Sorry to ramble, but I have a question for the discussion participants: have any of you tried to connect servers with artificial intelligence for calculations?
I just know how the problem with a huge event tree was solved for CA for Chinese poker (Pineapple)
Last edited by Gramazeka; 12-04-2024 at 06:07 PM.
"Don't Cast Your Pearls Before Swine" (Jesus)
Thanks, this is interesting! If I understand the column header notation correctly, when evaluating SPL#, you are computing split EVs assuming that you split, and resplit at every opportunity, up to a maximum of #+1 hands, right? (This is also the assumption in CDZ-, CDP1, and CDP[N].)Originally Posted by Cacarulo
I would be curious to learn how often the optimal strategy doesn't have this form, that is, in which of these 4840 scenarios do we split to a varying number of hands depending on the fall of the cards (and ideally in one of the larger shoes where the variability isn't due to avoidance of running out of cards).
Ok, now I understand. At first, it seemed like something your program was doing rather than discarding. I get that a brute-force analysis will go through a tree with all possibilities, including the pathological ones, which are logically discarded if they don’t contribute to maximizing the EV.
Yes, I understand now with the previous explanation. Clearly, this brute-force method, which is only applicable to these small subsets, is useful for evaluating other, faster implementations. This makes me happy because if brute force yields a maximum EV of 1.0097777, then 1.0093 isn’t bad at all. In fact, you can review the file I uploaded, and you’ll see that there are no cases where the difference is significant. This entire thread has helped me rule out the OP method I had been using until now. It’s not that the method is bad, but the differences in small subsets are considerable.I might misunderstand your question. If you are asking, "what is the expected value of splitting 8-8 vs. dealer 8?" then the answer hasn't changed; it's still the same 4854/4807=1.0097774079467443 value provided earlier. And the strategy that realizes this expected value would direct us to stand on an 86 drawn after the initial split.
If you're instead asking, "given that we have already split 8-8 vs. dealer 8, and drawn a 6 to the "first" hand, what is the expected value of the round resulting from standing on that 8+6=14?" then it's maybe worth noting that there are still arguably three possible relevant questions here, depending on how many resplits we might have already executed before eventually drawing that 6. Following are those results (with the optimal strategy being to stand in each of the three cases):
My new OP, let’s call it OPN, takes much more time than its predecessor, but it’s worth it for a deeper analysis.
The idea behind NDAS was to see if the differences would narrow by not doubling. But I understand that it might be complicated.NDAS will take some work. Remember, this implementation was simple-- partly because I wanted results quick, but with hindsight partly because the simplicity makes it hard for bugs to "hide" in there-- and I mean reeaallly simple. I can't even handle aces in the shoe, let alone H17, let alone NDAS.
Sincerely,
Cac
Luck is what happens when preparation meets opportunity.
Exactly. SPL1 means you split only once for a total of two hands. SPL2 means you can split up to twice for a total of three hands, and SPL3, three times for a total of four hands.
I'm curious about it too.I would be curious to learn how often the optimal strategy doesn't have this form, that is, in which of these 4840 scenarios do we split to a varying number of hands depending on the fall of the cards (and ideally in one of the larger shoes where the variability isn't due to avoidance of running out of cards).
Sincerely,
Cac
Luck is what happens when preparation meets opportunity.
Hi k_c,
While reviewing the thread, I noticed I overlooked this comment of yours. If it’s not too much trouble, could you tell me the result you got for 2,2 in 6D, S17, DOA, DAS, SPL1?
Thanks.
Sincerely,
Cac
Luck is what happens when preparation meets opportunity.
Hi Cac,
This is what I get. I've included the transformation to conditional EVs from unconditional which only applies to up cards of ten and ace. Transformation formula is condEV = (uncondEV + probDealerBJ) / (1 - probDealerBJ). There is no difference unless up card is T or A.
