose the rules allowed an unlimited number of resplits. If that was the rule then wouldn't we be obligated to explore all possibilities up to the point where all pair cards are eliminated and wouldn't this amount to splitting at every opportunity? In other words optimizing by declining to resplit may only apply to some limited number of allowed resplits and to find the limitations we'd need to split at every opportunity if rule was unlimited resplits.

k_c

[Cacarulo]

My fixed strategy CA computes split EVs efficiently by adapting to removal of generic non-pair cards in a a shoe composition, represented by np and to (generic) pair cards represented by p. I decided to see if I could implement the same methodology for optimal splits.

The fixed strategy algorithm requires the ability to compute EVx (= draw to a single pair card with 0 splits remaining) and EVPair (= draw to 2 unsplit pair cards) for shoe states of (2*allowedSplits-2) pair cards removed from initial shoe comp. Initial shoe comp is defined as shoe composition after 2 pair cards and dealer up card have been dealt. It requires removal of only p cards to compute split EVs.

if we let array pP[2*allowedSplits-2] = prob of drawing a pair card from each of relevant shoe states with (2*allowedSplits-2) pair cards removed,
underlying relationship for a fixed strategy is
EV(non-pair card) = (EVx[2*allowedSplits-2] - pP[2*allowedSplits-2]*EVPair[2*allowedSplits-2]) / (1 - pP[2*allowedSplits-2])
where EVPair[2*allowedSplits-2] refers to shoe state before matching p card is drawn
The same type of relationship applies also to optimal splits except EVx is replaced by optx as described below.

For fixed strategy drawing a matching p card to a single pair card results in one of 2 cases:
1, resplit occurs if more p cards & remaining splits are available
2. generic drawing sequence is concluded when no more p cards and/or splits remain
For optimal splits the value of an unsplit pair may apply when a matching p card is drawn to a single pair card. This is the only difference between fixed and optimal when a generic p card is drawn. Not addressing this difference results in output relative to splitting at every opportunity.

When a generic np card is drawn to a single p there is no difference since EV(non-pair card drawn) is the same whether the value of EVPair is for an unsplit or split pair. A generic np card is never algorithmically removed but is instead computed relative to not removing a p card so in the end only removal of p cards needs to be considered.

Difference in split EV comes from the handling of x hands (where no more splitting is allowed). Instead of drawing to a single pair card with a fixed strategy whose EV for each number of p removed is stored in EVx[pRem], we have optimal EV of drawing to (pRem + 2) p cards whose EV is stored in
optx[pRem].

                    number of hands     p removed     EV
Fixed strategy             n              pRem        n*EVx[pRem]
Optimal strategy           n              pRem        optx[pRem] where pRem = (n-2)

In order to compute SPL1 we need optimal EV of drawing to 2 p cards with 0 splits remaining (optx[0])
In order to compute SPL2 we need optimal EV of drawing to 2,3,&4 p cards with 0 splits remaining (optx[0] to optx[2])
In order to compute SPL3 we need optimal EV of drawing to 2,3,4,5,&6 p cards with 0 splits remaining (optx[0] to optx[4])
In order to compute SPLn we need optimal EV of drawing to 2,3,...,&2*n p cards with 0 splits remaining (optx[0] to optx[2*n-2])

Example: compute optimal SPL2, 6-6 v 6 dealt from shoe comp {0,0,0,0,0,11,0,5,0,0}

Pair cards present in shoe: 8   Non-pair cards present in shoe: 5
Number of allowed splits for which to display data (1 - 9): 2
Allowed splits: 2       Expected number of hands: 2.8717948718 = 112/39

P removed    optx(pRem + 2) multiplier    EVPair(pRem) multiplier

0            1.00000000000                -1.23076923077 = -16/13
1            0.82051282051 = 32/39        0.35897435897 = 14/39
2            -0.17948717949 = -7/39       0.00000000000

