What you are missing is that Eric is using CDZ- as strategy for split hands. CDZ- means strategy for split hands is determined prior to splitting by duplicating the strategy of non-split hands. My program has this option for a full shoe whereas Eric's can use this strategy for any shoe composition. In order to choose this option for a full shoe in my program you need to select 'Basic strat' under 'Compute mode' and 'Pre-split' under 'Depleted shoe split strat'. My program uses this as an option to compute basic strategy EVs for a depleted shoe. The default of my program is to use the optimal strategy of the first split hand and for a full shoe split EV for 2-2 v 6 s17 nDAS is more optimal using this strategy instead of using the strategy determined by pre-spilt hands.
It might sound complicated but it's simply a difference in the fixed strategy that is used for splitting.
k_c
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.
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.
Last edited by lij45o6; 06-16-2019 at 06:18 PM.
Correct.
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..."
I was assuming that we could compute 22 v 6 for SP2 using the single card removal method for SP1 by way of taking into account the removal of the 2 extra deuces and finding the overall weighted Expectation of splitting. Rather than one deuce removed, we remove two deuces and compute how many ways that split hand can be made up (hence, using the MC of the given hand state.) Since, there are 4 cards we are drawing to (3 deuces and an x), we compute the MC as 4!/(3! * 1!) = 4.
However; you raised an interesting (and correct) point: what do we do when we draw a third deuce on the second hand after drawing to the first. Lets say we split out hand {Px, Px}, and draw N for both: {PN, PN}. We are done here. But! What happend when we draw another P for either hand: {Px, PP} or {PP, PN}? Simply evaluating a SP2 hand using the SP1 method here is incorrect, as the number of ways a hand can be drawn is determined by which way the cards are drawn. (AKA, well no shit dogman.)
Maybe we are not all talking about the same program.
I'm talking about https://www.blackjackinfo.com/free-b...rial-analyzer/
from MGP
And not about http://www.bjstrat.net/programs.html
from KC (great program too)
KC agree with me. I quote
Peek AND early surrender seems rare (maybe only in Fun21 ?)"I agrre with you. My program displays unconditional EV in the cases of early surrender or ENHC even though the 'conditioned on no dealer blackjack where applicable' box is checked. To me conditional EV is not applicable in these cases although it could theoretically (and pretty meaninglesssly) still be computed"
And MGP choose to apply inconveniently 'conditioned on no dealer blackjack'' even for surrender.
Last edited by Phoebe; 06-14-2019 at 10:58 PM.
They aren't suboptimal, they simply make a different assumption about the allowed expressive power in specifying the player's strategy. CDZ- is in fact *more* optimal than the strategy assumed by Cacarulo in BJA3 in that composition-dependence doesn't stop at three cards, so to speak. And CDP1 (roughly what is assumed in BJA3, at least the part relevant to pair splitting) is also less optimal than, say, CDP (also evaluated at least by MGP's and my CAs), where the player can not only consider essentially *whether* he has split a given pair in modifying his strategy, but also *how many times*.
At any rate, for the purpose of this particular issue raised by dogman_1234, the distinction between CDZ- and CDP1 doesn't matter; the same assertion holds, namely, that the expected value of the two halves of the split are identical.
I forgot to mention in my most recent longer reply: I was misleading when I stated above that these various other strategies "aren't suboptimal." The intent was to emphasize the difference in allowed information assumed available to the player (composition vs. hand total, whether the hand is the result of a pair split, etc.), but for each of these various choices of allowed information, the numbers that are reported by all of our various CAs are indeed suboptimal... with the arguably lone exception of CDZ-, since that minus sign is essentially an implicit acknowledgment of suboptimality relative to CDZ (without the minus sign).
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