Optimal Counter Strategy & Counter-Strategy for Blackjack Without Upcards

[JohnGalt007]

I was wondering whether and to what extent an upcard-less blackjack variant would be analyzable and/or exploitable. Insurance is not available in this case, so any player naturals would have to be paid out immediately after inspection of the dealer's two downcards to see if the player wins or pushes. Furthermore, the player's decisions would be based solely on the value of his own hand rather than the probable strength of his hand relative to that of the dealer's. He would inevitably be hitting, standing, doubling down, and splitting in sub-optimal EV situations due to lack of information of one of the dealer's cards, so that his overall edge would be lower. Thus, if such a game was offered and the EV with basic strategy turned out to be much lower than that of ordinary blackjack, the house may consider offering slightly +EV rules to compensate and make the game more attractive to the ploppies (e.g. early surrender). I also suspect that combinatorial analysis of basic strategy would be substantially easier, since one wouldn't have to condition on nearly as many sub-cases and specific situations with a given hand (e.g. hard 12 vs. 3, hard 12 vs 4., etc.) because one doesn't have knowledge of that upcard. Thoughts?

[k_c]

I was wondering whether and to what extent an upcard-less blackjack variant would be analyzable and/or exploitable. Insurance is not available in this case, so any player naturals would have to be paid out immediately after inspection of the dealer's two downcards to see if the player wins or pushes. Furthermore, the player's decisions would be based solely on the value of his own hand rather than the probable strength of his hand relative to that of the dealer's. He would inevitably be hitting, standing, doubling down, and splitting in sub-optimal EV situations due to lack of information of one of the dealer's cards, so that his overall edge would be lower. Thus, if such a game was offered and the EV with basic strategy turned out to be much lower than that of ordinary blackjack, the house may consider offering slightly +EV rules to compensate and make the game more attractive to the ploppies (e.g. early surrender). I also suspect that combinatorial analysis of basic strategy would be substantially easier, since one wouldn't have to condition on nearly as many sub-cases and specific situations with a given hand (e.g. hard 12 vs. 3, hard 12 vs 4., etc.) because one doesn't have knowledge of that upcard. Thoughts?

My CA computes for case where only player hand is known, just doesn't output

Strategy no dealer cards known (S17, SPL1, SPLA1, NDAS, 1 card to split aces, double any 2 cards):

Nothing surrendered
Split A-A, 7-7, 8-8
Double Hard 10,11
Stand >= Soft 18, hit < 18
Stand >= Hard 15, hit < 15

Overall EV
1 deck -2.13%
6 decks -2.48%

k_c

[JohnGalt007]

My CA computes for case where only player hand is known, just doesn't output

Strategy no dealer cards known (S17, SPL1, SPLA1, NDAS, 1 card to split aces, double any 2 cards):

Nothing surrendered
Split A-A, 7-7, 8-8
Double Hard 10,11
Stand >= Soft 18, hit < 18
Stand >= Hard 15, hit < 15

Overall EV
1 deck -2.13%
6 decks -2.48%

k_c

Thanks for checking! Not at all surprised by the strategy recommendations. My CA doesn't compute for an unknown dealer card, but the law of total expectation should handle it in a pinch; e.g., E(stand on 17) = E(stand on 17|dealer's 1st card is A) + E(stand on 17|dealer's 1st card is 2) + ... + E(stand on 17|dealer's 1st card is T). Which CA are you using that allows you to turn off the dealer upcard setting?

[k_c]

Thanks for checking! Not at all surprised by the strategy recommendations. My CA doesn't compute for an unknown dealer card, but the law of total expectation should handle it in a pinch; e.g., E(stand on 17) = E(stand on 17|dealer's 1st card is A) + E(stand on 17|dealer's 1st card is 2) + ... + E(stand on 17|dealer's 1st card is T). Which CA are you using that allows you to turn off the dealer upcard setting?

It's my own CA.

It maintains a list of possible player hand compositions, 1 entry per hand comp. Hand comp parameters include EVs and strategies for each hand comp versus each up card as well as overall EVs and strategies for each comp.

The console version outputs overall EVs versus 1 or 0 up cards if computation of overall EV is the selected option. It doesn't output any of the individual component EVs/strategies for 0 up cards, only for 1 up card. The gui version computes values for 0 up cards but doesn't output them at all, only for 1 up card.

k_c

[JohnGalt007]

It's my own CA.

It maintains a list of possible player hand compositions, 1 entry per hand comp. Hand comp parameters include EVs and strategies for each hand comp versus each up card as well as overall EVs and strategies for each comp.

