Originally Posted by
ericfarmer
No. I see now that I likely added to the confusion by phrasing as a question at all, since it was intended to be rhetorical. That is, to the question "should we stand or double down with 6-8 vs. 8, in the split situation we've described, if we want to achieve an overall EV of 1548/715 for the round, no more and no less, using CDP strategy as computed by *either* "Eric's" or "MGP's" method?", the answer is, "neither, because if we always stand in that situation, or if we always double down in that situation, we have no hope of achieving the computed EV of 1548/715."
Note that I'm not "asking" anything here, I'm *claiming* that CDP-- *and CDPN*-- are broken as specifications of strategies, since they are unplayable, the situation described in this post being a concrete example. For some reason, the discussion keeps gravitating toward computational methods (what you label "MGP" and "Eric" in the tables of formulas you keep quoting)... it makes no difference which of these two computational approaches are used, the problem (with the *definition of the random variable*, which has a well-defined expected value no matter how we choose to compute it) remains.
This is exactly where the problem can arise, and it's still a problem for CDPN as well as CDP, which is what I seem to have not clearly communicated yet. This is where the problem arises in the example described in this post, albeit in the PPxxxx case for SPL3 instead of Pxxx for SPL2. The expected value that results from "all 3 hands are played the same way and all 3 take into account that 2 paircards are removed" cannot (in general) actually be achieved.
I say "in general" because finding examples of this sort of thing has turned out to be hard to find. That is, consider the example I've described so far ((0,0,0,0,0,11,0,5,0,0), splitting 6s vs. 8). I had to search quite a bit to find that example, first looking for "nice" shoes with the appropriate differences in EV as we move from CDZ- to CDP[n]... but even after that reasonably automatable search, I still had to "re-evaluate" those same EVs by brute-force enumeration and playout of shoe arrangements, looking for situations where the *indicated* strategy variations *didn't* yield the corresponding EV.
This is a detail that I'm not sure I've actually made explicit here yet. That is, it *is* possible to realize 1548/715 as an overall CDP EV for the split in this example... but to do so, not only do we need a dealer willing to deal all of our split hands out to two cards each before asking for a "non-split" strategy decision, but even then we still have to execute a strategy that looks surprisingly complicated. Coming back to the rhetorical question situation above, having split and resplit 6s to a maximum of 4 hands, and observing the first of those hands as 6-8, what should we do?
We've already said that we can't *always* stand, and we can't *always* double down. It turns out that we have to distinguish *two* of the other three split hands (any two will do, but they have to be *fixed*), and stand if those hands both get fleshed out to 6-6, otherwise (if we draw an 8 to either of them) double down.
But it's worse than that-- we have to execute the same "conditional stand/double" strategy for *each* of the other three split hands as well: for each, distinguish and fix two of the three "other slots," and stand only if both of those two other hands are 6-6.
E
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