Brick Waller: insurance sims

[pm]

Isn't variance decreased by decreasing the amount of money you could possibly lose at a time? If you lowered the insurance index, wouldn't it make it more likely that you'd lose an extra half-bet?

[Don Schlesinger]

> Isn't variance decreased by decreasing the
> amount of money you could possibly lose at a
> time? If you lowered the insurance index,
> wouldn't it make it more likely that you'd
> lose an extra half-bet?

That's a good question. And, it shows where my answer may, indeed, be wrong. I was forgetting the extra half unit bet for insurance.

When you take insurance, you can, indeed, lose 1.5 units, instead of just one unit. But, you can also lose zero units much more frequently. So, the question is, which becomes more important? Let's see.

Suppose, to simplify, that, without insurance, a certain hand would win one unit 25% of the time, lose one unit 65% and push 10%. Then, the average squared result is simply 0.9 units.

With insurance, let's say that when the dealer has a natural I would push 30% of the time (average squared result is zero), and when he didn't have a natural (the other 70% of the time), I would lose the insurance bet and the hand (-1.5 units) 65% of the time, lose insurance and win the bet 25% of the time (+.5), and lose the insurance and push the bet 10% of the time (-.5). Then, the overall average squared result from all possible outcomes of taking insurance is: .3*(0) + .7[.65(-1.5)^2 + .25(+.5)^2 + .10(-.5)^2] = 1.55.

Taking insurance adds to the variance! Why? Because the part that contributes the most (the 1.5^2) occurs so frequently (65% of the time). If the hand is a big winner, the multiplier for the 1.5^2 component is much smaller, and variance can, indeed, be reduced.

I learned something.

Don