Quote Originally Posted by dogman_1234 View Post
Okay, so:

1.) From what I am gathering here is that the method of computing exact split Expectation's for single-split hands is 2 times the overall weighted Expectations for each new hand drawn, conditioned on a pair card removed. That is, even when computing the overall split Expectation for 22 vs 6, we only need to be aware of one single hand conditioned on its pair-rank removed from a full deck. We don't need to be cognizant of the other card nor of the other hand's make-up. So, for splitting 22 vs 6 , we don't need to evaluate {2A, 2A} directly; taking into account the missing deuce and Ace in the second hand for the Expectation of the first. We simply need to do {2x, 2A} and add the new Expectation of 2A conditioned on the missing 2 pair-rank removed to properly find the correct E/Action for 2A. Summing for each new draw rank 1-10, we combine them (by their overall weighted Expectations as E[Split] = P(A) * E(A) + ...P(10) * E(10)) taking a factor of two (for the two hands) and we should properly derive the conditional expectation of splitting 22 vs 6 for 1D, S17, (n)DAS, SPL1.
Correct.

Quote Originally Posted by dogman_1234 View Post
2.) Further splitting gets more complicated as we most of the time hit a "wall" of non-pair ranks. So, to split 22 vs 6 for SP2: we can have {2x, 2x}, draw a 2 and develop {2x, 2x, 2x}, we then cycle through each rank x, from 1-10, for the right-most 2x, and times it by the number of ways that hand state can be ordered. But, from your paper, this is (not?) the way this is done. I would assume the multinomial coefficient of the given split rank values to determine the overall Expectation of splitting 22 v 6 for SP2. Assume we draw an Ace after splitting to 3 deuces, we have a MC of 4. We then take this MC and times it by the overall weighted expectation for each optimal action (similar to our SP1 example.) However; drawing a third deuce is not guaranteed and so this method is wrong, correct? As per your paper, what I just described is incorrect and there involves some level of detail that I am missing.
I'm not sure I understand the multinomial coefficient in your description, you may need to help me out with some more explicit detail here. I *think* the issue you're describing is that, although we can indeed split, and draw another deuce and re-split to the maximum of three hands (for SPL2), there are really two different sub-cases we need to consider: do we draw that third deuce immediately, so that even if we draw additional deuces to the resulting three split hands, we are already prohibited from re-splitting again? Or do we draw a non-deuce (with a correspondingly differently-conditioned EV calcluation for that split hand) first, and *then* draw the third deuce, so that only the last two split hands are "already prohibited" from re-splitting?

The relevant section of the paper that describes this situation and the resulting formula is here: "The second possibility is that the player splits the maximum number of hands, but completes (i.e., draws non-pair cards to) k of them (where 0 ? k ? n ? 2) before drawing additional pair cards to reach the maximum number of hands. The probability in this case is given by..."