The console version outputs overall EVs versus 1 or 0 up cards if computation of overall EV is the selected option. It doesn't output any of the individual component EVs/strategies for 0 up cards, only for 1 up card. The gui version computes values for 0 up cards but doesn't output them at all, only for 1 up card.

k_c

Interesting! Did you program it yourself? In what language?

I also have to wonder how the EOR for each rank is affected without knowledge of the dealer upcard. I suspect it's slightly lessened, given that the lack of upcard information results in a sample space that's not narrowed down quite as much as it would be with that information.

[k_c]

Interesting! Did you program it yourself? In what language?

I also have to wonder how the EOR for each rank is affected without knowledge of the dealer upcard. I suspect it's slightly lessened, given that the lack of upcard information results in a sample space that's not narrowed down quite as much as it would be with that information.

Language is c++. I took some programming courses a while back and applied what I learned to developing a blackjack combinatorial analyzer and it kind of evolved.
My website program is basically the same program with output channeled to a web page rather than somewhere else.

I suppose I could get eors for 0 dealer cards if I wanted. Not that interested though.

k_c

[JohnGalt007]

I suppose I could get eors for 0 dealer cards if I wanted. Not that interested though.

k_c

I ran a few sims with the aforementioned basic strategy, assuming flat betting and using 400 million rounds per sim, to approximate the effects of removal for each rank in both the 1-deck and 6-deck games. Here were my results:

EOR of Ace (1 deck): -0.639%
EOR of 2 (1 deck): +0.248%
EOR of 3 (1 deck): +0.363%
EOR of 4 (1 deck): +0.503%
EOR of 5 (1 deck): +0.666%
EOR of 6 (1 deck): +0.293%
EOR of 7 (1 deck): -0.078%
EOR of 8 (1 deck): -0.195%
EOR of 9 (1 deck): -0.181%
EOR of 10 (1 deck): -0.411%

EOR of Ace (6 decks): -0.082%
EOR of 2 (6 decks): +0.058%
EOR of 3 (6 decks): +0.084%
EOR of 4 (6 decks): +0.106%
EOR of 5 (6 decks): +0.131%
EOR of 6 (6 decks): +0.069%
EOR of 7 (6 decks): -0.002%
EOR of 8 (6 decks): -0.027%
EOR of 9 (6 decks): -0.020%
EOR of 10 (6 decks): -0.051%

I also obtained figures comparable to yours for the overall EV for the 1-deck and 6-deck games via sims: -2.115% and -2.396% respectively.

[Cacarulo]

I ran a few sims with the aforementioned basic strategy, assuming flat betting and using 400 million rounds per sim, to approximate the effects of removal for each rank in both the 1-deck and 6-deck games. Here were my results:

EOR of Ace (1 deck): -0.639%
EOR of 2 (1 deck): +0.248%
EOR of 3 (1 deck): +0.363%
EOR of 4 (1 deck): +0.503%
EOR of 5 (1 deck): +0.666%
EOR of 6 (1 deck): +0.293%
EOR of 7 (1 deck): -0.078%
EOR of 8 (1 deck): -0.195%
EOR of 9 (1 deck): -0.181%
EOR of 10 (1 deck): -0.411%

EOR of Ace (6 decks): -0.082%
EOR of 2 (6 decks): +0.058%
EOR of 3 (6 decks): +0.084%
EOR of 4 (6 decks): +0.106%
EOR of 5 (6 decks): +0.131%
EOR of 6 (6 decks): +0.069%
EOR of 7 (6 decks): -0.002%
EOR of 8 (6 decks): -0.027%
EOR of 9 (6 decks): -0.020%
EOR of 10 (6 decks): -0.051%

I also obtained figures comparable to yours for the overall EV for the 1-deck and 6-deck games via sims: -2.115% and -2.396% respectively.

Hi John,

There's an important point about EORs: they must sum to zero.
If you calculated them through simulations (CVData), you have to set the number of rounds to a specific value.
In 6D, for example, 40 rounds per shoe or fewer. In 1D, 3 rounds or fewer. This is very important to avoid the CCE (Cut Card Effect).

Sincerely,
Cac

[JohnGalt007]

Hi John,

There's an important point about EORs: they must sum to zero.
If you calculated them through simulations (CVData), you have to set the number of rounds to a specific value.
In 6D, for example, 40 rounds per shoe or fewer. In 1D, 3 rounds or fewer. This is very important to avoid the CCE (Cut Card Effect).

