https://preterhuman.net/docs/index.php?title=Optimal_Wagering&feed=atom&action=historyOptimal Wagering - Revision history2021-05-07T17:11:08ZRevision history for this page on the wikiMediaWiki 1.35.0https://preterhuman.net/docs/index.php?title=Optimal_Wagering&diff=1537&oldid=prevNetfreak: Created page with "<pre> Optimal Wagering Copyright 1991, Michael Hall Permission to repost, print for own use. I think I've got som..."2021-02-15T23:47:58Z<p>Created page with "<pre> Optimal Wagering Copyright 1991, Michael Hall Permission to repost, print for own use. I think I've got som..."</p>
<p><b>New page</b></p><div><pre><br />
Optimal Wagering<br />
Copyright 1991, Michael Hall<br />
Permission to repost, print for own use.<br />
<br />
I think I've got some good discoveries here... even if you don't<br />
follow the math, you can get some useful blackjack information here.<br />
<br />
The question of optimal wagering has been brewing on rec.gambling<br />
for a while. I rephrase this question as the following:<br />
<br />
* What's the optimal win per hand as a portion of bankroll and<br />
what is the betting pattern necessary for this?<br />
<br />
That is, we want to maximize E/a' where E is the win per hand<br />
and a' is the required bankroll.<br />
<br />
E is simply defined by:<br />
<br />
E=sum{WiPiEi}<br />
<br />
where i is the situation<br />
Wi is the wager for that situation<br />
Pi is the probability of that situation<br />
Ei is the expected value of that situation<br />
<br />
I defined a' in previous articles. Unfortunately, I made a<br />
slight error, in that I left out a couple of sqrt's. I<br />
hope the following is correct...<br />
<br />
log((1/R) - 1)<br />
a'= ----------------------------(sqrt(s^2 + E^2))<br />
/sqrt(s^2 + E^2) + E\<br />
log| ------------------- |<br />
\sqrt(s^2 + E^2) - E/<br />
<br />
where R is the risk of ruin<br />
E is the win per hand<br />
s^2 is the variance of E<br />
a' is the necessary units of blackjack bankroll<br />
<br />
[Incidentally, the Kelly criterion leads to a bankroll formula<br />
proportional to the one above, and so Kelly betting produces the<br />
same optimal wagering schemes as the ones shown below.]<br />
<br />
I tried to maximize E/a' by taking the derivatives wrt Wi and setting<br />
them to 0. That got really ugly. Then I tried to maximize E or minimize<br />
R using various formulations of Lagrange multipliers. That got really ugly <br />
too. I did come up with the partial derivatives, which are ugly themselves,<br />
but solving for the Wi's is where it gets *really* ugly.<br />
<br />
So I gave up and just wrote a program to evaluate the function given<br />
Wi's as input, and then I wrote a program to do a simple hill-climbing<br />
on this function in the space of integers between 1 and some maximum<br />
bet like 4 or 8. My intuition is that hill-climbing should converge to<br />
the global maximum and not a local maximum of this function, but I don't<br />
have any proof of this. BTW: my program does adjust for the basic<br />
variance of blackjack, increasing the effective bet size by 1.1 and other<br />
such things.<br />
<br />
For a downtown Vegas single deck 75% penetration (Snyder's tables in<br />
"Fundamentals of Blackjack" by Chambliss and Rogenski), here is the<br />
optimal betting patterns I found for spreads of 1-2, 1-4 and 1-8:<br />
<br />
SINGLE DECK<br />
DOWNTOWN VEGAS<br />
<br />
1-2 1-4 1-8 <br />
ADV FREQ HI-LO BET BET BET<br />
Ei Pi Wi Wi Wi<br />
-.026 .065 -5 1 1 1<br />
-.021 .030 -4 1 1 1<br />
-.016 .055 -3 1 1 1<br />
-.011 .070 -2 1 1 1<br />
-.006 .100 -1 1 1 1<br />
-.001 .200 0 1 1 1<br />
+.004 .095 +1 1 1 1<br />
+.009 .075 +2 1 1 2<br />
+.014 .050 +3 2 2 3<br />
+.019 .045 +4 2 3 5<br />
+.024 .040 +5 2 4 6<br />
+.029 .035 +6 2 4 7<br />
+.034 .030 +7 2 4 8<br />
+.039 .030 +8 2 4 8<br />
+.044 .080 +9 2 4 8<br />
<br />
The Hi-Lo column shows the approximate High-Low (or Hi-Opt I) count for<br />
each advantage, though you should adjust for the extra advantage from<br />
strategy deeper into the deck. Note that the bet should not be raised<br />
until a true count of 3, unless you are using a very wide spread. <br />
You might fool a few pit critters by your low bet at a true count of 2.<br />
(Or at least you won't get nailed when you increase your bet at a true<br />
count of 2, like I did once.) For the 1-2 and 1-4 spreads, the betting<br />