I've also included fixed strategy (CDP1) EVs for comparison. There doesn't seem to be much gain available when number of cards is relatively large.
unconditional EV:
conditional EV:Code:Shoe comp (A-5): {24, 24, 24, 24, 24} Shoe comp (6-T): {24, 24, 24, 24, 96} Please input pair value. (a or A,2,3,4,5,6,7,8,9,t or T): Pair value = 2 Please input allowed splits (1, 2, 3): 1 Single split EVs (full peek), S17, DAS, no DAN, no HSA, no auto win Double any 2 cards Unconditional EV, Allowed splits: 1 ...Computing, please wait... SPL1 (split & play without knowledge of dealer up card): -0.258597888511985 SPL1 versus 2: -0.0826509839841486 SPL1 versus 3: -0.0166698337010412 SPL1 versus 4: 0.0539259405857191 SPL1 versus 5: 0.148864253205873 SPL1 versus 6: 0.208396572339893 SPL1 versus 7: -0.00518731072051802 SPL1 versus 8: -0.174269908583176 SPL1 versus 9: -0.36425809678629 SPL1 versus T: -0.51199175809251 SPL1 versus A: -0.593410384796985 Press c or C for EV conditioned on no dealer blackjack, any other key to exit
fixed strategy (CDP1 unconditional EV)Code:Shoe comp (A-5): {24, 24, 24, 24, 24} Shoe comp (6-T): {24, 24, 24, 24, 96} Pair value: 2 Single split EVs (full peek), S17, DAS, no DAN, no HSA, no auto win Double any 2 cards EV conditioned on no dealer BJ, Allowed splits: 1 SPL1 (split & play without knowledge of dealer up card): -0.22113017635677 SPL1 versus 2: -0.0826509839841486 SPL1 versus 3: -0.0166698337010412 SPL1 versus 4: 0.0539259405857191 SPL1 versus 5: 0.148864253205873 SPL1 versus 6: 0.208396572339893 SPL1 versus 7: -0.00518731072051802 SPL1 versus 8: -0.174269908583176 SPL1 versus 9: -0.36425809678629 SPL1 versus T: -0.470896327195037 SPL1 versus A: -0.410158727240696 Press u or U to display unconditional EV, any other key to exit
k_cCode:SPL1 versus 2: -0.082685246976023224 SPL1 versus 3: -0.016669833701041109 SPL1 versus 4: 0.053387202411827761 SPL1 versus 5: 0.14885828503349974 SPL1 versus 6: 0.20839657233989284 SPL1 versus 7: -0.0051873107205179514 SPL1 versus 8: -0.17426990858317637 SPL1 versus 9: -0.36425809678643595 SPL1 versus T: -0.51207104532617520 SPL1 versus A: -0.59341815057856440
"Perfection is the enemy of success."
-Elon Musk-
Thank you, k_c. With your data, I put together a comparative chart that I find very interesting.
As you mentioned, with so many cards in the shoe, the differences in EV are not very significant.
The next step would be to do the same but with 1D, H17, DOA, DAS, SPL1.
Code:k_c Exact Cac OPN Cac CDP k_c CDP1 (uEV) 2,2 vs A -41.0158727240696% -41.0163718899804% -41.0169993092847% -59.3418150578564% 2,2 vs T -47.0896327195037% -47.0939938231909% -47.0982291248380% -51.2071045326175% 2,2 vs 9 -36.4258096786290% -36.4258096786436% -36.4258096786436% -36.4258096786436% 2,2 vs 8 -17.4269908583176% -17.4269908583176% -17.4269908583176% -17.4269908583176% 2,2 vs 7 -0.5187310720518% -0.5187310720518% -0.5187310720518% -0.5187310720518% 2,2 vs 6 20.8396572339893% 20.8396572339893% 20.8396572339893% 20.8396572339893% 2,2 vs 5 14.8864253205873% 14.8858285033500% 14.8858285033500% 14.8858285033500% 2,2 vs 4 5.3925940585719% 5.3724334126691% 5.3387202411828% 5.3387202411828% 2,2 vs 3 -1.6669833701041% -1.6669833701041% -1.6669833701041% -1.6669833701041% 2,2 vs 2 -8.2650983984149% -8.2685246976023% -8.2685246976023% -8.2685246976023%
They are sorted from highest to lowest (left to right), and I highlighted the matches between the different algorithms in blue.
