I'm pretty late to the thread. There are a lot of interesting things to discuss here, and always so little time. Just focusing on item 1 (and item 4 at the end) to start: you're right that it seems like there has been some confusion on this point, in part because I think it's important to clarify under exactly what conditions it's actually true.
For example, playing strategy matters; expected return is invariant with respect to shoe depth only if the playing strategy does not depend on shoe composition. The following figure compares the expected return for a round played with fixed, basic, total-dependent strategy (in red), and for a round played with Hi-Opt II with full indices (in blue), as a function of number of decks remaining in a 6-deck shoe. (Six decks are not ideal for a discussion of floating advantage, but it's the setup for which I have data readily available.)
Attachment 3277
To interpret this plot: start with a fully-shuffled 6-deck shoe, observe, for example, 5 decks' worth of playing, then sit down and play a heads-up round from the remainder. Do this millions of times (i.e., shuffle the shoe, watch 5 decks go by, then play a round), and average the resulting outcomes. For the player utilizing basic *playing* strategy (whether he is counting for the purpose of *betting* or not), the expected return of about -0.48% is no different than it would be from the top of the shoe. For the Hi-Opt II player, expected return does improve (in this case, to about +0.15%), even "on average" over all encountered true counts, as the shoe is depleted.
The point here is that advantage is only independent of shoe depletion if the playing strategy doesn't depend on the composition of the depleted shoe. For example, even for an otherwise fixed, "basic," total-dependent strategy player, if he is using a count *only* to take insurance, then this independence no longer holds, albeit with a smaller extent of change in EV.
Also, particularly in discussion of FA where *extremely* depleted shoes are important (unfortunately I only have readily available data to 5/6 pen), we must be precise about exactly "how depleted" the shoe can get and still preserve constant EV for the fixed-strategy player. We can't run out of cards, or we risk the cut-card effect: suppose that instead of watching 5 decks get burned before playing a round, we instead watch, say, 48 heads-up *rounds* get played before we sit down to play, then watch another 48 rounds from another full shoe before playing, rinse and repeat, and average the results. The following figure shows the result.
Attachment 3278
The problem is that our "experiment" may fail during some repetitions: for a burn card at 5/6 penetration, some shoes may not even make it past 43-ish rounds. That effective conditioning on a 49th round even *existing* skews the uniformity of the distribution of possible arrangements of cards in the shoe.
Have to stop for now, but one last comment on item 4: I may misunderstand what's being said here, but I'm not sure I agree with this. The *exact* invariance of fixed-strategy EV should also apply to any linear combination of shoe rank probabilities, of which a true count is an example. A proof of this is
here, including mathematically-gory details about what "fixed" strategy means, and what "not running out of cards risking the cut-card effect" means.
Originally Posted by ericfarmer
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