beyondbj,
You asked what the change in the Basic Strategy EV would be for a BJ game in which any unbusted dealer's hand containing 6 cards would be an automatic winner for the dealer. You failed to specify the house rules and conditions, so I arbitrarily chose a SD, H17, D10, NoDAS game for a heads-up B.S. player playing one spot and getting Rule of 1. I also assumes that a player's BJ would automatically win, so the dealer would NOT attempt to draw a 6-card hand to beat him.
Now the change in BS EV consists of two terms: pushes (under "normal" rules) that the player loses, and wins (under "normal" rules) that the player loses. For example, if the player has 17, and the dealer draws to a 6-card 17, then the player loses, instead of pushes. This costs him one bet. Similarly, if the player has 17 and the dealer draws to a 6-card 15, and the next card busts the dealer, then the player loses a hand that he would have won: this costs him two bets.
Below is the output compiled from a half-dozen CVData sims, each of 400-million rounds, using "normal" BJ rules. Column 1 shows the player's hand, ranging from "Stiff" to 21 (non-BJ). These groups correspond to the half-dozen CVData runs... they are grouped according to the player's total we are considering. The next two columns show the dealer's hand: Column 2 is for when the dealer completes her hand on card 6 (so four hit cards); Column 3 is for when more than 6 cards are required. In each case, the dealer's result is shown: either a total of 17-21, or else a "bust". Column 4 shows the total number of hands played for each of the half-dozen sims: it is more than 400-million due to splits. Column 5 shows the total number of times the indicated result was achieved. For example, if we look down to a player's total of 19, we see that, when the player totaled 19:
- the dealer drew to a 6-card 19 10,832 times
- the dealer drew to a more-than-6-card 19 558 times
- the dealer drew to a 6-card hand with a total of 18 or hard 17 19,081 times (not a soft 17, since I assumed the game is H17)
- the dealer drew to a more-than-6-card bust 5,146 times.
Ok... for your hypothetical 6-card-autoloser game, the player would have lost every one of these 4 cases. Since under "normal" rules, he would have pushed under cases 1 and 2, each of those losses cost him 1 unit (shown as "-1" in Column 7); and since he would have won under cases 3 and 4, each of those losses cost him 2 units (shown as "-2" in Column 7).
Column 6 is the ratio of Column 5 to Column 4, expressed as percentage of outcomes. Column 8 is the product of Columns 6 and 7, and shows the change in BS EV for each of these cases. Since they are all independent, we sum Column 8 to find the answer:
The BS EV of the game is decreased by just over 0.07%.
Hope this helps!
Dog Hand
HTML:
Player Dealer Dealer Hands "Wins" Percent Result Product
6-card >6 cards
Stiff bust 407,469,536 16,872 0.00414% -2 -0.00828%
17 17 407,470,484 9,385 0.00230% -1 -0.00230%
17 451 0.00011% -1 -0.00011%
bust 5,219 0.00128% -2 -0.00256%
18 18 407,475,224 9,401 0.00231% -1 -0.00231%
18 470 0.00012% -1 -0.00012%
<18 8,879 0.00218% -2 -0.00436%
bust 5,007 0.00123% -2 -0.00246%
19 19 407,480,408 10,832 0.00266% -1 -0.00266%
19 558 0.00014% -1 -0.00014%
<19 19,081 0.00468% -2 -0.00937%
bust 5,146 0.00126% -2 -0.00253%
20 20 407,471,208 16,670 0.00409% -1 -0.00409%
20 884 0.00022% -1 -0.00022%
<20 43,009 0.01056% -2 -0.02111%
bust 7,378 0.00181% -2 -0.00362%
21 noBJ 21 407,471,420 2,478 0.00061% -1 -0.00061%
21 137 0.00003% -1 -0.00003%
<21 8,495 0.00208% -2 -0.00417%
bust 992 0.00024% -2 -0.00049%
%Tie 0.01258% DelEV -0.07152%
%Win 0.02947%