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 ∑kE/k x ∑13Y2/ ∑13YE
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 ∑kE/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, ∑9Y = -4 x tag10 , expression 1) simplifies to this:
2) -∑13Y2 / (12 x tag10)
I have dubbed this quantity w, for ace weight. w is generally a positive number, since tag10 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 – tagi x TC / ∑13Y2
Specifically, for the tens (i = 10 to 13), the expected number is:
5) (4 – tag10 x TC / ∑13Y2 ) x 4
And the expected number of non-tens is:
6) 52 – (4 – tag10 x TC / ∑13Y2 ) 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 – tag10 x TC / ∑13Y2 ) x 4 x 2 – {52 – (4 – tag10 x TC / ∑13Y2 ) x 4} > 0
Solving for TC, our insurance index is:
8) TC = -4 x ∑13Y2 / (12 x tag10)
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 = -∑13Y2 / (12 x tag10), 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 (∑13Y2 , remembering to multiply by 4 for the tens tag) is: 22 x 5 + 32 + (-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