k_c said:
I can give one example where optimal play would vary from fixed strategy play when only one split is made but I'm sure there are more.
Say a pair of 8s are split versus a dealer 7 dealt from 2 full decks. On the first hand player draws 4-1-1-1-1 so his full hand is 8-4-1-1-1-1. Correct strategy is to hit, so player draws. Possible draws are 1,2,3,4,5,6,7,8,9,10.
On the second hand player again draws 4-1-1-1-1 for a hand of 8-4-1-1-1-1. (If player draws an ace on the first 8-4-1-1-1-1 then this hand is not possible.) However, at this point no more aces remain, 2 4s are removed, 2 8s are removed, and the hit card that was drawn on the first hand is also removed. The best strategy for the second hand is to stand, which differs from the strategy of the first hand.
So if a fixed strategy is used that has determined 8-4-1-1-1-1 vs 7 should be hit then both hands would be hit even though EV is greater for hitting the first and standing on the second.
Optimal strategy is to hit the first hand and stand on the second, which increases EV to its maximum value.
Say a pair of 8s are split versus a dealer 7 dealt from 2 full decks. On the first hand player draws 4-1-1-1-1 so his full hand is 8-4-1-1-1-1. Correct strategy is to hit, so player draws. Possible draws are 1,2,3,4,5,6,7,8,9,10.
On the second hand player again draws 4-1-1-1-1 for a hand of 8-4-1-1-1-1. (If player draws an ace on the first 8-4-1-1-1-1 then this hand is not possible.) However, at this point no more aces remain, 2 4s are removed, 2 8s are removed, and the hit card that was drawn on the first hand is also removed. The best strategy for the second hand is to stand, which differs from the strategy of the first hand.
So if a fixed strategy is used that has determined 8-4-1-1-1-1 vs 7 should be hit then both hands would be hit even though EV is greater for hitting the first and standing on the second.
Optimal strategy is to hit the first hand and stand on the second, which increases EV to its maximum value.
this is the output from my CA (2 decks S17 DAS, 8,8 vs 7 1 split)
Code:
player's probabilities for standing
p_-1 = 0.746823618412
p_0 = 0
p_+1= 0.253176381588
p_+1.5 = 0
EV for standing= -0.493647236824 ± 0.869662236489
player's probabilities for doubling
p_-2 = 0.672573911737
p_0 = 0.0582195766871
p_+2= 0.269206511576
EV for doubling= -0.80673480032 ± 1.76530469189
player's probabilities for hitting
p_-1 = 0.672573911737
p_0 = 0.0582195766871
p_+1= 0.269206511576
EV for hitting= -0.40336740016 ± 0.882652345944
player's probabilities for splitting
p_-4 = 0.00416281658921
p_-3 = 0.0509584335589
p_-2 = 0.186420099874
p_-1 = 0.109646729082
p_0 = 0.202007416303
p_+1= 0.116494516984
p_+2 = 0.230490196456
p_+3 = 0.0901480191297
p_+4 = 0.00967177202379
EV for splitting= 0.234592559516 ± 1.82484526862
0.234592559516 (ICNT)
0.231816082001 (K_C)
0.231816011870 (MGP)