Blackjack Forum |
Poker Tournaments |

## A THEORETICAL BASIS FOR THE VICTOR INSURANCE PARAMETER |
|||||

COMPUTER POWER TO THE PEOPLE!!
FROM ET FAN: Best Casino Bonuses Online |
A Theoretical
Basis for the Victor Insurance Parameter©Copyright ETFan 2007 The Victor Insurance
Parameter (VIP) system for the insurance decision is breathtaking in its
simplicity and power. It
almost makes me want to switch to an
ace-neutral balanced count, instead of my not-broken-don't-fix-it hi-lo.
However, I had some difficulty understanding the arguments behind it in the
library here. (I have trouble going directly from verbiage to hard
numbers, without the intervening equations.) The essence of the
VIP method, as put forth by its inventor Rich Victor, is that the insurance
decision shall rely solely on the running count (RC) divided by the number
of unseen aces. If this results in a value greater than "the threshold,"
then insurance is taken. This threshold is the same, regardless of the
number of decks in the shoe, so it works the same for 8 decks as it works
for pitch. With the VIP system there is
no need
whatsoever to estimate the size of the
remaining shoe, and no need to do a true count conversion!
Yet accuracy is actually improved for the insurance decision. The purpose of this
article is to put the VIP on a sound mathematical footing by sketching a proof
showing it is precisely equivalent to the conventional (Griffin) method of
combining a primary count with an ace side count. In the process I
learned some interesting things, including a very simple way to calculate
optimal threshold values. People uninterested in abstruse mathematics may
wish to skip down to expression 11) where the main conclusions are presented
along with thresholds for some popular counts.
We'll start with the
Griffin method, and simplify to VIP. We have a balanced, ace-neutral
primary count (e.g. Hi-Opt I or the Victor APC) and we keep a side count of
aces. According to the expression on pg. 64 of
The Theory of
Blackjack by Griffin, the correct adjustment of the primary running count
for each extra or deficient blocked card is:1) 52/(52-k) x S
^{k}E/k x S^{13}Y^{2}/ S^{13}YE Now we are looking at
the insurance decision, so the E's are just the effects of removal for
insurance. Ie. 4/221
4/221 4/221 4/221 4/221 4/221 4/221 4/221 4/221 -9/221 for the ten ranks. k = 4 for the four aces, so S
^{k}E/k = 4/221, and the Y's are just the tags for
our primary count. Using these facts, plus the fact that for a
balanced count, S^{9}Y = -4 x tag_{10 }, expression 1) simplifies to
this:2) -S
^{13}Y^{2
}/ (12 x tag_{10}) I have dubbed
this quantity w, for ace weight. w is generally a positive number, since
tag
_{10} is generally negative. Our insurance
criterion then looks like this:
3) {RC - (surplus
aces) x w} / d > index
Where RC is the
Running Count, w is the weight from 2), d is the number of unseen decks
remaining (not necessarily an integer), and index is our insurance index.
Looks simple enough, eh?
However, this index
should not be exactly the same as the regular insurance index sold with the
primary count. That insurance index is generally dependent on the number
of decks we're facing, in order to accurately adjust for effects of removal of
one ace. In other words, you know the dealer is showing an ace when you're
making an insurance decision! But we are already counting all the aces in
our side count. So the correct index to use is the index for an infinite
deck. In that way, the one ace will have no effect (until you bring
in your secondary side count), but your true count will still have an effect,
since you've obviously counted infinitely many cards to achieve that TC (math
head guffaw here).
