Proving CA algorithms incorrect (or at least inexact)

**Cacarulo** · 12-02-2024, 09:47 AM

Originally Posted by iCountNTrack

Hi Cac, you are exactly right for SPL1 and SPL2 the CA returns the same value because splitting is not optimal. My algorithm would take that walk of re-splitting compute the EV compare it to other playing decisions and disregard it. As far as your other ask, unfortunately I cannot compute overall EVs, while it's faster than Eric's python implementation, it's still really slow and can only compute specific player hand against a dealer upcard.

Hi ICNT,

There's a small difference I still can't reconcile. For 8,8 vs. 8, my EV is slightly lower than yours and the maximum calculated by Eric. Your EV is 1.009777407946744, while mine is 1.0093134195637896. I'll keep looking to figure out where this difference might come from.

Sincerely,
Cac

**Cacarulo** · 12-02-2024, 10:31 AM

Originally Posted by ericfarmer

Yeah, I tried to make the gist reasonably machine-readable, the idea being to automate evaluation of each of those split EVs-- with an eye toward comparison with the E[X_max] value in each case, searching for similar exceedances like I did in finding this particular one with E[X_CDP]. Let me know if a different format is easier to work with, or if I can otherwise help.

Maybe a more convenient import format would be a space-separated text file, and instead of fractions, I’d prefer decimal numbers. For example:
0 0 0 0 0 14 0 13 0 0 8 8 1.0108481503689 1.0097774079467

I should maybe clarify what's easy and what's hard here. The E[X_max] values, i.e. the truly maximum EV among all possible strategies, computed using either my recent Python code or iCountNTrack's recent fixes to his code, indeed take a long time to evaluate. But CDP, on the other hand, is just as fast as CDP1, CDZ-, mimic-the-dealer, etc., so that I can evaluate the entire CDP strategy and overall expected return for an arbitrary shoe in a fraction of a second, whether for multiple decks, SPL3, etc.

Agree that for most players here, the details of this "theoretical" discussion are likely of little interest. But I do think there are practical motivations here: for example, several years ago Gronbog did some really interesting analysis of multi-parameter counting systems: imagine a spectrum of possible win rates, ranging from conventional hi-lo or HO2-ish single-parameter systems at the lower end, and the "bring-a-laptop-to-the-table maximum EV" speed of light at the upper end-- where do those multi-parameter systems live in between?

If that analysis had been done by evaluating the upper bound performance using CDP for each round's depleted-shoe, instead of CDZ- (as it actually was, albeit for other reasons), we now know that the "speed of light" performance upper bound could have looked more distant, and incorrectly so, so that perhaps the takeaway from that analysis might have been different, with possibly less accurate or downright incorrect conclusions.

Certainly, there are numerous practical motivations behind this theoretical discussion, and we are in full agreement on that. The objective, of course, is to identify the optimal solution and achieve it within a reasonable timeframe.

Sincerely,
Cac

**ericfarmer** · 12-02-2024, 12:14 PM

Originally Posted by Cacarulo

Hi ICNT,

There's a small difference I still can't reconcile. For 8,8 vs. 8, my EV is slightly lower than yours and the maximum calculated by Eric. Your EV is 1.009777407946744, while mine is 1.0093134195637896. I'll keep looking to figure out where this difference might come from.

Sincerely,
Cac

Optimal strategy (after splitting 8-8) is to almost always stand on 14, except double down in the following few situations (and note even some of these are not reachable via optimal strategy):

Code:

double on 86 vs dealer 8:
    868  88   86#  8    
    868  868  88   86#  
    868  88   88   86#  
    88   88   868  86#  
    88   868  86#  8    
    88   868  88   86#  
    868  88   868  86#  
    88   868  868  86#  
    88   88   88   86#  
    88   88   86#  8

Maybe one of these are the difference?

**ericfarmer** · 12-02-2024, 12:25 PM

Originally Posted by Cacarulo

Maybe a more convenient import format would be a space-separated text file, and instead of fractions, I’d prefer decimal numbers. For example:
0 0 0 0 0 14 0 13 0 0 8 8 1.0108481503689 1.0097774079467

I've put this format here.

