Skeptic Tank Text Archive File
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 hillclimbing
on this function in the space of integers between 1 and some maximum
bet like 4 or 8. My intuition is that hillclimbing 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 12, 14 and 18:
SINGLE DECK
DOWNTOWN VEGAS
12 14 18
ADV FREQ HILO 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 HiLo column shows the approximate HighLow (or HiOpt 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 12 and 14 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 HiLo 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)
14 18 116
ADV FREQ HILO 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)
18 116 132
ADV FREQ HILO 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)
08 016 032
ADV FREQ HILO 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'
*
1Deck FLAT .001420% .0050? 1.27 352
1Deck 12 .008027% .0165 2.47 206
1Deck 14 .014170% .0348 6.16 245
1Deck 18 .018132% .0695 19.19 383
2Deck 14 .002765% .0170 7.55 615
2Deck 18 .006787% .0433 19.92 638
2Deck 116 .009916% .0946 65.16 955
8Deck 18 .000251% .0064 11.77 2550
8Deck 116 .000673% .0162 28.00 2401
8Deck 132 .001033% .0328 75.24 3177
8Deck 08 .000675% .0086 7.82 1263
8Deck 016 .001047% .0169 19.33 1600
8Deck 032 .001288% .0326 59.57 2532
Some things to conclude, given the above table:
* A 12 spread on a single deck is more than 6 times more profitable
than a 032 spread on 8 decks! Even flat betting a single deck
is probably better. 8 decks stink!
* It takes a 116 spread on double decks to beat a 12 spread on single
decks! (Can this be true?)
* A 18 spread buys you 29% more income over a 14 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
14 on a 75% penetration downtown Vegas game.
* You need about a 132 spread on 8 decks before you can get away
with playing through negative counts. A 18 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.
EMail Fredric L. Rice / The Skeptic Tank
