From: jacobs@cs.utah.edu (Steven R Jacobs)
Subject: Re: BJ strategy diff. (need help on  h,dd,s)
In-reply-to: breid@pyrhard2.eng.pyramid.com's message of 23 Jul 92 02:46:51 GMT
Organization: University of Utah CS Dept
	<JACOBS.92Jul22140307@cells.cs.utah.edu> <182751@pyramid.pyramid.com>
Date: 23 Jul 92 08:34:42

In article <182751@pyramid.pyramid.com> breid@pyrhard2.eng.pyramid.com (Bill Reid) writes:
>>In article <JACOBS.92Jul18101647@cells.cs.utah.edu> jacobs@cs.utah.edu (Steven R Jacobs) writes:
>
>>The 12.9% is the original bet size, doubling gives 25.8% of total
>>bankroll at risk.  You are right, I didn't make this as clear as
>>I should have.
>
>Now I may not understand the true Kelly strategy (as opposed to the 
>stripped down version typically used), but 25.8% of your bankroll 
>sounds high to bet on a 25% ev hand (maybe I really DON'T want 
>the Kelly strategy).

Actually, this is a 48% e.v. hand, and betting 25.8% is the best
only for doubling.  For standing (which is also 48% e.v.), the
best bet is 54% of bankroll.  But these numbers are worthless in
any real situation, because you need to know in advance that you
will be dealt A8 vs. 6 in order to place that optimal bet.

>Now without casting any aspersions, is it 
>remotely possible that pulling this number out of what you have 
>now clarified as a utility function may have given a wrong answer? 

That is remotely possible, yes.  But I've checked the numbers, and
they appear to be correct.

>It certainly is an answer that most people would feel uncomfortable 
>putting on in actual play.  

With good reason -- when bets are placed, you virtually never have
the 48% expectation needed to justify such huge bets.

>>Actually, you've (quite accidently) made a valid point here.  
>
> Accidents will happen!  Now we can really compare relative utility, 
> as follows:
>
>>The optimal bet size should be plugged into the equations to find out
>>what the actual utility is each case, and the "best" bet chosen by
>>taking the option which gives the most utility.
>>So, utility if we choose the "double down" game (with bet sizes
>>that are optimal for doubling):
>>   U(.129) = .586*log(1 + .258) + .345*log(1 - .258) = 0.03155
>>and the utility if we choose the "stand" game (with bet sizes
>>that are optimal for standing):
>>   U(.5416)= .686*log(1+.5416) + .204*log(1-.5416) = 0.1378
>>So, we get much more utility by standing instead of doubling, even
>>if we have the luxury of changing the bet size in each case.

It is important to be sure that we understand what this comparison
means, because it is sort of an apples-to-oranges comparison.  This
compares the maximum attainable utility for doubling to the maximum
attainable utility for standing.  The problem with comparing these
two numbers is that they occur at different bet sizes.  In any "real"
situation, the bet will be placed before we are faced with the playing
decision.  Then the comparison that matters is "given *this* bet size,
which play option will give the most utility".  It is important to
understand that the answer to this "correct" comparison will change
as the bet size changes -- for very small bet sizes, the answer will
be the same as basic strategy, but for larger bet sizes the answer
will sometimes be different than basic strategy.  It is also important
to understand that this dividing line between "very small bet sizes"
and "larger bet sizes" will be different for each situation (player
cards and dealer upcard) the the player can face.  This dividing line
also changes with the count, and a full set of strategy indices (where
"strategy index" == "this dividing line") based on the player's chosen
counting system, would be the important thing for a card counter to
know.

> Now aside from how the .129 and .5416 numbers were originally 
> generated (not that important), I am wondering if we can settle 
> on this as the correct method of evaluating relative utility in 
> this case.  No flames, just a question.  

I would say that the correct way to evaluate relative utility is
to simply compute, given the size of the bet, the utility for each
playing option, and choose the play that gives the most utility.
Once the bet size has been fixed by placing the bet, no derivatives
are needed.

> I ask this for two reasons:
>
> a) it looks reasonable to me (I know that means nothing to you) 
> b) if this is the way, those good at math (not me!) can come up with a 
>    simpler equation in which to plug in the expectation and variance 
>    numbers, and find the crossover point "at home", without resorting 
>    to iterating the bet size down until you're blue in the face 
>    (it looks like we are one derivative away from that)

I've outlined the approach in an article that was posted with the
title "Notes on Log Optimal Strategy" (or something close to that).
It gives the utility functions for hitting, standing, and doubling.
The main difficulty in using these utility functions is finding the
probabilities of winning, losing, pushing, etc. that are needed to
compute the utility.  Unfortunately, finding these probabilities is
no easy chore.  In short, there just isn't any simple formula for
solving these kinds of problems.

>>Sigh.  You've accidently found some problems in my derivation, but
>>don't get (more) cocky (than you already are).  It is clear to me
>>that you'd never _really_ know that you'd found a problem unless
>>I helped point it out for you.
>
> I'll drop the attitude if you drop yours (actually I'll drop it anyway, 
> and have on numerous occasions).

I honestly appreciate that.  I'll be calm and reasonable as long as
you are.  I think this sort of exchange is much more productive than
the flame wars.

>In another article Jacko writes:
>
>>I'm sure this stuff is super-neat-o to all the non-playing experts
>>out there, but from the perspective of a practicing player, what I want
>>to know is at what (positive) true count is the crossover point where
>>doubling A-8 vs 6 becomes more "utilitarian" so to speak?  For example,
>>according to Griffin (Peter, not Merv), the removal effect of a 3, 4, or 5
>>from the deck for this play is greater than 3%, in favor of doubling.  

Short answer: we gotta learn to walk before we can learn to run.

Long answer: Just as basic strategy was developed before card counting,
I think we need to understand "log optimal" equivalent of basic strategy
before we can really know how to compute the log optimal strategy indices.
Computing log-optimal decisions is not too difficult for hit/stand/double
decisions, but it is much more difficult for splitting because there are
shortcuts that can be used to find basic strategy splitting decisions,
and these shortcuts no longer apply when a non-linear utility function is
used.

Another point to keep in mind is that strategy indices based on the count
will be a little different for *each* counting system.  Be patient, we're
working on it.

>>So my wild-ass-guess on this play is that a TC of only +2 or so is going to 
>>see doubling as better than standing.

The A8 vs. 6 is probably borderline enough that a +1 count will swing
the decision for small bet sizes.

>>If it helps your calculations, you
>>can assume that my maximum initial bet on a given hand is always <= 1% of
>>my bankroll (since it is).  (This stuff about 25%-30% bankroll being bet
>>per hand makes a nice strawman but has little practical value.)

There are several basic strategy changes at the 1% level, and more at
about 2%.  I think a Kelly bettor would occasionally make bets as large
are 5% of bankroll on the rare occasion of a +10 count.  I agree that
the 25%+ bet sizes aren't really of much practical value to a "sane"
gambler, but they would be useful under tournament conditions.

I also suspect that the log-optimal strategy indices will get shifted
around quite a bit for some plays, mostly soft doubling and some hard
doubling.  I have no real feel for what will happen with pair splitting,
although I don't expect the changes to be as drastic (but we may be
surprised).
--
Steve Jacobs  ({bellcore,hplabs,uunet}!utah-cs!jacobs, jacobs@cs.utah.edu)
"Never try to teach a pig to sing.  It wastes your time and annoys the pig."