Press x to exit, i to reinput p & np, any other key to reinput num splits

optx[0] = -0.013986013986014 = -2/143 = EV(draw to 2 pair cards, 0 splits remain)     EVPair[0] = -1/132
optx[1] = -1/33 = EV(draw to 3 pair cards, 0 splits remain)                           EVPair[1] = 1/33
optx[2] = -2/33 = EV(draw to 4 pair cards, 0 splits remain)

EV(SPL2_forcedResplit) = 1*optx[0] - 16/13*EVPair[0] + 32/39*optx[1] + 14/39*EVPair[1] - 7/39*optx[2]
EV(SPL2_forcedResplit) = -2/143-32/39*1/33+7/39*2/33+16/13*1/132+14/39*1/33 = -0.0077700077700077700077700077700078

The algorithm gets the correct optimal SPL2 value of the simple example for splitting at every opportunity. I only have an algorithm to compute EV of drawing to 2 p cards with 0 splits remaining so I can't check values in general for more than 1 allowed split. For the simple example I am able to determine the needed values.

If it turns out that the algorithm works in general this would mean that computing optimal splits efficiently would only depend upon being able to compute optimal EV of drawing to a string of individual pair cards (with 0 splits remaining) efficiently as computation time of other needed values is negligible.

Regarding further optimization by declining to resplit at some point rather than resplitting at every opportunity: what happens here is that number of expected hands is variable as opposed to fixed as would be the case for forced resplitting. Suppose the rules allowed an unlimited number of resplits. If that was the rule then wouldn't we be obligated to explore all possibilities up to the point where all pair cards are eliminated and wouldn't this amount to splitting at every opportunity? In other words optimizing by declining to resplit may only apply to some limited number of allowed resplits and to find the limitations we'd need to split at every opportunity if rule was unlimited resplits.

k_c

Hi k_c,

The truth is I've been distracted with other matters and haven't been able to dedicate more time to this thread. That doesn't mean I won't come back to it at some point :-)
I see that at least now there are two of us calculating optimal splits using the method of "forced-resplits".

If you continue with this method and manage to calculate SPL3, you should get the following EV:
SPL3: 0.004662004662005

Sincerely,
Cac

[ericfarmer]

The algorithm gets the correct optimal SPL2 value of the simple example for splitting at every opportunity. I only have an algorithm to compute EV of drawing to 2 p cards with 0 splits remaining so I can't check values in general for more than 1 allowed split. For the simple example I am able to determine the needed values.

If it turns out that the algorithm works in general this would mean that computing optimal splits efficiently would only depend upon being able to compute optimal EV of drawing to a string of individual pair cards (with 0 splits remaining) efficiently as computation time of other needed values is negligible.

This sounds really interesting. I don't fully understand your description yet, will need to read through this some more. In the meantime, I've updated my blackjack_split.exe (the latest release at the previously posted GitHub link) to support the option of forcing resplits at every opportunity, but otherwise making optimal (EV-maximizing) strategy decisions, if it's helpful to check against your potentially much more efficient algorithm.

I would be particularly interested to see what this algorithm computes for the SPL3 case, since I get a different value than Cacarulo's. Following is my output, with inputs if you want to reproduce/experiment yourself:

Memory allocation exponent (29=12GB, 30=24GB, etc.): 29
European no hole card rule (1=yes, 0=no): 0
Dealer hits soft 17 (1=yes, 0=no): 0
Double down after split (1=yes, 0=no): 0
Maximum number of splits (3=SPL3): 3
Re-split aces (1=yes, 0=no): 0
Force re-splits (1=yes, 0=no): 1
Shoe (ace through ten, -1 to exit): 0 0 0 0 0 11 0 5 0 0
Pair card: 6
Dealer up card: 6
Output count of all states (1=yes, 0=no): 0
Output count of optimal states (1=yes, 0=no): 1
Output list of optimal states (1=yes, 0=no): 1

Split 6 vs. 6:
    Searched 871 states to depth 17, 0% capacity.
    Solving depth 0, 100% complete.
    Elapsed time 1.432 seconds.

