Since May '99
œKȓq@
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.

Twenty-one@geocities.com