Sincerely,
Cac

Thank you for pointing that out! I will redo my sims and get back to you ASAP.

[k_c]

I found I could get EORs for the case of 0 known dealer cards from the console version of my program without any additional programming by inputting removal of 1 of each rank followed by computing the overall EV for each input.

S17, SPL1, SPLA1, NDAS, 1 card to split aces, double any 2 cards, no dealer cards known

1 deck full shoe EV: -2.12742%

Rem      Overall EV    EOR
 A       -2.76959%     -0.64217%
 2       -1.80373%      0.32369%
 3       -1.67418%      0.45324%
 4       -1.50989%      0.61753%
 5       -1.30989%      0.81753%
 6       -1.70053%      0.42689%
 7       -1.88761%      0.23981%
 8       -2.23505%     -0.10763%
 9       -2.28742%     -0.16000%
 T       -2.53659%     -0.40917%

6 decks full shoe EV: -2.47821%

Rem      Overall EV    EOR
 A       -2.58454%     -0.10633%
 2       -2.42162%      0.05659%
 3       -2.40166%      0.07655%
 4       -2.37697%      0.10124%
 5       -2.34727%      0.13094%
 6       -2.42395%      0.05426%
 7       -2.47876%     -0.00055%
 8       -2.50512%     -0.02691%
 9       -2.50451%     -0.02630%
 T       -2.54286%     -0.06465%

k_c

[JohnGalt007]

I found I could get EORs for the case of 0 known dealer cards from the console version of my program without any additional programming by inputting removal of 1 of each rank followed by computing the overall EV for each input.

S17, SPL1, SPLA1, NDAS, 1 card to split aces, double any 2 cards, no dealer cards known

1 deck full shoe EV: -2.12742%

Rem      Overall EV    EOR
 A       -2.76959%     -0.64217%
 2       -1.80373%      0.32369%
 3       -1.67418%      0.45324%
 4       -1.50989%      0.61753%
 5       -1.30989%      0.81753%
 6       -1.70053%      0.42689%
 7       -1.88761%      0.23981%
 8       -2.23505%     -0.10763%
 9       -2.28742%     -0.16000%
 T       -2.53659%     -0.40917%

6 decks full shoe EV: -2.47821%

Rem      Overall EV    EOR
 A       -2.58454%     -0.10633%
 2       -2.42162%      0.05659%
 3       -2.40166%      0.07655%
 4       -2.37697%      0.10124%
 5       -2.34727%      0.13094%
 6       -2.42395%      0.05426%
 7       -2.47876%     -0.00055%
 8       -2.50512%     -0.02691%
 9       -2.50451%     -0.02630%
 T       -2.54286%     -0.06465%

k_c

I just finished re-calculating my own EORs from the sims, setting 2 rounds per shoe for 1-deck games and 30 rounds per shoe for 6-deck games. All of my EORs sum to zero, and I obtained similar numbers to yours, so I'm confident in both of our findings.

Overall EV of 1 deck: -2.113%
EOR of Ace (1 deck): -0.669%
EOR of 2 (1 deck): +0.331%
EOR of 3 (1 deck): +0.437%
EOR of 4 (1 deck): +0.598%
EOR of 5 (1 deck): +0.775%
EOR of 6 (1 deck): +0.372%
EOR of 7 (1 deck): -0.008%
EOR of 8 (1 deck): -0.151%
EOR of 9 (1 deck): -0.137%
EOR of 10 (1 deck): -0.387%

Overall EV of 6 decks: -2.403%
EOR of Ace (6 decks): -0.097%
EOR of 2 (6 decks): +0.051%
EOR of 3 (6 decks): +0.077%
EOR of 4 (6 decks): +0.100%
EOR of 5 (6 decks): +0.135%
EOR of 6 (6 decks): +0.056%
EOR of 7 (6 decks): -0.003%
EOR of 8 (6 decks): -0.025%
EOR of 9 (6 decks): -0.032%
EOR of 10 (6 decks): -0.067%

[Cacarulo]

I found I could get EORs for the case of 0 known dealer cards from the console version of my program without any additional programming by inputting removal of 1 of each rank followed by computing the overall EV for each input.