E(split)= 0.0062160062160064051

148 states reachable using optimal strategy.

cards=[1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0], hands=[6, 6, 0, 0], current=0, E(hit)=0.0062160062160064051
cards=[2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0], hands=[12, 6, 0, 0], current=0, E(split)=-0.026515151515151353
cards=[2, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0], hands=[14, 6, 0, 0], current=0, E(hit)=0.058585858585858817
cards=[1, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0], hands=[22, 6, 0, 0], current=1, E(hit)=-0.8484848484848484
cards=[1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0], hands=[6, 6, 6, 0], current=0, E(hit)=-0.026515151515151353
cards=[3, 0, 0, 0, 0, 0, 4, 0, 1, 0, 0], hands=[20, 6, 0, 0], current=0, E(stand)=0.51212121212121242
cards=[2, 0, 0, 0, 0, 0, 4, 0, 1, 0, 0], hands=[14, 6, 6, 0], current=0, E(hit)=-0.027272727272727094
cards=[1, 0, 0, 0, 0, 0, 4, 0, 1, 0, 0], hands=[20, 6, 0, 0], current=1, E(hit)=0.51212121212121242
cards=[2, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0], hands=[12, 6, 6, 0], current=0, E(split)=-0.02597402597402582
cards=[2, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0], hands=[22, 14, 0, 0], current=1, E(hit)=-0.71111111111111103
cards=[2, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0], hands=[22, 12, 0, 0], current=1, E(split)=-0.90000000000000002
cards=[3, 0, 0, 0, 0, 0, 4, 0, 3, 0, 0], hands=[22, 20, 0, 0], current=1, E(stand)=-0.3888888888888889
cards=[2, 0, 0, 0, 0, 0, 5, 0, 1, 0, 0], hands=[20, 12, 0, 0], current=1, E(split)=0.4714285714285717
cards=[3, 0, 0, 0, 0, 0, 5, 0, 1, 0, 0], hands=[20, 6, 6, 0], current=0, E(stand)=0.4714285714285717
cards=[2, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0], hands=[20, 14, 0, 0], current=1, E(hit)=0.58333333333333348
cards=[1, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0], hands=[22, 6, 6, 0], current=1, E(hit)=-0.90000000000000002
cards=[1, 0, 0, 0, 0, 0, 3, 0, 4, 0, 0], hands=[22, 22, 0, 0], current=2, E(dealer)=-2
cards=[1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0], hands=[6, 6, 6, 6], current=0, E(hit)=-0.02597402597402582
cards=[2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0], hands=[12, 6, 6, 6], current=0, E(hit)=0.067460317460317623
cards=[3, 0, 0, 0, 0, 0, 5, 0, 2, 0, 0], hands=[20, 20, 0, 0], current=1, E(stand)=1.0000000000000002
cards=[2, 0, 0, 0, 0, 0, 5, 0, 1, 0, 0], hands=[14, 6, 6, 6], current=0, E(hit)=-0.13809523809523797
cards=[1, 0, 0, 0, 0, 0, 4, 0, 3, 0, 0], hands=[20, 22, 0, 0], current=2, E(dealer)=-0.3888888888888889