**ericfarmer** · 12-02-2024, 12:33 PM

Originally Posted by iCountNTrack

The following is the link to my old CDCA that I built in 2011, this is the console version. I unfortunately cannot find the source for the UI. But this will do. This has the bug fix where playing decisions return the same EV discovered earlier. Please dont ask me for the source code. It's really embarrassing, it would be the perfect example of spaghetti code. To my defense it was the first full blown full coding experience for me but I am sure that the code works well. This is the CA that uses my "locally optimum" algorithm and ZERO recursion.

https://drive.google.com/file/d/13Ey...ew?usp=sharing

Thanks for this, stdin/out makes automation a lot easier. I ran this through the gist of small (shoe, split hand) scenarios, and compared each output with the corresponding E[X_max] values. The following figure shows a scatter plot of results, organized (on the x-axis) by the total number of cards in the depleted shoe:

cdca_comparison.jpg

Originally Posted by iCountNTrack

The following is the source for my totally new implementation using brute force. I do use a memoization map to avoid re-calculating outcomes for the same split hands. The code uses recursion but it's really simple. It matches my older CA results and Eric's python code for almost everything except a couple of deck compositions and plays where it's yielding higher values. I cant seem to figure out whether it's a bug or real.

What compiler (version) are you using? I'm getting compiler errors with both MSVS and g++ (w64devkit).

Thanks,
E

**iCountNTrack** · 12-02-2024, 03:38 PM

Originally Posted by ericfarmer

Thanks for this, stdin/out makes automation a lot easier. I ran this through the gist of small (shoe, split hand) scenarios, and compared each output with the corresponding E[X_max] values. The following figure shows a scatter plot of results, organized (on the x-axis) by the total number of cards in the depleted shoe:

cdca_comparison.jpg

E

Nice work! Did you use Igor Pro to plot this? Do you have the results in a table including the deck composition used? I had hoped our results would be less scattered

Originally Posted by ericfarmer

Thanks for this, stdin/out makes automation a lot easier. I ran this through the gist of small (shoe, split hand) scenarios, and compared each output with the corresponding E[X_max] values. The following figure shows a scatter plot of results, organized (on the x-axis) by the total number of cards in the depleted shoe:

E

I am using VS 2022. I suspect you are using the default C++14 standard, you will need to use the C++17 or C++20 (I am using C++20 standard)

**iCountNTrack** · 12-02-2024, 06:49 PM

Originally Posted by Cacarulo

Hi ICNT,

There's a small difference I still can't reconcile. For 8,8 vs. 8, my EV is slightly lower than yours and the maximum calculated by Eric. Your EV is 1.009777407946744, while mine is 1.0093134195637896. I'll keep looking to figure out where this difference might come from.

Sincerely,
Cac

Hi Cac, please see below for the final states for all the combinations of split hands (last column indicates the multiplicity) used in the EV calculation. You can see that there are certain doubled hands which are impacting the results in your code.

Code:

Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0)	1
Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0)  	3
Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0)	10
Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0)	10
Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0)	10
Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0)	10
Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0)	7
Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0)	7
Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0)	7
Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0)	7
Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0)	4
Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0)	4
Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0)	4
Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0)	4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,6 (dblChk: 1) || Hand:8,6 (dblChk: 0)	4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,6 (dblChk: 1) || Hand:8,8 (dblChk: 0)	4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,8 (dblChk: 1) || Hand:8,6,6 (dblChk: 1)	4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,8 (dblChk: 1) || Hand:8,6,8 (dblChk: 1)	4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,8 (dblChk: 1) || Hand:8,8 (dblChk: 0)	4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,6 (dblChk: 1)	4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,8 (dblChk: 1)	4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0)	4

**Cacarulo** · 12-03-2024, 02:59 AM

Originally Posted by ericfarmer

Optimal strategy (after splitting 8-8) is to almost always stand on 14, except double down in the following few situations (and note even some of these are not reachable via optimal strategy):

I don’t quite understand; what does it mean that some are not reachable via optimal strategy?

Code:

double on 86 vs dealer 8:
    868  88   86#  8    
    868  868  88   86#  
    868  88   88   86#  
    88   88   868  86#  
    88   868  86#  8    
    88   868  88   86#  
    868  88   868  86#  
    88   868  868  86#  
    88   88   88   86#  
    88   88   86#  8

Maybe one of these are the difference?

I’ll review the code later to see if I can find the differences. Thanks.