S17, SPL1, SPLA1, NDAS, 1 card to split aces, double any 2 cards, no dealer cards known

1 deck full shoe EV: -2.12742%

Rem      Overall EV    EOR
 A       -2.76959%     -0.64217%
 2       -1.80373%      0.32369%
 3       -1.67418%      0.45324%
 4       -1.50989%      0.61753%
 5       -1.30989%      0.81753%
 6       -1.70053%      0.42689%
 7       -1.88761%      0.23981%
 8       -2.23505%     -0.10763%
 9       -2.28742%     -0.16000%
 T       -2.53659%     -0.40917%

6 decks full shoe EV: -2.47821%

Rem      Overall EV    EOR
 A       -2.58454%     -0.10633%
 2       -2.42162%      0.05659%
 3       -2.40166%      0.07655%
 4       -2.37697%      0.10124%
 5       -2.34727%      0.13094%
 6       -2.42395%      0.05426%
 7       -2.47876%     -0.00055%
 8       -2.50512%     -0.02691%
 9       -2.50451%     -0.02630%
 T       -2.54286%     -0.06465%

k_c

Hi k_c,

These EoRs are not adding up to zero and I think I know why. Clearly, you are using an 'optimal strategy' for each of the
removed cards and even for the mean. For the calculation of EoRs, it is necessary to use a 'fixed strategy' that can be
either CD or TD and apply that strategy to each of the calculated EoRs including the mean. In that way, they will add up to zero.
Therefore, if we apply the same strategy you calculated previously to each of the removals including the mean,
the following EoRs can be derived:

1) 6D,S17,DOA,NDAS,SPA1,SPL1,NS

EoR [A] =  -0.1064591156
EoR [2] =   0.0565686956
EoR [3] =   0.0764859973
EoR [4] =   0.1010893701
EoR [5] =   0.1306842940
EoR [6] =   0.0539411353
EoR [7] =  -0.0011815396
EoR [8] =  -0.0268070877
EoR [9] =  -0.0261811532
EoR [T] =  -0.0645351491
Mean    =  -2.4783263122
SS      =   0.0686553706
CHS     =   0.0000000000

2) 1D,S17,DOA,NDAS,SPA1,SPL1,NS

EoR [A] =  -0.6657028874
EoR [2] =   0.3252436383
EoR [3] =   0.4442687610
EoR [4] =   0.5904081218
EoR [5] =   0.7709422304
EoR [6] =   0.3594745906
EoR [7] =  -0.0034053944
EoR [8] =  -0.1327485370
EoR [9] =  -0.1432482903
EoR [T] =  -0.3863080582
Mean    =  -2.1777174105
SS      =   2.4535636517
CHS     =   0.0000000000

As you can see, the sum is zero. It is important to note that a simulator (such as CVDATA, for example) should obtain the same results.

Sincerely,
Cac

[k_c]

Hi k_c,

These EoRs are not adding up to zero and I think I know why. Clearly, you are using an 'optimal strategy' for each of the
removed cards and even for the mean. For the calculation of EoRs, it is necessary to use a 'fixed strategy' that can be
either CD or TD and apply that strategy to each of the calculated EoRs including the mean. In that way, they will add up to zero.
Therefore, if we apply the same strategy you calculated previously to each of the removals including the mean,
the following EoRs can be derived:

1) 6D,S17,DOA,NDAS,SPA1,SPL1,NS

EoR [A] =  -0.1064591156
EoR [2] =   0.0565686956
EoR [3] =   0.0764859973
EoR [4] =   0.1010893701
EoR [5] =   0.1306842940
EoR [6] =   0.0539411353
EoR [7] =  -0.0011815396
EoR [8] =  -0.0268070877
EoR [9] =  -0.0261811532
EoR [T] =  -0.0645351491
Mean    =  -2.4783263122
SS      =   0.0686553706
CHS     =   0.0000000000

2) 1D,S17,DOA,NDAS,SPA1,SPL1,NS

EoR [A] =  -0.6657028874
EoR [2] =   0.3252436383
EoR [3] =   0.4442687610
EoR [4] =   0.5904081218
EoR [5] =   0.7709422304
EoR [6] =   0.3594745906
EoR [7] =  -0.0034053944
EoR [8] =  -0.1327485370
EoR [9] =  -0.1432482903
EoR [T] =  -0.3863080582
Mean    =  -2.1777174105
SS      =   2.4535636517
CHS     =   0.0000000000

As you can see, the sum is zero. It is important to note that a simulator (such as CVDATA, for example) should obtain the same results.

Sincerely,
Cac

I was just about to post that EORs sum to 0 as long as they are relative to a fixed strategy when I saw your post. I have to admit I learned that at some point but had forgotten and have just remembered. You are right that the EORs I posted were for compute mode of 'optimal'. If I change program compute mode to 'basic strategy' EORs should sum to 0.

k_c