cards=[1, 0, 0, 0, 0, 0, 5, 0, 1, 0, 0], hands=[20, 6, 6, 0], current=1, E(hit)=0.4714285714285717
cards=[2, 0, 0, 0, 0, 0, 4, 0, 3, 0, 0], hands=[22, 14, 6, 0], current=1, E(hit)=-0.66666666666666652
cards=[2, 0, 0, 0, 0, 0, 5, 0, 2, 0, 0], hands=[22, 12, 6, 0], current=1, E(split)=-1
cards=[3, 0, 0, 0, 0, 0, 6, 0, 1, 0, 0], hands=[20, 6, 6, 6], current=0, E(stand)=0.43650793650793673
cards=[3, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0], hands=[18, 6, 6, 6], current=0, E(stand)=-0.30158730158730152
cards=[1, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0], hands=[22, 22, 6, 0], current=2, E(hit)=-1.875
cards=[2, 0, 0, 0, 0, 0, 5, 0, 2, 0, 0], hands=[20, 14, 6, 0], current=1, E(hit)=0.52380952380952417
cards=[2, 0, 0, 0, 0, 0, 6, 0, 1, 0, 0], hands=[20, 12, 6, 0], current=1, E(split)=0.43650793650793673
cards=[1, 0, 0, 0, 0, 0, 5, 0, 2, 0, 0], hands=[20, 20, 0, 0], current=2, E(dealer)=1.0000000000000002
cards=[1, 0, 0, 0, 0, 0, 5, 0, 2, 0, 0], hands=[22, 6, 6, 6], current=1, E(hit)=-1
cards=[3, 0, 0, 0, 0, 0, 5, 0, 3, 0, 0], hands=[22, 20, 6, 0], current=1, E(stand)=-0.32142857142857129
cards=[2, 0, 0, 0, 0, 0, 5, 0, 4, 0, 0], hands=[22, 22, 12, 0], current=2, E(split)=-2
cards=[2, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0], hands=[22, 12, 6, 6], current=1, E(hit)=-1.0714285714285714
cards=[1, 0, 0, 0, 0, 0, 5, 0, 3, 0, 0], hands=[20, 22, 6, 0], current=2, E(hit)=-0.32142857142857129
cards=[2, 0, 0, 0, 0, 0, 5, 0, 3, 0, 0], hands=[22, 14, 6, 6], current=1, E(hit)=-0.85714285714285698
cards=[1, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0], hands=[18, 6, 6, 6], current=1, E(hit)=-0.30158730158730152
cards=[3, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0], hands=[20, 20, 6, 0], current=1, E(stand)=0.94642857142857173
cards=[2, 0, 0, 0, 0, 0, 4, 0, 5, 0, 0], hands=[22, 22, 14, 0], current=2, E(hit)=-1
cards=[1, 0, 0, 0, 0, 0, 6, 0, 1, 0, 0], hands=[20, 6, 6, 6], current=1, E(hit)=0.43650793650793673
cards=[2, 0, 0, 0, 0, 0, 6, 0, 3, 0, 0], hands=[20, 22, 12, 0], current=2, E(split)=-0.476190476190476
cards=[3, 0, 0, 0, 0, 0, 6, 0, 3, 0, 0], hands=[22, 20, 6, 6], current=1, E(stand)=-0.476190476190476
cards=[2, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0], hands=[18, 12, 6, 6], current=1, E(hit)=0.12500000000000008
cards=[1, 0, 0, 0, 0, 0, 5, 0, 4, 0, 0], hands=[22, 22, 6, 6], current=2, E(hit)=-2
cards=[3, 0, 0, 0, 0, 0, 5, 0, 5, 0, 0], hands=[22, 22, 20, 0], current=2, E(stand)=-1
