In the game of blackjack, I'd like to know what the player's edge
would be if he were playing with 'perfect information' but not varying
his bets. That is, if a player could precisely calculate the odds of
the next card at each juncture by remembering all the previous cards
dealt, and adjusted his strategy accordingly, what would the odds of
the game be. I'm not picky about the exact rules used(something like a
6 deck shoe preffered).

I am NOT looking for odds based on card-counting strategies, basic
strategy, or any other system. This is really more of a theoretical,
game-theory question.

Hello and thank you for your interesting question.

Fortunately the "Wizard of Odds" has done our work for us and has
calculated the odds for shoes holding 1,2,4,5,6,8 and an infinite
number of decks.  The Wizard also explains a bit about the recursive
method that he used to derive his results.  I am unable to copy and
paste the posted results here (because they originate in a .pdf
document) but you will find them at
http://www.thewizardofodds.com/game/bjapx9.html
Wizard of Odds
http://www.thewizardofodds.com/game/bj.html

Does the Wizard know what he is talking about?  
Yes!  A theoretical calculation agrees with the Wizard's figures to
three decimal places.
Blackjack: A Game Theoretic Monte Carlo Analysis
http://math.ucsd.edu/~tmcelroy/bjacktalk.pdf

If you find any of the above to be unclear, please request
clarification.  I would appreciate it if you would hold off on rating
my answer until I have an opportunity to reply.

Google search terms used:
blackjack game-theory

Sincerely,
Google Answers Researcher
richard-ga

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Clarification of Answer by richard-ga on 04 Feb 2003 17:10 PST

I misspoke in saying that I could not copy and paste the Wizard's
results.  Rather his tabular calculation is too long to print.  But
here is the summary result for the 6-deck shoe:
[Assumptions]
Dealer hits on soft 17 
Player may double on any first two cards 
Player may resplit to four hands 
Numbers are presented both ways for whether the player may or may not
double after a split
[Results]
Expected return of game:
Player may double after split: -0.006156 
Player may not double after split: -0.007601
http://www.thewizardofodds.com/game/bjapx9j.html

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Request for Answer Clarification by mtrauch-ga on 04 Feb 2003 17:56 PST

Close...

However, I'm looking for the answer to a more complex (massively more
complex, i imagine) question.

The 'composition dependent returns' analyzed in these pages address
the expected return based on the composition of the deck after a
*single* hand has been dealt.

My question is, if we play through an entire 'round' (e.g., 6 decks),
and adjust strategy for the compostion at every deal (not just the
first deal, as Mr. Shackley (the man behind the wizard of odds)has
done ), what is the expected return.

I know the answer for simple card-counting systems is positive, and
these systems discard large amounts of information. Intuitively, I
would guess at an answer between +3% and +5% for 4 decks, if not more.

I am assuming someone has already done this work for us, too...I just
can't find it!

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Clarification of Answer by richard-ga on 05 Feb 2003 07:14 PST

Hello again:

I understand what you're saying.

In effect you're assuming that the player has a perfect memory at each
point in time as the deck is played out, and an appropriate perfect
strategy based on the known cards remaining in the deck.  I haven't
found anybody who claims to have run a simulation at that level.

Obviously if we assume an unlimited number of decks the difference is
zero, since perfect knowledge and no knowledge are the same thing when
an infinite number of cards remain in the shoe.

Here's something I have found in the Baccarat context that touches on
this issue:
"In March, 1982 the "Gambling Times" published a series of six-card
subsets which could give the player an advantage at the very end of
the deck. This inspired Joel Friedman to investigate all possible
six-card subsets. He discovered that a player with computer-perfect
knowledge of the last six-cards could gain an average profit of 26% on
that hand. By raising his bets fourfold (or more), his profits could
outweigh the loss from making the approximately 80 "waiting" bets on
the banker the player would have to make in order to earn the right to
bet on this last hand. More than 24% of his gain would come from the
tie bet. This finding seemed strange considering the tie bet is almost
always disregarded by expert opinion as a frivolous wager that only a
fool would make, since it is fourteen times less favourable than the
bank or player bets. But, as Friedman's study showed, the tie
advantage changes much more rapidly than the player or the banker. The
tie is like a golden chalice in a snake pit."
Baccarat for the Clueless
http://ourworld.compuserve.com/homepages/greenbaize21/extract.htm

There's some qualitative information about your issue (search for
'perfect' in the document) in
The Intelligent Gambler
http://conjelco.com/IG/IG3.pdf

Searching "Composition dependent" and "composition independent" turns
up discussions mostly addressing the question of individual card
combinations in the hand.
http://groups.google.com/groups?hl=en&lr=&ie=ISO-8859-1&q=%
22composition+dependent%22+%22composition+independent%22+blackjack&sa=N&tab=wg

That and systems that exploit the imperfect shuffle:
Shuffling Cards
http://www.dartmouth.edu/~chance/course/topics/winning_number.html

I think that's all we're going to find that is pertinent to your
question.

Finally, here's an article about a book you might enjoy, although I
doubt it will offer any substance:
Hacking Las Vegas
http://www.wired.com/wired/archive/10.09/vegas.html

Good luck and thanks for allowing me to work on your question.
richard-ga

Not sure how to rate this! I think the researcher did a great job, but
the answer is likely either not on the net, or buried very deeply. A
little misread of the question at first (IMHO), but researcher was
very responsive. Would recommend both research and service, though
maybe not for questions as arcane as this.