Analysis of C.S.M. by Persi Diaconus, Ph.D. et al

[ZenMaster_Flash]

https://www.dropbox.com/s/hmwr68ikthv2udk/Analysis%20of%20C.S.M.pdf?dl=0

[DSchles]

Damn! Persi certainly has come a long way since we used to go elementary school together!

Don

[ZenMaster_Flash]

Dr. Persi of Princeton University PROVED that it takes
seven (7) riffle shuffles to truly randomize a deck of cards.

I have wondered:
How much less random an eight (8) deck shoe can be.

[Norm]

He showed that for a deck with 52 different cards. For most games, blackjack has ten different cards.

[ZenMaster_Flash]

52 different cards

Excellent point. Did he revisit the null hypothesis
with what one can simply entitle a 'blackjack deck?"

[Three]

Some variants have suit dependent payoffs.

[Norm]

Which is why I said: " For most games, blackjack has ten different cards."

[ZenMaster_Flash]

"For most games, blackjack has ten different cards."

Correct, as Spanish21 has only nine (9) different cards.

[Norm]

More if Super Bonus or Match the Dealer rules are included as suits come into play.

[ZenMaster_Flash]

" ... if Super Bonus or Match the Dealer rules are included as suits come into play."

The (free) multi-card 21, 6-7-8, and 7-7-7 bonuses are intrinsic to the game.

[Three]

Correct, as Spanish21 has only nine (9) different cards.

SP21 still has ten value cards but the 4 suits also matter in the 678 and 777 bonus possibilities. So it has 10*4 or 40 different cards. A-9 and Faces of 4 different suits. BJ has A-9 and T's with suits being irrelevant.

[ericfarmer]

Dr. Persi of Princeton University PROVED that it takes
seven (7) riffle shuffles to truly randomize a deck of cards.

I have wondered:
How much less random an eight (8) deck shoe can be.

This is a pretty popular rule of thumb, probably because it is easy to state, but I think it requires some clarification (even beyond the blackjack-specific issues Norm has already noted).

First, for a deck of 52 distinct cards (vs. a blackjack deck where suits and face cards are indistinguishable), seven riffle shuffles, using the GSR model, are required for the total variation distance (TVD) from uniform to be less than 1/2. That's not nearly as fun to say at parties; as Diaconis writes in the OP's linked paper: "The engineers and executives who consulted us found it hard to understand the total variation distance." At any rate, this is not the same thing as the usual popular interpretation of something along the lines of, "Seven shuffles are needed to 'fully randomize' a deck of cards." The problem is defining what we mean by "fully random," since even with seven shuffles, or eight, or nine, etc., the probability distribution of possible arrangements of cards is *never* going to be *exactly* uniform. So how much non-uniformity are we willing to allow?

I think a more accurate statement is, "At least six riffle shuffles are necessary, and 8 or 9 shuffles are recommended." The 8-9 recommendation is justified by the asymptotic "cutoff" behavior of TVD at 3/2lg(52), or around 8.55 shuffles. Granted, that sounds rather complicated as well. But the "at least six" part is easier to explain: using a simple counting argument, it can be shown that you need at least *five* shuffles to even have a chance of realizing all 52! possible arrangements of cards in the deck... and using a slightly more refined counting argument, you need at least *six* shuffles to even possibly realize a *particular* arrangement-- namely, an exact reversal of the order of cards. That is, if you want a reversal of the cards in the deck to be one of your possible arrangements, you need at least six riffle shuffles to get there.

As Norm points out, things are different with blackjack, since the cards are not all distinct, and so there are fewer possible permutations. There is another good paper by Diaconis et. al., "Riffle shuffles of a deck with repeated cards," here (PDF). TL;DR: ignoring suit and face card distinctions saves you a few shuffles...

But again, this analysis focuses on 52-card decks. For an 8-deck shoe, things are doubly complicated: first, there are simply more cards, and some of the formulas get unwieldy with 8x52=416 cards. But also, the GSR model of the riffle shuffle doesn't quite apply directly in that case anyway: the dealer doesn't break the whole shoe into just two roughly equal huge halves, and riffle them together.

[ZenMaster_Flash]

I apologize for my silliness about how many cards there are.
I was thinking of ranks, not cards;
but these really new Indica-Dominant Hybrids have my minor
intellectual capacity reduced to (virtual) cognitive neuro-sludge.