                         Optimal Wagering
                     Copyright 1991, Michael Hall
               Permission to repost, print for own use.

I think I've got some good discoveries here... even if you don't
follow the math, you can get some useful blackjack information here.

The question of optimal wagering has been brewing on rec.gambling
for a while.  I rephrase this question as the following:

* What's the optimal win per hand as a portion of bankroll and
  what is the betting pattern necessary for this?

That is, we want to maximize E/a' where E is the win per hand
and a' is the required bankroll.

E is simply defined by:

E=sum{WiPiEi}

where i is the situation
     Wi is the wager for that situation
     Pi is the probability of that situation
     Ei is the expected value of that situation

I defined a' in previous articles.  Unfortunately, I made a
slight error, in that I left out a couple of sqrt's.  I
hope the following is correct...

         log((1/R) - 1)
a'= ----------------------------(sqrt(s^2 + E^2))
        /sqrt(s^2 + E^2) + E\
    log| ------------------- |
        \sqrt(s^2 + E^2) - E/

where R is the risk of ruin
      E is the win per hand
      s^2 is the variance of E
      a' is the necessary units of blackjack bankroll

[Incidentally, the Kelly criterion leads to a bankroll formula
 proportional to the one above, and so Kelly betting produces the
 same optimal wagering schemes as the ones shown below.]

I tried to maximize E/a' by taking the derivatives wrt Wi and setting
them to 0.  That got really ugly.  Then I tried to maximize E or minimize
R using various formulations of Lagrange multipliers.  That got really ugly 
too.  I did come up with the partial derivatives, which are ugly themselves,
but solving for the Wi's is where it gets *really* ugly.

So I gave up and just wrote a program to evaluate the function given
Wi's as input, and then I wrote a program to do a simple hill-climbing
on this function in the space of integers between 1 and some maximum
bet like 4 or 8.  My intuition is that hill-climbing should converge to
the global maximum and not a local maximum of this function, but I don't
have any proof of this.  BTW: my program does adjust for the basic
variance of blackjack, increasing the effective bet size by 1.1 and other
such things.

For a downtown Vegas single deck 75% penetration (Snyder's tables in
"Fundamentals of Blackjack" by Chambliss and Rogenski), here is the
optimal betting patterns I found for spreads of 1-2, 1-4 and 1-8:

     SINGLE DECK
    DOWNTOWN VEGAS

                    1-2   1-4   1-8 
  ADV  FREQ  HI-LO  BET   BET   BET
   Ei   Pi          Wi    Wi    Wi
-.026  .065   -5     1     1     1
-.021  .030   -4     1     1     1
-.016  .055   -3     1     1     1
-.011  .070   -2     1     1     1
-.006  .100   -1     1     1     1
-.001  .200    0     1     1     1
+.004  .095   +1     1     1     1
+.009  .075   +2     1     1     2
+.014  .050   +3     2     2     3
+.019  .045   +4     2     3     5
+.024  .040   +5     2     4     6
+.029  .035   +6     2     4     7
+.034  .030   +7     2     4     8
+.039  .030   +8     2     4     8
+.044  .080   +9     2     4     8

The Hi-Lo column shows the approximate High-Low (or Hi-Opt I) count for
each advantage, though you should adjust for the extra advantage from
strategy deeper into the deck.  Note that the bet should not be raised
until a true count of 3, unless you are using a very wide spread. 
You might fool a few pit critters by your low bet at a true count of 2.
(Or at least you won't get nailed when you increase your bet at a true
count of 2, like I did once.)  For the 1-2 and 1-4 spreads, the betting
pattern is easy to remember - true count minus 1 (minimum of 1, maximum
of 2 or 4.) [More exact results using simulations for the input data
showed that the optimal spread for Hi-Lo here is actually to bet equal
to the true count.]

