Optimal (RA) Insurance Strategy

From Stanford Wong's BJ21

Posted by MathProf on 29 Oct 1998, 6:01 pm, in response to Re: Insurance and Reduced Deviation, posted by Grimy Fellow on 29 Oct 1998, 9:27 am

Grimmy wrote:

Interesting, in that it would seem that this is only true when the original hand is likely to win against the dealer's ace. When the original hand is likely to lose the deviation is increased, and thus the insurance bet produces more risk.

First, that statement, as it stands is not true. If you are in borderline situation (1/3 or remaining cards are 10s) then you should take "All or Nothing Insurance" when your hand has -25% conditional expectation, or better. That is, there are lot of losing hands for which you should still take full insurance.

Mathematically your optimal insurance bet in this situation is

           B =  (-R+CE)/3 

where CE is the conditional expectation of your hand, assuming the dealer does not have blackjack. . R is the result that you will be paid if the dealer does have BJ. R is -1, unless you have a natural in which case R=0.

This insurance bet will minimize your overall variance. Here, full insurance corresponds to b=1/2. If you can take partial insurance, this formula tells you what wager to make.

Now if you have a choice between 2 possible insurance bets, you should "round to the nearest". So if your choice is 0 insurance or 1/2 insurance, then your critical value is 1/4. Take insurance when b>=1/2. That is how I came up with the CE figure of -25% above.

Note that some partial insurance is always justified. With a late surrender hand, the optimal bet is 1/6. If you had wagered 6 units, you would insure with 1.

If you have a possiblity of taking "half-insurance" (25% of your bet), then your critical value is 1/8. You will always be above this, so half-insurance always decreases variance in borderline cases. If you are playing 2 spots, insure one of them, no matter how bad they are.

I learned of this equation during conversation with Steve H last year. Since that time, I expanded on these equation and figured out the general RA insurance strategy. It tells you what your optimal ins bet is in any situation, based on bet size, CE, and 10s density. These equations allow you to calculate shifts in your insurance index:

I actually use this in play when I'm more likely to insure 20's in close counts and less likely to insure stiffs.

I think there are many people who do, and who lower their index for insuring 20s downward. Unfortunately, my studies suggested that this usually wrong (unless you use a 10-count). The most common 20 is a pair of 10s. Now RA considerations do cause a lowering of the insurance index. The amount depends on your bet size, and is nearly proportional to it. However, the base-line index for a pair of 10s is slighlty higher than the normal index. (I recall it being about .25 Hi Lo points in 6 decks. For single deck, is is 6 times greater!) ; I can look that up if you want.) You have to a fairly large bet out for the RA effect to overcome the composition-effect. I'll doule check the calcualtions if you want, but I think the index shift is about .26 HiLo points per 1% bet. So in multi-deck play you need a 1% bet to overcome the composition effect, and in singled deck you need over 5%!

Now stiffs a different matter. That is because Ten-Low has a higher (composition-based) insurance index, and RS considerations raise it even further. That is here, both tendencies are cooperating to raise the index. Similarly for A9, bot tendencies cooperate to lower the index.

Don may be aware of this problem being addressed someplace in the literature. If not, and if people are interested, I'll post these equations and supporting theory sometime.