Actually, in these examples, CDP is exactly the same as CDP1 in all cases.
Sincerely,
Cac
Last edited by Cacarulo; 12-12-2024 at 08:23 PM.
Luck is what happens when preparation meets opportunity.
Here are same calculations for 1D, H17, DOA, DAS, SPL1.
unconditional EV:
conditional EV:Code:Shoe comp (A-5): {4, 4, 4, 4, 4} Shoe comp (6-T): {4, 4, 4, 4, 16} Please input pair value. (a or A,2,3,4,5,6,7,8,9,t or T): Pair value = 2 Single split EVs (full peek), H17, DAS, no DAN, no HSA, no auto win Double any 2 cards Unconditional EV, Allowed splits: 1 ...Computing, please wait... SPL1 (split & play without knowledge of dealer up card): -0.238720247661669 SPL1 versus 2: -0.0381300776241396 SPL1 versus 3: 0.0306399925523454 SPL1 versus 4: 0.120799718036259 SPL1 versus 5: 0.278576911576813 SPL1 versus 6: 0.312424257822685 SPL1 versus 7: 0.00778533732817593 SPL1 versus 8: -0.176447876864187 SPL1 versus 9: -0.357570152056584 SPL1 versus T: -0.495377112467691 SPL1 versus A: -0.655314312303188 Press c or C for EV conditioned on no dealer blackjack, any other key to exit
fixed strategy (CDP1 unconditional EV):Code:Shoe comp (A-5): {4, 4, 4, 4, 4} Shoe comp (6-T): {4, 4, 4, 4, 16} Pair value: 2 Single split EVs (full peek), H17, DAS, no DAN, no HSA, no auto win Double any 2 cards EV conditioned on no dealer BJ, Allowed splits: 1 SPL1 (split & play without knowledge of dealer up card): -0.196754783277816 SPL1 versus 2: -0.0381300776241396 SPL1 versus 3: 0.0306399925523454 SPL1 versus 4: 0.120799718036259 SPL1 versus 5: 0.278576911576813 SPL1 versus 6: 0.312424257822685 SPL1 versus 7: 0.00778533732817593 SPL1 versus 8: -0.176447876864187 SPL1 versus 9: -0.357570152056584 SPL1 versus T: -0.450521744687041 SPL1 versus A: -0.48819397887443 Press u or U to display unconditional EV, any other key to exit
k_cCode:SPL1 versus 2: -0.039003239280159890 SPL1 versus 3: 0.029964794486866675 SPL1 versus 4: 0.11809451370183771 SPL1 versus 5: 0.27844300621648843 SPL1 versus 6: 0.31221302615446223 SPL1 versus 7: 0.0063296898639954857 SPL1 versus 8: -0.17688223642192197 SPL1 versus 9: -0.35852352354280476 SPL1 versus T: -0.49601741386821585 cev: -0.451218961767612814 SPL1 versus A: -0.65625290625447130 cev: -0.489587648680881627
"Perfection is the enemy of success."