Sincerely,
Cac

**Cacarulo** · 12-03-2024, 03:02 AM

Originally Posted by iCountNTrack

Hi Cac, please see below for the final states for all the combinations of split hands (last column indicates the multiplicity) used in the EV calculation. You can see that there are certain doubled hands which are impacting the results in your code.

Code:

Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0)    1
Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0)      3
Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0)    10
Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0)    10
Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0)    10
Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0)    10
Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0)    7
Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0)    7
Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0)    7
Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0)    7
Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0)    4
Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0)    4
Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0)    4
Hand:8,8 (dblChk: 0) || Hand:8,6 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0)    4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,6 (dblChk: 1) || Hand:8,6 (dblChk: 0)    4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,6 (dblChk: 1) || Hand:8,8 (dblChk: 0)    4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,8 (dblChk: 1) || Hand:8,6,6 (dblChk: 1)    4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,8 (dblChk: 1) || Hand:8,6,8 (dblChk: 1)    4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,8 (dblChk: 1) || Hand:8,8 (dblChk: 0)    4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,6 (dblChk: 1)    4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,6,8 (dblChk: 1)    4
Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0) || Hand:8,8 (dblChk: 0)    4

Thanks ICNT. Later, I will review the code to see where the difference is coming from.

Sincerely,
Cac

**Cacarulo** · 12-03-2024, 03:10 AM

Originally Posted by ericfarmer

I've put this format here.

Thank you! In a while, I will share some examples from the list that I found very interesting.

Sincerely,
Cac

**Cacarulo** · 12-03-2024, 10:24 AM

Gentlemen, here are some interesting examples I’d like you to recalculate, especially the maximum expected value:

0 0 0 0 0 4 7 0 0 0 | 7, 7 vs 6 |
0 0 0 0 0 4 7 0 0 0 | 6, 6 vs 6 |
0 0 0 0 0 6 4 0 0 0 | 7, 7 vs 6 |
0 0 0 0 0 7 5 0 0 0 | 7, 7 vs 6 |

For now, just these.

Sincerely,
Cac

**ericfarmer** · 12-03-2024, 03:56 PM

Originally Posted by iCountNTrack

Nice work! Did you use Igor Pro to plot this? Do you have the results in a table including the deck composition used? I had hoped our results would be less scattered

This figure was created using Mathematica; that's what I typically reach for when doing visualization.

The split EVs are here, in the same order as the previously linked table (in original format here, and updated for Cacarulo here).

Originally Posted by iCountNTrack

I am using VS 2022. I suspect you are using the default C++14 standard, you will need to use the C++17 or C++20 (I am using C++20 standard)

Yeah, I couldn't get g++ to deduce the specialization of your hash<GameStateKey> without an explicit typename, even with C++20. But I did get MSVS to work; following is a log of the output for the original 6-6 vs. 8 example (starting from 11 sixes and 5 eights), where your split EV is 3.818181818181817. Compare with my value of 2.295104895104895, which agrees with your local-optimum value from the above table.

Code:

Composition Dependent Combinatorial Analyzer 2025

Do you want to:
1) Enter the number of decks
2) Enter the composition of every rank for a depleted deck
Enter your choice (1 or 2): 2
Enter the composition of each rank (number of cards for A, 2-10, J, Q, K), separated by spaces:
(Order: A 2 3 4 5 6 7 8 9 10/J/Q/K): 0 0 0 0 0 11 0 5 0 0
Enter game rules:
Does the dealer hit on soft 17? (y/n): n
Is double after split allowed? (y/n): y
Is resplitting allowed? (y/n): y
Enter maximum number of resplits: 3
Is European No Hole Card (ENHC) rule in effect? (y/n): n
Enter dealer's upcard (A, 2-10, J, Q, K): 8
Enter player's cards (A, 2-10, J, Q, K), separated by spaces: 6 6
Elapsed time: 1.37719 seconds

Outcome Statistics:

Action: Stand
Expectation Value: 0.0769231
Standard Deviation: 0.997037
Net Outcome Probabilities:
  Net Outcome: -1 units, Probability: 0.461538
  Net Outcome: 1 units, Probability: 0.538462

Action: Hit
Expectation Value: 0.244755
Standard Deviation: 0.878785
Net Outcome Probabilities:
  Net Outcome: -1 units, Probability: 0.293706
  Net Outcome: 0 units, Probability: 0.167832
  Net Outcome: 1 units, Probability: 0.538462

