From: jacobs@cs.utah.edu (Steven R Jacobs)
Newsgroups: rec.gambling
Subject: Re: maximum expected gain?
Date: 28 Sep 92 18:12:39 GMT
Organization: University of Utah CS Dept
In-reply-to: huh@wccf.mit.edu's message of Mon, 28 Sep 1992 00:20:00 GMT

In article <27SEP199219200531@wccf.mit.edu> huh@wccf.mit.edu (GENE SIMON HUH) writes:
>
>Let's suppose a purely hypothetical situation wherein a person has a 
>perfect memory and a perfect ability to calculate probabilities instantly.  
>What would be the maximum expected gain for this person in a single-deck
>game of blackjack, given that this person can keep track of every card and
>make decisions accordingly?  The many counting strategies outlined in the FAQ 
>approach this ideal with varying degrees of success, but it would be 
>interesting to know what perfect counting strategy would yield in terms of 
>expected gain (besides getting you barred, I mean).

This is a difficult question to answer in a meaningful way, because it
requires a lot of assumptions about the playing conditions.  If you use
Fairyland playing conditions (no heat from the pit, infinite spread possible,
Wonging allowed, 100% penetration, etc.) then you can make the expectation
almost as large as you want by waiting for a huge count and betting
appropriately.  You'll only actually play a hand once every three weeks,
but when you do play, you will have a very high probability of winning.

I guess what you are really asking is "compared to modern card counters,
how much better could a 'perfect' player do"?  Roughly speaking, a good
card counter will have an edge of about 1.5% under typically favorable
conditions, and maybe, MAYBE get 2.0% with conditions that are extremely
favorable.  I would estimate that a "perfect" player could gain an
additional 0.5% under typical conditions and maybe 0.75% more under
extremely favorable conditions.  Of course, as conditions get closer and
closer to the fairyland ideal, the perfect player will do better.  The
point is that card counting is reasonably efficient.  There is room for
a fair amount of improvement, but the cost in added complexity is so high
that it isn't practical to try to emulate the "perfect" player.

> Would it be easy to
> train a computer to play blackjack using a "perfect" strategy?  It doesn't
> seem to me as if it would be too hard, but I'm not computer-literate, so
> what do I know?

It depends on what you mean by "easy".  If you really want absolutely
perfect playing/betting decisions, then I'd say the answer is "no, this
is by no means easy and possibly completely intractible".  However, if
you are willing to accept some approximations for betting decisions and
for pair splitting decisions, then the answer is "it can be (and has
been) done".  I know of nobody that has done exact computations for
pair splitting decisions, as these tend to take a lot of memory and
CPU cycles.  This limitation is bypassed by making approximations
and/or using simulation.  Perfect betting decisions are expensive to
make because they require a complete analysis of the current deck state.
Approximations in betting strategy are necessary for a practical solution.
Most playing decisions (non-pairs) can be computed in less that a second,
and pair splitting based approximations can be computed in a few seconds
in most cases (splitting 10s vs. dealer 10 is fast, splitting 2s vs. dealer
ace takes much longer).
--
Steve Jacobs  ({bellcore,hplabs,uunet}!utah-cs!jacobs, jacobs@cs.utah.edu)
"Be afraid.  Be very afraid.  I have K-10 offsuit...."