pattern is easy to remember - true count minus 1 (minimum of 1, maximum<br />
of 2 or 4.) [More exact results using simulations for the input data<br />
showed that the optimal spread for Hi-Lo here is actually to bet equal<br />
to the true count.]<br />
<br />
Here's the same stuff, but for 2 decks:<br />
<br />
DOUBLE DECK<br />
(BSE of -0.2% assumed)<br />
<br />
1-4 1-8 1-16<br />
ADV FREQ HI-LO BET BET BET<br />
Ei Pi Wi Wi Wi<br />
-.027 .060 -5 1 1 1<br />
-.022 .040 -4 1 1 1<br />
-.017 .060 -3 1 1 1<br />
-.012 .080 -2 1 1 1<br />
-.007 .110 -1 1 1 1<br />
-.002 .200 0 1 1 1<br />
+.003 .110 +1 1 1 2<br />
+.008 .085 +2 3 3 5<br />
+.013 .055 +3 4 5 8<br />
+.018 .045 +4 4 7 11<br />
+.023 .040 +5 4 8 14<br />
+.028 .030 +6 4 8 16<br />
+.033 .025 +7 4 8 16<br />
+.038 .020 +8 4 8 16<br />
+.043 .040 +9 4 8 16<br />
<br />
<br />
<br />
Here's the same stuff, but for 8 decks:<br />
<br />
EIGHT DECKS<br />
(NEGATIVE COUNTS PLAYED)<br />
1-8 1-16 1-32<br />
ADV FREQ HI-LO BET BET BET<br />
Ei Pi Wi Wi Wi<br />
-.030 .010 -5 1 1 1<br />
-.025 .010 -4 1 1 1<br />
-.020 .020 -3 1 1 1<br />
-.015 .060 -2 1 1 1<br />
-.010 .130 -1 1 1 1<br />
-.005 .510 0 1 1 1<br />
.000 .130 +1 1 1 1<br />
+.005 .060 +2 8 8 10<br />
+.010 .030 +3 8 15 20<br />
+.015 .015 +4 8 16 30<br />
+.020 .010 +5 8 16 32<br />
+.025 .010 +6 8 16 32<br />
+.030 .005 +7 8 16 32<br />
<br />
<br />
EIGHT DECKS<br />
(NEGATIVE COUNTS NOT PLAYED)<br />
0-8 0-16 0-32<br />
ADV FREQ HI-LO BET BET BET<br />
Ei Pi Wi Wi Wi<br />
-.030 .010 -5 0 0 0<br />
-.025 .010 -4 0 0 0<br />
-.020 .020 -3 0 0 0<br />
-.015 .060 -2 0 0 0<br />
-.010 .130 -1 0 0 0<br />
-.005 .510 0 1 1 1<br />
.000 .130 +1 1 1 1<br />
+.005 .060 +2 4 5 8<br />
+.010 .030 +3 8 10 16<br />
+.015 .015 +4 8 15 24<br />
+.020 .010 +5 8 16 31<br />
+.025 .010 +6 8 16 32<br />
+.030 .005 +7 8 16 32<br />
<br />
What follows are statistics on all these different optimal spreads.<br />
The bankroll requirements assume we want to have a 20% chance of<br />
losing *half* the bankroll before winning *half* the bankroll.<br />
One you lose half the bankroll, I'd advise cutting the bet size<br />
in half. (Note that the desired risk of ruin has absolutely no effect<br />
on the optimal betting pattern - it just changes the bankroll<br />
by a constant amount.)<br />
<br />
UNIT^2 UNITS<br />
% BANK GAIN UNIT GAIN VARIANCE REQUIRED<br />
PER HAND PER HAND PER HAND BANKROLL<br />
DECKS SPREAD| E/(2a') E s^2 2*a' <br />
-------------*--------------------------------------------<br />
1-Deck FLAT |.001420% .0050? 1.27 352<br />
1-Deck 1-2 |.008027% .0165 2.47 206<br />
1-Deck 1-4 |.014170% .0348 6.16 245<br />
1-Deck 1-8 |.018132% .0695 19.19 383<br />
2-Deck 1-4 |.002765% .0170 7.55 615<br />
2-Deck 1-8 |.006787% .0433 19.92 638<br />
2-Deck 1-16 |.009916% .0946 65.16 955<br />
8-Deck 1-8 |.000251% .0064 11.77 2550<br />
8-Deck 1-16 |.000673% .0162 28.00 2401<br />
8-Deck 1-32 |.001033% .0328 75.24 3177<br />
8-Deck 0-8 |.000675% .0086 7.82 1263<br />
8-Deck 0-16 |.001047% .0169 19.33 1600<br />
8-Deck 0-32 |.001288% .0326 59.57 2532<br />
<br />
<br />
Some things to conclude, given the above table:<br />
<br />
* A 1-2 spread on a single deck is more than 6 times more profitable<br />
than a 0-32 spread on 8 decks! Even flat betting a single deck<br />
is probably better. 8 decks stink!<br />
<br />
* It takes a 1-16 spread on double decks to beat a 1-2 spread on single<br />
decks! (Can this be true?)<br />
<br />
* A 1-8 spread buys you 29% more income over a 1-4 spread on<br />
a single deck, but you'll probably lose more than that from<br />
the extra countermeasures.<br />
<br />
* Given a $6,125 bankroll, you could spread $25-$100 on a single<br />
deck, making $86.8/hour (.014170%*6125*100). This is probably<br />
overly optimistic, since it rare that you can freely spread<br />
1-4 on a 75% penetration downtown Vegas game.<br />
<br />
* You need about a 1-32 spread on 8 decks before you can get away<br />
with playing through negative counts. A 1-8 spread gets killed<br />
sitting through negative counts, as the high bankroll requirement<br />
shows.<br />
<br />
One thing that might be fun is playing around with the above<br />
betting spreads. They are optimal, but how weird can you get<br />
without sacrificing much of the E/a'?<br />
<br />
I'd like to acknowledge Blair for getting me to think in terms of<br />
percent bankroll win.<br />
</pre><br />
<br />
[[Category:Unsorted]]</div>Netfreak