Action: Double Down
Expectation Value: 0.48951
Standard Deviation: 1.75757
Net Outcome Probabilities:
  Net Outcome: -2 units, Probability: 0.293706
  Net Outcome: 0 units, Probability: 0.167832
  Net Outcome: 2 units, Probability: 0.538462

Action: Split
Expectation Value: 3.818181818181817
Standard Deviation: 3.114880308911239
Net Outcome Probabilities:
  Net Outcome: -4.000000000000000 units, Probability: 0.001398601398601
  Net Outcome: -3.000000000000000 units, Probability: 0.004195804195804
  Net Outcome: -2.000000000000000 units, Probability: 0.020979020979021
  Net Outcome: -1.000000000000000 units, Probability: 0.020979020979021
  Net Outcome: 0.000000000000000 units, Probability: 0.034965034965035
  Net Outcome: 4.000000000000000 units, Probability: 0.086713286713287
  Net Outcome: 5.000000000000000 units, Probability: 0.005594405594406
  Net Outcome: 8.000000000000000 units, Probability: 0.440559440559440

Best Action: Split

**iCountNTrack** · 12-03-2024, 04:03 PM

Originally Posted by ericfarmer

This figure was created using Mathematica; that's what I typically reach for when doing visualization.

The split EVs are here, in the same order as the previously linked table (in original format here, and updated for Cacarulo here).

Yeah, I couldn't get g++ to deduce the specialization of your hash<GameStateKey> without an explicit typename, even with C++20. But I did get MSVS to work; following is a log of the output for the original 6-6 vs. 8 example (starting from 11 sixes and 5 eights), where your split EV is 3.818181818181817. Compare with my value of 2.295104895104895, which agrees with your local-optimum value from the above table.

Code:

Composition Dependent Combinatorial Analyzer 2025

Do you want to:
1) Enter the number of decks
2) Enter the composition of every rank for a depleted deck
Enter your choice (1 or 2): 2
Enter the composition of each rank (number of cards for A, 2-10, J, Q, K), separated by spaces:
(Order: A 2 3 4 5 6 7 8 9 10/J/Q/K): 0 0 0 0 0 11 0 5 0 0
Enter game rules:
Does the dealer hit on soft 17? (y/n): n
Is double after split allowed? (y/n): y
Is resplitting allowed? (y/n): y
Enter maximum number of resplits: 3
Is European No Hole Card (ENHC) rule in effect? (y/n): n
Enter dealer's upcard (A, 2-10, J, Q, K): 8
Enter player's cards (A, 2-10, J, Q, K), separated by spaces: 6 6
Elapsed time: 1.37719 seconds

Outcome Statistics:

Action: Stand
Expectation Value: 0.0769231
Standard Deviation: 0.997037
Net Outcome Probabilities:
  Net Outcome: -1 units, Probability: 0.461538
  Net Outcome: 1 units, Probability: 0.538462

Action: Hit
Expectation Value: 0.244755
Standard Deviation: 0.878785
Net Outcome Probabilities:
  Net Outcome: -1 units, Probability: 0.293706
  Net Outcome: 0 units, Probability: 0.167832
  Net Outcome: 1 units, Probability: 0.538462

Action: Double Down
Expectation Value: 0.48951
Standard Deviation: 1.75757
Net Outcome Probabilities:
  Net Outcome: -2 units, Probability: 0.293706
  Net Outcome: 0 units, Probability: 0.167832
  Net Outcome: 2 units, Probability: 0.538462

Action: Split
Expectation Value: 3.818181818181817
Standard Deviation: 3.114880308911239
Net Outcome Probabilities:
  Net Outcome: -4.000000000000000 units, Probability: 0.001398601398601
  Net Outcome: -3.000000000000000 units, Probability: 0.004195804195804
  Net Outcome: -2.000000000000000 units, Probability: 0.020979020979021
  Net Outcome: -1.000000000000000 units, Probability: 0.020979020979021
  Net Outcome: 0.000000000000000 units, Probability: 0.034965034965035
  Net Outcome: 4.000000000000000 units, Probability: 0.086713286713287
  Net Outcome: 5.000000000000000 units, Probability: 0.005594405594406
  Net Outcome: 8.000000000000000 units, Probability: 0.440559440559440

Best Action: Split

Aaaa, sorry I should have told you about this. The number of resplits means the number of splits AFTER the initial split. So when you input 3 that means you will end up with 5 split hands.