-Elon Musk-
Thank you again, k_c. This table provides a much clearer view of the differences between the various algorithms:
Code:k_c Exact Cac OPN Cac CDP CDP1 (unEV) 2,2 vs A -48.8193978874430% -48.9212718550763% -48.9587648680882% -48.9587648680882% 2,2 vs T -45.0521744687041% -45.1021398516231% -45.1218961767613% -45.1218961767613% 2,2 vs 9 -35.7570152056584% -35.7967348646956% -35.8523523542805% -35.8523523542805% 2,2 vs 8 -17.6447876864187% -17.6797029049566% -17.6882236421922% -17.6882236421922% 2,2 vs 7 0.7785337328176% 0.7079141663035% 0.6329689863995% 0.6329689863995% 2,2 vs 6 31.2424257822685% 31.2298260083870% 31.2213026154462% 31.2213026154462% 2,2 vs 5 27.8576911576813% 27.8494857709317% 27.8443006216488% 27.8443006216488% 2,2 vs 4 12.0799718036259% 11.8713418133971% 11.8094513701838% 11.8094513701838% 2,2 vs 3 3.0639992552345% 3.0248178406537% 2.9964794486867% 2.9964794486867% 2,2 vs 2 -3.8130077624140% -3.8504029199489% -3.9003239280160% -3.9003239280160%
Once again, we see that Cac CDP is exactly the same as CDP1.
What remains now is to determine the overall EV of a perfect game to compare it with the other approaches.
Obviously, your algorithm will be the winner despite the machine times.
Sincerely,
Cac
Luck is what happens when preparation meets opportunity.
Hi Cac,
Getting back to Eric's original depleted shoe example shows that CDP may be the same as CDP1 for SPL1, but not necessarily. In the depleted shoe example my program indicates that it's the same for 8-8 but different for 6-6. Algorithmically I think it's the same because the premise of CDP is to use a fixed strategy until you somehow you "know better" because of consideration of number of pair cards that have been removed.
6-6
8-8Code:Shoe comp (A-5): {0, 0, 0, 0, 0} Shoe comp (6-T): {11, 0, 5, 0, 0} Please input pair value. (a or A,2,3,4,5,6,7,8,9,t or T): Pair value = 6 Single split EVs (full peek), S17, DAS, no DAN, no HSA, no auto win Double any 2 cards Unconditional EV, Allowed splits: 1 ...Computing, please wait... SPL1 (split & play without knowledge of dealer up card): 0.367632367632368 SPL1 versus 2: X X X X X SPL1 versus 3: X X X X X SPL1 versus 4: X X X X X SPL1 versus 5: X X X X X SPL1 versus 6: 0.0217560217560218 SPL1 versus 7: X X X X X SPL1 versus 8: 1.04055944055944 SPL1 versus 9: X X X X X SPL1 versus T: X X X X X SPL1 versus A: X X X X X Press c or C for EV conditioned on no dealer blackjack, any other key to exit
fixed strategy (CDP1)Code:Shoe comp (A-5): {0, 0, 0, 0, 0} Shoe comp (6-T): {11, 0, 5, 0, 0} Please input pair value. (a or A,2,3,4,5,6,7,8,9,t or T): Pair value = 8 Single split EVs (full peek), S17, DAS, no DAN, no HSA, no auto win Double any 2 cards Unconditional EV, Allowed splits: 1 ...Computing, please wait... SPL1 (split & play without knowledge of dealer up card): 0.10989010989011 SPL1 versus 2: X X X X X SPL1 versus 3: X X X X X SPL1 versus 4: X X X X X SPL1 versus 5: X X X X X SPL1 versus 6: 0.083916083916084 SPL1 versus 7: X X X X X SPL1 versus 8: 0.205128205128205 SPL1 versus 9: X X X X X SPL1 versus T: X X X X X SPL1 versus A: X X X X X Press c or C for EV conditioned on no dealer blackjack, any other key to exit
k_cCode:SPL1 6-6 v 6: -0.013986013986013898 SPL1 6-6 v 8: 1.0349650349650350 SPL1 8-8 v 6: 0.083916083916084072 SPL1 8-8 v 8: 0.20512820512820487
"Perfection is the enemy of success."
-Elon Musk-
Hi k_c,
Yes, CDP is the same as CDP1 for SPL1. I’m not sure which case you’re referring to where CDP differs from CDP1.