cards=[3, 0, 0, 0, 0, 0, 7, 0, 2, 0, 0], hands=[22, 18, 6, 6], current=1, E(stand)=-1.4285714285714286
cards=[2, 0, 0, 0, 0, 0, 7, 0, 1, 0, 0], hands=[20, 12, 6, 6], current=1, E(hit)=0.50000000000000022
cards=[2, 0, 0, 0, 0, 0, 5, 0, 4, 0, 0], hands=[20, 22, 14, 0], current=2, E(hit)=0.14285714285714285
cards=[2, 0, 0, 0, 0, 0, 7, 0, 1, 0, 0], hands=[18, 14, 6, 6], current=1, E(hit)=-0.64285714285714279
cards=[1, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0], hands=[20, 20, 6, 0], current=2, E(hit)=0.94642857142857173
cards=[2, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0], hands=[20, 14, 6, 6], current=1, E(hit)=0.35714285714285743
cards=[2, 0, 0, 0, 0, 0, 5, 0, 5, 0, 0], hands=[22, 22, 14, 6], current=2, E(hit)=-1
cards=[3, 0, 0, 0, 0, 0, 8, 0, 1, 0, 0], hands=[20, 18, 6, 6], current=1, E(stand)=0.14285714285714296
cards=[2, 0, 0, 0, 0, 0, 6, 0, 3, 0, 0], hands=[20, 20, 14, 0], current=2, E(hit)=1.0952380952380956
cards=[1, 0, 0, 0, 0, 0, 6, 0, 3, 0, 0], hands=[20, 22, 6, 6], current=2, E(hit)=-0.476190476190476
cards=[1, 0, 0, 0, 0, 0, 7, 0, 2, 0, 0], hands=[18, 22, 6, 6], current=2, E(hit)=-1.4285714285714286
cards=[3, 0, 0, 0, 0, 0, 8, 0, 1, 0, 0], hands=[18, 20, 6, 6], current=1, E(stand)=0.14285714285714296
cards=[2, 0, 0, 0, 0, 0, 6, 0, 4, 0, 0], hands=[22, 22, 12, 6], current=2, E(hit)=-2.1666666666666665
cards=[3, 0, 0, 0, 0, 0, 6, 0, 4, 0, 0], hands=[20, 22, 20, 0], current=2, E(stand)=0.33333333333333337
cards=[3, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0], hands=[18, 18, 6, 6], current=1, E(stand)=0.095238095238095274
cards=[2, 0, 0, 0, 0, 0, 7, 0, 2, 0, 0], hands=[20, 20, 12, 0], current=2, E(split)=0.85714285714285743
cards=[1, 0, 0, 0, 0, 0, 5, 0, 5, 0, 0], hands=[20, 22, 22, 0], current=3, E(dealer)=-1
cards=[3, 0, 0, 0, 0, 0, 7, 0, 2, 0, 0], hands=[20, 20, 6, 6], current=1, E(stand)=0.85714285714285743
cards=[3, 0, 0, 0, 0, 0, 7, 0, 4, 0, 0], hands=[22, 22, 18, 6], current=2, E(stand)=-2.3999999999999999
cards=[3, 0, 0, 0, 0, 0, 6, 0, 5, 0, 0], hands=[22, 22, 20, 6], current=2, E(stand)=-1
cards=[3, 0, 0, 0, 0, 0, 7, 0, 3, 0, 0], hands=[20, 20, 20, 0], current=2, E(stand)=1.4000000000000004
cards=[1, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0], hands=[18, 18, 6, 6], current=2, E(hit)=0.095238095238095274
cards=[2, 0, 0, 0, 0, 0, 7, 0, 3, 0, 0], hands=[18, 22, 14, 6], current=2, E(hit)=-1.5333333333333332
cards=[2, 0, 0, 0, 0, 0, 6, 0, 4, 0, 0], hands=[20, 22, 14, 6], current=2, E(hit)=1.4802973661668753e-16
cards=[1, 0, 0, 0, 0, 0, 8, 0, 1, 0, 0], hands=[18, 20, 6, 6], current=2, E(hit)=0.14285714285714296
