From: breid@pyrhard2.eng.pyramid.com (Bill Reid)
Newsgroups: rec.gambling
Subject: Re: BJ strategy diff. (need help on  h,dd,s)
Date: 21 Jul 92 16:05:23 GMT
Organization: Pyramid Technology Corporation, Mountain View

In article <JACOBS.92Jul18101647@cells.cs.utah.edu> jacobs@cs.utah.edu (Steven R Jacobs) writes:

I'm confused (pause for oh-so-witty comeback)
>
>                         distribution of outcome
>                        --------------------------  
>    play    value       -2    -1     0    +1    +2
>----------------------------------------------------
>   double:  0.482626   .345        .069        .586
>    stand:  0.482354         .204  .110   .686
>
>If you double down, you expect to win 58.6% of the time and lose 34.5% of
>the time.  If you stand, you win about 10% more often (68.6% of the time)
>and lose about 14% less often (20.4% of the time).  This tradeoff almost
>exactly balances out the expectation for these two options.
>
>Let's see how Kelly would bet on these two plays, if adjusting the bet
>size were allowed.  Note that the expectations here are very large, so
>we cannot use the "bet proportionally to expectation" rule to get an
>accurate answer, because that rule is an approximation that only applies
>when your advantage is small.
>
>First, lets find the optimal bet size for doubling down.  The log
>utility for betting a fraction 'b' of the total bankroll is:
>
>   U(b) = .586*log(1 + 2b) + .345*log(1 - 2b)
      ^                 ^^                 ^^
What is b?  Is b our bet size, bankroll, or Kelly fraction (I guess 
U(b) is our Kelly fraction, as in f(b)?)?  And why are 
we doubling it?  Because we are doubling the bet?  Isn't that begging 
the question?  Aren't we trying to find the optimal size to bet on the 
expectation and variance of "A8 v 6 with one more card"? How can we 
do that if we have already arbitrarily doubled our a) bankroll b) bet 
size c) Kelly fraction d) whatever b is supposed to be (do-be-do-wah)? 
  
>The coefficients of .586 and .345 are the probabilities of winning or
>losing two bets in this situation.  The maximum can be found by taking
>the derivative of this equation and setting dU/db = 0.
>
We'll take your word for it, as the rules of derivations in calculus is 
the kind of grunt work I leave for the hoi polloi to soil their hands 
with.  

>      2(.586)/(1+2b) -2(.345)/(1-2b) = 0
>   => 1.172*(1-2b) = .690(1+2b)
>   => (1.172 - .690) = 2*(1.172 + .690)*b
>   => 0.482 = 3.724b
>   => b = .1294
       ^   ^^^^^
Ah, so b IS the Kelly fraction, not f(b), or U(b)?  And the algebra  
works out, so long as b is a fraction of a unity (bankroll), because plug 
in a number over 1, and we got problems. 
 
>So, Kelly says to bet 12.9% of your bankroll if you are allowed to
>double down.  Not quite the 18% that Bill claimed, but still a pretty
>good chunk of the bankroll.  

Now is that the doubled amount, or the undoubled amount?  I really AM 
confused here.  If we have 6.45% of our bankroll out, do we  
acheive perfect log utility by doubling it, or are you saying that 
we should double 12.9% to 25.8%?  Please clarify.  Who gave you 
permission to do this?

>Now let's see what Kelly would say about
>standing with these cards.  The utility equation in this case is:
>
>   U(b) = .686*log(1 + b) + .204*log(1 - b)
    ^^^                 ^                 ^
Okay, here we are looking at solving the Kelly fraction U(b) (f(b)?) for 
the expectation and variance of "A8 vs 6, no cards taken".  And although 
in this case we have not doubled our b, we seem to be solving for 
b (bankroll unity?) or b (bet size) or wwwwwwhat?  Are you saying 
f(b)=b above, then solving for b, or have I been exposed to too much 
EMF here?

>  dU/db = .686/(1+b) - .204/(1-b)
                   ^Kelly?      ^Kelly?
>
>   =>  .686*(1-b) = .204*(1+b)
>   =>  (.686 - .204) = (.686 + .204)*b
>   =>  .482 = .890b
>   =>  b = .5416
        ^   ^^^^^Yeah its Kelly allright, so 
                 f(.5416)=.686*log(1+.5416)+.204*log(1-.5416)
                 spurk-sparkle-spuzz-spatter-sbam-sputtttttttttth! 

