Quote Originally Posted by BackCounter View Post
Hello; Don, you have a lot of footsteps to follow in! The progressive levels of confidence that you used was a big help. I run ten threads for each player hand-upcard combination and use the mean of the results. I used each thread as a measurement and generated 99% confidence intervals for each action, then if the top two actions overlapped I redid it with more iterations. Starting at 100,000, then 1,000,000, then 10,000,000 if necessary. It takes less than an hour for one data point and typically all but a few make 99%.

The data points used integer true counts only. Generating decks ahead of time with the desired count saves a lot of time; it just takes a few minutes until the count gets to about 10, then there is an exponential-like increase, close to two hours for 14. I use the pattern of card counts, how many 0, +1, and -1 cards are before and after the deck position, then populate and shuffle them internally based on the pattern to get the decks, then shuffle the cards after the deck position to get more. 100,000 iterations looking for a count of 14 generates only about 25 deck patterns. I hope this does not introduce any sort of bias.

Then running a sim to get the hand frequencies, advantage, insurance, variance, and covariance takes another 45 minutes or so; it takes 1,000,000,000 hands to get consistent data. 11 data points took a total of about 18 hours of running time. Not as efficient as Norm, but hey, I did it. (Using floating-point math and lots of multi-dimensional arrays.)

The problem: the basic strategy comes out correct, but I do not know if the graphs for all the counts come out right. For 8,8 vs. T, the lines for split and stand have a reverse slope from the ones in the back of the Hi-Lo book. And the indices do not all match. And the variance/s.d. seems to increase faster than it should at high counts. That's what makes me think there is a bias somewhere.

Four decks, and the rules specified for the illustrious 18 in BJA3.

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Hi, I agree with Don about the indices 10vT and 15vT being very low. I'll give you some indices for 4D, Hi-Lo so you can see the difference that exists:

10 vs T = +3.8
64 vs T = +4.0
73 vs T = +3.5
82 vs T = +3.7
55 vs T = +3.9

15 vs T = +4.2
T5 vs T = +4.1
96 vs T = +4.0
87 vs T = +4.1

Hope this helps.

Sincerely,
Cac