cards=[2, 0, 0, 0, 0, 0, 7, 0, 3, 0, 0], hands=[20, 22, 12, 6], current=2, E(hit)=-0.66666666666666652
cards=[1, 0, 0, 0, 0, 0, 7, 0, 2, 0, 0], hands=[20, 20, 6, 6], current=2, E(hit)=0.85714285714285743
cards=[2, 0, 0, 0, 0, 0, 8, 0, 2, 0, 0], hands=[18, 22, 12, 6], current=2, E(hit)=-1.3499999999999999
cards=[1, 0, 0, 0, 0, 0, 6, 0, 4, 0, 0], hands=[20, 20, 22, 0], current=3, E(dealer)=0.33333333333333337
cards=[1, 0, 0, 0, 0, 0, 7, 0, 4, 0, 0], hands=[18, 22, 22, 6], current=3, E(hit)=-2.3999999999999999
cards=[3, 0, 0, 0, 0, 0, 9, 0, 2, 0, 0], hands=[18, 22, 18, 6], current=2, E(stand)=-1.5999999999999999
cards=[2, 0, 0, 0, 0, 0, 9, 0, 1, 0, 0], hands=[18, 20, 12, 6], current=2, E(hit)=0.73333333333333339
cards=[2, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0], hands=[18, 18, 12, 6], current=2, E(hit)=1.6666666666666667
cards=[2, 0, 0, 0, 0, 0, 8, 0, 2, 0, 0], hands=[18, 20, 14, 6], current=2, E(hit)=-0.29999999999999988
cards=[3, 0, 0, 0, 0, 0, 8, 0, 3, 0, 0], hands=[18, 22, 20, 6], current=2, E(stand)=-1.0999999999999999
cards=[2, 0, 0, 0, 0, 0, 9, 0, 1, 0, 0], hands=[18, 18, 14, 6], current=2, E(stand)=-0.53333333333333333
cards=[1, 0, 0, 0, 0, 0, 6, 0, 5, 0, 0], hands=[20, 22, 22, 6], current=3, E(hit)=-1
cards=[2, 0, 0, 0, 0, 0, 7, 0, 3, 0, 0], hands=[20, 20, 14, 6], current=2, E(hit)=0.86666666666666703
cards=[2, 0, 0, 0, 0, 0, 8, 0, 2, 0, 0], hands=[20, 20, 12, 6], current=2, E(hit)=0.8500000000000002
cards=[3, 0, 0, 0, 0, 0, 8, 0, 3, 0, 0], hands=[20, 22, 18, 6], current=2, E(stand)=-1.0999999999999999
cards=[3, 0, 0, 0, 0, 0, 7, 0, 4, 0, 0], hands=[20, 22, 20, 6], current=2, E(stand)=0.20000000000000018
cards=[1, 0, 0, 0, 0, 0, 7, 0, 3, 0, 0], hands=[20, 20, 20, 0], current=3, E(dealer)=1.4000000000000004
cards=[3, 0, 0, 0, 0, 0, 10, 0, 1, 0, 0], hands=[18, 20, 18, 6], current=2, E(stand)=1.2
cards=[2, 0, 0, 0, 0, 0, 8, 0, 4, 0, 0], hands=[18, 22, 22, 12], current=3, E(hit)=-2.75
cards=[3, 0, 0, 0, 0, 0, 8, 0, 3, 0, 0], hands=[20, 20, 20, 6], current=2, E(stand)=1.2000000000000004
cards=[3, 0, 0, 0, 0, 0, 9, 0, 2, 0, 0], hands=[20, 20, 18, 6], current=2, E(stand)=0.50000000000000011
cards=[3, 0, 0, 0, 0, 0, 10, 0, 1, 0, 0], hands=[18, 18, 20, 6], current=2, E(stand)=1.2
cards=[1, 0, 0, 0, 0, 0, 7, 0, 4, 0, 0], hands=[20, 20, 22, 6], current=3, E(hit)=0.20000000000000018
cards=[1, 0, 0, 0, 0, 0, 9, 0, 2, 0, 0], hands=[18, 18, 22, 6], current=3, E(hit)=-1.5999999999999999
cards=[1, 0, 0, 0, 0, 0, 9, 0, 1, 0, 0], hands=[16, 18, 18, 6], current=3, E(hit)=-0.53333333333333333