>So, Kelly says to bet over 54% of your bankroll in this case.  

So KELLY gave you permission.  I should take this up with him.

>This
>is a little over 4 times as much as Kelly would say to bet on doubling
>down.  Since the expectation in about the same in both cases, a Kelly
>bettor will make 4 times as much money by ignoring basic strategy
>and standing instead of doubling down.  

How does this follow?  I thought the precise Kelly fraction would yield 
the log optimal bankroll growth, and that any other bet would yield 
less utility.  Did Kelly give you permission to make the giant leap of 
logic and math and say that if the Kelly fraction is lower for one 
bet over another, we just divide the two fractions and claim that as 
our relative utility?  Prove this mathematically.

>This would seem to imply
>rather strongly that standing is a _much_ better play, at least for
>*some* bet sizes.
>
For the bet sizes (b) above, which is also the bankroll size (even though 
we double it arbitrarily, so now we have TWO bankrolls), 
which is also the Kelly bankroll fraction, all at 
the same time.  It is also true in flatland, I hear.

>OK, we aren't generally allowed to see in advance which cards will
>be dealt to us, so we have to make bets that are much smaller than
>54% of the bankroll.  

We are also making bets smaller than 12.9% if we use log optimal betting 
before we begin to play.  In fact, with Basic Strategy, we are making bets 
around 0.042% (roughly 0.0006 = 1.4b, as above, feel free to correct).  So 
if 12.9% is our riskless optimal bankroll growth log fraction for doubling, 
how can be possibly be "going crazy" if we are using log optimal betting 
rigorously throughout the game? 

>Let's say we've just bet 1% of our bankroll,
>and see which of these play options will give us greater utility. 

Oh yes, let's do.  This gets better all the time.  Watch the hands 
carefully folks.  

>THAT is how you decide which play to make, if you want to optimize
>utility (this applies to _any_ utility function, not just log).
>
So THAT'S how you do it!!!!

>For doubling:
>
>   U(b) = .586*log(1 + 2b) + .345*log(1 - 2b)
      ^Kelly right?     ^^Kelly*2 right?   ^^Kelly*2 right?
>
>so
>
>   U(.01) = .586*log(1.02) + .345*log(0.98) = 0.00463
      ^^^NOT KELLY!!! ^^^^OH NO!!!!    ^^^^YIKES!!!!

And with a hearty "PRESTO"!:
>
>For standing:
>
>   U(b) = .686*log(1 + b) + .204*log(1 - b)
      ^now bank frac    ^now bank frac    ^now bank frac
>
>so
>
>   U(.01) = .686*log(1.01) + .204*log(0.99) = 0.00478
      ^^^             ^^^^             ^^^^    ^^^^^^^So the Kelly fraction
                                                      of the bankroll 
                                                      fraction which was 
                                                      the Kelly fraction 
                                                      of the bankroll unity 
                                                      is now the bankroll 
                                                      fraction of the 
                                                      Kelly fraction, leading 
                                                      us ineluctably to 
                                                      the conclusion: 

>So, standing will still give us more utility in this case, and
>it is the better play.  

NOW I UNDERSTAND!!! 

>Note that it is only _slightly_ better,
>but we knew it was somewhat borderline since the expectations
>are so close for the two plays.  

Gee, I wonder how close it would be if you didn't start off by doubling 
our bankroll fraction/Kelly fraction/bet size/smoke and mirrors b in the 
first place?

>So, if you are a Kelly bettor, you need a
>bankroll of more than 5000 units to ever make this play
>profitable.
>
So, we can take a 500 unit bankroll, plug in an A8 v 6 as the hand, 
fly through two multi-million hand simulations playing it out both 
ways, and wind up with a higher ROI for not doubling?  Let's do it.

>I would encourage anyone who wishes to refute this argument to SHOW
>where I have made a mistake in the numbers.  Replies of the form
>"too much smoke, I'm right, please don't respond" just won't cut
>it in a public forum.
>
I would encourage anyone who actually bought this analysis to seek 
analysis.  

>Yes, you can learn from the net, but it helps to pay attention to what
>is really being said.
>
Yes, indeedy.

Oh, by the way, I LOVE double after splits, unlimited splitting, any 
kind of unrestricted rules, just because it's a lot more fun to spread 
out a whole bunch of bets on just one starting hand.  What's my optimal 
strategy for THAT utility function?
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