Here's the same stuff, but for 2 decks:

     DOUBLE DECK
  (BSE of -0.2% assumed)

                    1-4   1-8   1-16
  ADV  FREQ  HI-LO  BET   BET   BET
   Ei   Pi          Wi    Wi    Wi
-.027  .060   -5     1     1     1
-.022  .040   -4     1     1     1
-.017  .060   -3     1     1     1
-.012  .080   -2     1     1     1
-.007  .110   -1     1     1     1
-.002  .200    0     1     1     1
+.003  .110   +1     1     1     2
+.008  .085   +2     3     3     5
+.013  .055   +3     4     5     8
+.018  .045   +4     4     7    11
+.023  .040   +5     4     8    14
+.028  .030   +6     4     8    16
+.033  .025   +7     4     8    16
+.038  .020   +8     4     8    16
+.043  .040   +9     4     8    16



Here's the same stuff, but for 8 decks:

      EIGHT DECKS
(NEGATIVE COUNTS PLAYED)
                    1-8   1-16   1-32
  ADV  FREQ  HI-LO  BET   BET    BET
   Ei   Pi          Wi    Wi     Wi
-.030  .010   -5     1     1      1
-.025  .010   -4     1     1      1
-.020  .020   -3     1     1      1
-.015  .060   -2     1     1      1
-.010  .130   -1     1     1      1
-.005  .510    0     1     1      1
 .000  .130   +1     1     1      1
+.005  .060   +2     8     8     10
+.010  .030   +3     8    15     20
+.015  .015   +4     8    16     30
+.020  .010   +5     8    16     32
+.025  .010   +6     8    16     32
+.030  .005   +7     8    16     32


      EIGHT DECKS
(NEGATIVE COUNTS NOT PLAYED)
                    0-8   0-16   0-32
  ADV  FREQ  HI-LO  BET   BET    BET
   Ei   Pi          Wi    Wi     Wi
-.030  .010   -5     0     0      0
-.025  .010   -4     0     0      0
-.020  .020   -3     0     0      0
-.015  .060   -2     0     0      0
-.010  .130   -1     0     0      0
-.005  .510    0     1     1      1
 .000  .130   +1     1     1      1
+.005  .060   +2     4     5      8
+.010  .030   +3     8    10     16
+.015  .015   +4     8    15     24
+.020  .010   +5     8    16     31
+.025  .010   +6     8    16     32
+.030  .005   +7     8    16     32

What follows are statistics on all these different optimal spreads.
The bankroll requirements assume we want to have a 20% chance of
losing *half* the bankroll before winning *half* the bankroll.
One you lose half the bankroll, I'd advise cutting the bet size
in half.  (Note that the desired risk of ruin has absolutely no effect
on the optimal betting pattern - it just changes the bankroll
by a constant amount.)

                                       UNIT^2    UNITS
              % BANK GAIN  UNIT GAIN  VARIANCE  REQUIRED
               PER HAND     PER HAND  PER HAND  BANKROLL
 DECKS SPREAD|  E/(2a')        E        s^2       2*a'   
-------------*--------------------------------------------
1-Deck FLAT  |.001420%      .0050?      1.27      352
1-Deck 1-2   |.008027%      .0165       2.47      206
1-Deck 1-4   |.014170%      .0348       6.16      245
1-Deck 1-8   |.018132%      .0695      19.19      383
2-Deck 1-4   |.002765%      .0170       7.55      615
2-Deck 1-8   |.006787%      .0433      19.92      638
2-Deck 1-16  |.009916%      .0946      65.16      955
8-Deck 1-8   |.000251%      .0064      11.77     2550
8-Deck 1-16  |.000673%      .0162      28.00     2401
8-Deck 1-32  |.001033%      .0328      75.24     3177
8-Deck 0-8   |.000675%      .0086       7.82     1263
8-Deck 0-16  |.001047%      .0169      19.33     1600
8-Deck 0-32  |.001288%      .0326      59.57     2532


Some things to conclude, given the above table:

  * A 1-2 spread on a single deck is more than 6 times more profitable
    than a 0-32 spread on 8 decks!  Even flat betting a single deck
    is probably better. 8 decks stink!

  * It takes a 1-16 spread on double decks to beat a 1-2 spread on single
    decks!  (Can this be true?)

  * A 1-8 spread buys you 29% more income over a 1-4 spread on
    a single deck, but you'll probably lose more than that from
    the extra countermeasures.

  * Given a $6,125 bankroll, you could spread $25-$100 on a single
    deck, making $86.8/hour (.014170%*6125*100).  This is probably
    overly optimistic, since it rare that you can freely spread
    1-4 on a 75% penetration downtown Vegas game.

  * You need about a 1-32 spread on 8 decks before you can get away
    with playing through negative counts.  A 1-8 spread gets killed
    sitting through negative counts, as the high bankroll requirement
    shows.

One thing that might be fun is playing around with the above
betting spreads.  They are optimal, but how weird can you get
without sacrificing much of the E/a'?

I'd like to acknowledge Blair for getting me to think in terms of
percent bankroll win.