Here I have another comparative table:
Code:k_c Exact Cac OPN Cac CDP CDP1 6,6 vs 6 2.1756021756022% 1.1655011655012% -1.3986013986014% -1.3986013986014% 6,6 vs 8 104.0559440559440% 103.4965034965035% 103.4965034965035% 103.4965034965035% 8,8 vs 6 8.3916083916084% 8.3916083916084% 8.3916083916084% 8.3916083916084% 8,8 vs 8 20.5128205128205% 20.5128205128205% 20.5128205128205% 20.5128205128205%
Sincerely,
Cac
Luck is what happens when preparation meets opportunity.
I guess I'm not being so articulate.
What I'm seeing is that CDP is the same as a fixed strategy for SPL1 although it is supposed to be more optimal than a fixed strategy. However, that behavior seems a bit anomalous because it can't be manifested in SPL1.
Just my $.02 worth.
k_c
"Perfection is the enemy of success."
-Elon Musk-
Concur, CDP and CDP1 are, by definition, the same for SPL1, and will only ever (possibly) yield different results for SPL2 or SPL3. Or at least as long as we are all on the same page of definitions-- the details for both CDP1 and CDP are described, in the context of this example, here, but focusing on the distinctions and similarities among SPL1, SPL2, and SPL3:
For CDP1, initially focusing on SPL1, split a pair of 2s, and compute the strategy that maximizes the expected return of the first half of the split, using the knowledge of that additional 2 removed from the shoe (for the eventual second half of the split). Then play that same strategy for the second half of the split as well-- which will, as a result, have exactly the same expected return as the first half. (MGP repeatedly disagreed with me on this, but this is part of what makes calculation of CDP1 so efficient.)
For CDP1, with SPL2 or SPL3, start the same way: split a pair of 2s, and compute the strategy that maximizes the expected return of the first split hand, accounting for the (currently) single additional pair card removed... but assuming that we don't draw another 2. Then evaluate the overall expected return, assuming that we split and resplit at every opportunity up to a maximum of 3 or 4 hands as dictated by the rules, and otherwise play the "non-split" strategy determined as above.
There are some mathematical details involved in that conditioning on drawing an additional pair card or not, described in the paper linked through above, that involve computing expected returns conditioned on multiple "extra" pair cards removed. This is where CDP comes in: for CDP1, we still use the same fixed strategy when evaluating each of those conditional expected returns; but CDP-- which was originally described to me by MGP-- allows us to "re-optimize" a strategy for each of those possible number of additional pair cards removed from the shoe. That is, remove one extra 2 from the shoe, and compute strategy that maximizes "single-hand" expected return; then remove two extra 2s from the shoe, and compute a (possibly different) strategy that maximizes single-hand expected return; etc.
This might seem sketchy-- for SPL3, for example, we don't necessarily "just" know the composition of our current hand and the number of "eventual" additional pair cards that we will draw in the round, so how does it make sense to optimize a strategy based on that information? You're right, it doesn't make sense; hence the linked thread, and the (0,0,0,0,0,11,0,5,0,0) example showing why CDP should be abandoned ... but only for multiple resplits, since the infeasibility-of-execution-at-the-table problem only exists for multiple resplits: for SPL1, CDP "reduces to" CDP1: we can think of CDP1 as "strategy accounting for one additional pair card (2 in this case) removed", but it's perhaps clearer to think of CDP1 as "strategy that is allowed to vary depending on whether I have split 2s or not".
By "Cac OPN", you mean the strategy/algorithm whose results you have been reporting for most of this thread, right? I might have lost track somewhere, but I don't recall seeing the separate/additional "Cac CDP"? Is this CDP in the sense I've tried to describe above, or something else? (There are two sources of CDP numbers that you can use to verify: either my CA, which I've linked earlier, or MGP's here, although his doesn't work for depleted shoes like this example. But maybe we could compare notes for e.g. 1D?)
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