cards=[3, 0, 0, 0, 0, 0, 9, 0, 2, 0, 0], hands=[18, 20, 20, 6], current=2, E(stand)=0.50000000000000011
cards=[3, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0], hands=[18, 18, 18, 6], current=2, E(stand)=4
cards=[2, 0, 0, 0, 0, 0, 7, 0, 5, 0, 0], hands=[18, 22, 22, 14], current=3, E(hit)=-1
cards=[2, 0, 0, 0, 0, 0, 7, 0, 5, 0, 0], hands=[20, 22, 22, 12], current=3, E(hit)=-1
cards=[1, 0, 0, 0, 0, 0, 8, 0, 3, 0, 0], hands=[18, 20, 22, 6], current=3, E(hit)=-1.0999999999999999
cards=[2, 0, 0, 0, 0, 0, 8, 0, 4, 0, 0], hands=[18, 20, 22, 14], current=3, E(hit)=-0.99999999999999978
cards=[3, 0, 0, 0, 0, 0, 8, 0, 5, 0, 0], hands=[18, 22, 22, 20], current=3, E(stand)=-1
cards=[1, 0, 0, 0, 0, 0, 10, 0, 1, 0, 0], hands=[18, 18, 20, 6], current=3, E(hit)=1.2
cards=[1, 0, 0, 0, 0, 0, 9, 0, 2, 0, 0], hands=[18, 20, 20, 6], current=3, E(hit)=0.50000000000000011
cards=[2, 0, 0, 0, 0, 0, 10, 0, 1, 0, 0], hands=[16, 18, 18, 12], current=3, E(hit)=1.2
cards=[1, 0, 0, 0, 0, 0, 8, 0, 3, 0, 0], hands=[20, 20, 20, 6], current=3, E(hit)=1.2000000000000004
cards=[1, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0], hands=[18, 18, 18, 6], current=3, E(hit)=4
cards=[3, 0, 0, 0, 0, 0, 9, 0, 4, 0, 0], hands=[18, 22, 22, 18], current=3, E(stand)=-3.333333333333333
cards=[3, 0, 0, 0, 0, 0, 8, 0, 5, 0, 0], hands=[20, 22, 22, 18], current=3, E(stand)=-1
cards=[2, 0, 0, 0, 0, 0, 9, 0, 3, 0, 0], hands=[18, 20, 22, 12], current=3, E(hit)=-1.1666666666666665
cards=[2, 0, 0, 0, 0, 0, 8, 0, 4, 0, 0], hands=[20, 20, 22, 12], current=3, E(hit)=-0.24999999999999978
cards=[2, 0, 0, 0, 0, 0, 10, 0, 2, 0, 0], hands=[18, 18, 22, 12], current=3, E(hit)=-0.5
cards=[2, 0, 0, 0, 0, 0, 9, 0, 2, 0, 0], hands=[16, 18, 18, 14], current=3, E(hit)=-1.3999999999999999
cards=[2, 0, 0, 0, 0, 0, 9, 0, 3, 0, 0], hands=[18, 18, 22, 14], current=3, E(hit)=-2.333333333333333
cards=[2, 0, 0, 0, 0, 0, 7, 0, 5, 0, 0], hands=[20, 20, 22, 14], current=3, E(hit)=2
cards=[1, 0, 0, 0, 0, 0, 8, 0, 5, 0, 0], hands=[18, 20, 22, 22], current=4, E(dealer)=-1
cards=[1, 0, 0, 0, 0, 0, 9, 0, 4, 0, 0], hands=[18, 18, 22, 22], current=4, E(dealer)=-3.333333333333333
cards=[3, 0, 0, 0, 0, 0, 11, 0, 2, 0, 0], hands=[18, 18, 22, 18], current=3, E(stand)=2
cards=[3, 0, 0, 0, 0, 0, 9, 0, 4, 0, 0], hands=[18, 20, 22, 20], current=3, E(stand)=-0.99999999999999978
cards=[2, 0, 0, 0, 0, 0, 11, 0, 1, 0, 0], hands=[18, 18, 18, 14], current=3, E(stand)=4
cards=[2, 0, 0, 0, 0, 0, 10, 0, 2, 0, 0], hands=[18, 18, 20, 14], current=3, E(stand)=0.5
