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Originally Posted by dogman_1234
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.

Originally Posted by dogman_1234
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..."

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Originally Posted by ericfarmer
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.)

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Originally Posted by ericfarmer
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.
ericfarmer,

Would you post a primer defining all the different split methods: CDZ-, CDP, etc.? I find myself getting lost in the terminology.

Dog Hand

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Originally Posted by Dog Hand
ericfarmer,

Would you post a primer defining all the different split methods: CDZ-, CDP, etc.? I find myself getting lost in the terminology.

Dog Hand
You might also want to reread BJA3, pp. 387-393, to see how Cacarulo and I decided to handle the subject.

Don

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Originally Posted by DSchles
You might also want to reread BJA3, pp. 387-393, to see how Cacarulo and I decided to handle the subject.

Don
Trying to sell more books?

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Originally Posted by BoSox
Trying to sell more books?
Obviously!

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Originally Posted by Dog Hand
ericfarmer,

Would you post a primer defining all the different split methods: CDZ-, CDP, etc.? I find myself getting lost in the terminology.

Dog Hand
All of the various notation is intended to specify what information a player is allowed to use when determining playing strategy (whether to stand, hit, double, etc.). For any particular such notation, the intent is that it refers to the playing strategy that maximizes expected return of the round... but I say "intent," because as we will see, this intent is almost uniformly not attainable.

CDZ is, I think, the easiest to describe: it stands for "composition-dependent zero memory." The idea is to allow the player to use (1) the dealer's up card and (2) the composition of the player's current hand only, in determining whether to stand/hit/double/split/surrender. When we say that the player can use the "composition" of his hand, we mean that the player is allowed to distinguish, for example, 8-8 vs. 10-6 vs. 7-3-6 vs. 2-2-2-2-2-2-2-2, etc. And by "current hand only," we mean that the player must apply the same strategy for a given up card and hand, independent of whether it is the sole hand in the round, or the result of a split pair. If strategy is to hit 2-6 vs. dealer 5, then we must always hit 2-6 vs. dealer 5, even if we just split 2s and drew the 6 to one of the hands, or split 6s and drew the 2.

The good news is that CDZ is easy to describe... the bad news is that CDZ is (at least as far as I know) still intractable to actually compute. That is, I believe that there does not exist an implementation of any CA that is able to compute EV-maximizing CDZ strategy for a given shoe and rules. And splitting pairs are what make this hard to do: should we always hit 2-6 vs. dealer 5, both "pre-split" and "post-split," or should we consider always doubling instead, since there are shoes and rules where the latter yields a strictly higher overall EV for the round?

So CDZ is the specification of a generally unattainable goal; CDZ-, on the other hand (note the minus sign), refers to a strategy that we can compute efficiently: we can compute optimal EV-maximizing composition-dependent strategy for "pre-split" hands (or, if you like, temporarily assuming that splitting pairs is not even a thing)... and then blanket-apply that pre-split strategy to all post-split hands as well.

To summarize so far, we know that E(CDZ-) <= E(CDZ), even when we can only actually compute the value of E(CDZ-), and not E(CDZ). But this isn't entirely academic: in most of my past analyses, we have found, mostly by accident, explicit examples of situations where we stumble across a composition-dependent strategy whose EV-- call it E(CDZ*)-- is strictly better than E(CDZ-) for the same situation. In other words, we have been able to prove the existence of situations where E(CDZ-) < E(CDZ) (note the strict inequality), by demonstration of an intermediate CDZ* such that E(CDZ-)<E(CDZ*)<=E(CDZ).

This is already long, so I'll take another crack later at expanding on this. But some interesting food for thought: you can imagine how we can specify things like "TDZ" similarly, where we only get to know the (soft/hard) total for the hand, not its composition. It is perhaps counter-intuitive, but TDZ suffers from the same problem of being intractable to compute... in fact, TDZ is actually worse, in the sense that even if splitting pairs were not a thing, TDZ would still be very difficult to compute. (To take this one step further, I have argued that there does not exist a CA that I know of that is able to compute EV-maximizing TDZ strategy.)

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Originally Posted by ericfarmer
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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