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.Originally Posted by Dog Hand
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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