To make things
simple, I'll calculate this infinite deck index based on a 52 card pack,
and just ignore the removal of any aces. The calculation works out
proportionately, for any number of decks up to infinity.
According to the
principle of proportional deflection (see Grffin pg. 109, or the argument on pg.
63) the expected number of cards of rank i, for a given true count with 52 cards
unseen, is:
4) 4 - tag
_{i}
x TC / S^{13}Y^{2} Specifically, for the
tens (i = 10 to 13), the expected number is:
5) (4 -
tag
_{10} x TC / S^{13}Y^{2} ) x 4 And the expected
number of non-tens is:
6) 52 - (4 -
tag
_{10} x TC / S^{13}Y^{2} ) x 4 If we take insurance,
we win 2 bets when a ten is in the hole, and lose 1 bet when a non-ten is in the
hole. We would like to take insurance when our expectation is
positive. So our criterion for taking insurance is:
7) (4 -
tag
_{10} x TC / S^{13}Y^{2} ) x 4 x 2 - {52 - (4 -
tag_{10} x TC / S^{13}Y^{2} ) x 4} >
0 Solving for TC, our
insurance index is:
8) TC = -4 x S
^{13}Y^{2 }/
(12 x tag_{10}) Exactly 4 times as
large as 2) -- our expression for w. So we can rewrite 3)
thus:
9) {RC - (surplus
aces) x w} / d > 4w
We're almost
there! Naturally now, the expected number of aces is 4d. Let's call
the total number of unseen aces remaining in the shoe: aces (original,
huh?) So the number of surplus aces is (aces - 4d), and 9)
becomes:
10) {RC - (aces - 4d)
x w} / d > 4w
Which simplifies
to:
(RC - aces x w) / d + 4w
> 4w
(RC - aces x w) / d >
0
And since d is always
positive ...
11) RC > aces x
w
Ta Dah!!!! d
drops out, and we no longer have to estimate the number of decks in the hopper
to make our insurance decisions. And the "threshold" is simply w -- the
weight prescribed by Griffin for adjusting a balanced ace-neutral count for a
side count of aces. We can calculate w very easily for any set of tags,
using expression 2).
It seems appropriate
to use the Victor APC as an example. To find the threshold, w = -S
^{13}Y^{2 }/ (12 x tag_{10}), we
note the tags for the
VAPC are: 0 2 2 2 3 2 2 0 -1 -3, so the sum of squares for the tags (S^{13}Y^{2 },
remembering to multiply by 4 for the tens tag) is: 2^{2 }x 5 + 3^{2
}+ (-1)^{2 }+ (-3)^{2 }x 4 = 66. So w = -66 /
(12 x (-3)) = 11/6 = 1.8333... We will take
insurance any time our Running Count is more than 1.8 times the number of unseen
aces in the shoe. (It may be permissable to round the threshold down
slightly, since insurance is often a variance reducer.)
Now that we have
shown that the VIP is equivalent to the conventional Griffin method, we can
assert that the VIP insurance correlations can be calculated with the formula
for multiple correlations exemplified on pg. 62 of
Theory of Blackjack,
assuming we have valid thresholds. It turns out the insurance
correlations are increased from the primary counts by approximately 2% in each
case. The thresholds for a
few popular counts are listed below (without rounding):
Canfield (0 0 1 1 1 1 1 0 -1
-1): 5/6
Hi-Opt I (0 0 1 1 1 1 0
0 0 -1): 2/3
Hi-Opt II (0 1 1 2 2 1 1 0 0
-2): 7/6
Omega II (0 1 1 2 2 2 1 0 -1
-2): 4/3
Uston APC (0 1 2 2 3 2 2 1
-1 -3): 16/9
Victor APC (0 2 2 2 3 2 2 0
-1 -3): 11/6
For fun, here is a
level 1 count with a threshold of exactly one:
(0 1 1 1 1 1 1 -1 -1
-1): 1
I don't recommend
this count, but if you use it, you can take insurance any time your RC exceeds
the number of unseen aces. Same goes for this level 2 count:
(0 1 1 1 1 1 1 1 1
-2): 1
Or here's one with
threshold two:
(0 2 2 2 2 2 2 -2 -2
-2): 2
All right, that one
is silly.
Good night, and Q.E.D.
Gracy. ETF
Return to the Professional Gambling Library Return to Arnold Snyder's Blackjack Forum Online Home |

© 2004-2007 Blackjack Forum Online, All Rights Reserved |
||||||