cards=[2, 0, 0, 0, 0, 0, 10, 0, 2, 0, 0], hands=[18, 20, 20, 12], current=3, E(hit)=1.5
cards=[2, 0, 0, 0, 0, 0, 8, 0, 4, 0, 0], hands=[20, 20, 20, 14], current=3, E(hit)=1.5000000000000002
cards=[3, 0, 0, 0, 0, 0, 10, 0, 3, 0, 0], hands=[18, 20, 22, 18], current=3, E(stand)=-1.3333333333333333
cards=[3, 0, 0, 0, 0, 0, 8, 0, 5, 0, 0], hands=[20, 20, 22, 20], current=3, E(stand)=2
cards=[2, 0, 0, 0, 0, 0, 9, 0, 3, 0, 0], hands=[18, 20, 20, 14], current=3, E(hit)=-0.16666666666666646
cards=[2, 0, 0, 0, 0, 0, 11, 0, 1, 0, 0], hands=[18, 18, 20, 12], current=3, E(stand)=4
cards=[1, 0, 0, 0, 0, 0, 9, 0, 3, 0, 0], hands=[16, 18, 18, 22], current=4, E(dealer)=-2.6666666666666665
cards=[3, 0, 0, 0, 0, 0, 10, 0, 3, 0, 0], hands=[18, 18, 22, 20], current=3, E(stand)=-1.3333333333333333
cards=[2, 0, 0, 0, 0, 0, 9, 0, 3, 0, 0], hands=[20, 20, 20, 12], current=3, E(hit)=1.0000000000000002
cards=[3, 0, 0, 0, 0, 0, 9, 0, 4, 0, 0], hands=[20, 20, 22, 18], current=3, E(stand)=-0.99999999999999978
cards=[3, 0, 0, 0, 0, 0, 11, 0, 1, 0, 0], hands=[16, 18, 18, 18], current=3, E(stand)=4
cards=[3, 0, 0, 0, 0, 0, 10, 0, 2, 0, 0], hands=[16, 18, 18, 20], current=3, E(stand)=0.5
cards=[1, 0, 0, 0, 0, 0, 11, 0, 2, 0, 0], hands=[18, 18, 18, 22], current=4, E(dealer)=2
cards=[1, 0, 0, 0, 0, 0, 11, 0, 1, 0, 0], hands=[16, 18, 18, 18], current=4, E(dealer)=4
cards=[1, 0, 0, 0, 0, 0, 8, 0, 5, 0, 0], hands=[20, 20, 20, 22], current=4, E(dealer)=2
cards=[3, 0, 0, 0, 0, 0, 9, 0, 4, 0, 0], hands=[20, 20, 20, 20], current=3, E(stand)=1.3333333333333337
cards=[3, 0, 0, 0, 0, 0, 10, 0, 3, 0, 0], hands=[18, 20, 20, 20], current=3, E(stand)=0.66666666666666685
cards=[1, 0, 0, 0, 0, 0, 9, 0, 4, 0, 0], hands=[18, 20, 20, 22], current=4, E(dealer)=-0.99999999999999978
cards=[1, 0, 0, 0, 0, 0, 11, 0, 1, 0, 0], hands=[16, 18, 18, 20], current=4, E(dealer)=4
cards=[3, 0, 0, 0, 0, 0, 11, 0, 2, 0, 0], hands=[18, 20, 20, 18], current=3, E(stand)=4
cards=[1, 0, 0, 0, 0, 0, 10, 0, 3, 0, 0], hands=[18, 18, 20, 22], current=4, E(dealer)=-1.3333333333333333
cards=[1, 0, 0, 0, 0, 0, 10, 0, 2, 0, 0], hands=[16, 18, 18, 20], current=4, E(dealer)=0.5
cards=[3, 0, 0, 0, 0, 0, 10, 0, 3, 0, 0], hands=[20, 20, 20, 18], current=3, E(stand)=0.66666666666666685
cards=[1, 0, 0, 0, 0, 0, 11, 0, 2, 0, 0], hands=[18, 18, 20, 20], current=4, E(dealer)=4
cards=[1, 0, 0, 0, 0, 0, 10, 0, 3, 0, 0], hands=[18, 20, 20, 20], current=4, E(dealer)=0.66666666666666685
cards=[1, 0, 0, 0, 0, 0, 9, 0, 4, 0, 0], hands=[20, 20, 20, 20], current=4, E(dealer)=1.3333333333333337
========================================