Re: Moss's Algebraic Approximation Formula

From Stanford Wong's BJ21

Posted by Pete Moss on 12 Nov 1997, at 6:51 p.m., in response to Re: Moss's Algebraic Approximation Formula, posted by Cacarulo on 12 Nov 1997, at 2:48 p.m.

Hmm... I'm going to have to think about this a bit more. I had to puzzle a bit to figure out what those numbers are, 1.81 and -4.07. It turns out they are 100 times the difference between the expecation for betting a ten will be pulled from a full deck (no dealer's ace removed) and making the same bet when one card is removed, giving a 51 card pack. All the information we need is there, but it is in a somewhat inconvenient form for the best-linear-predictor algorithm. I'll bet that's where the N and N-1 in the other formula come from. They are artifacts of how the "effects of removal" were calculated, and have nothing to do with the number of decks you are calculating the indices for. That's just a guess.

I'll get back to you on using "effects of removal".

If instead of the "effects of removal" vector, you use the unbalanced insurance count, ten = -2, others = +1, it is a piece of cake.

You get a = -.0342052, b = 1.25553. You know insurance will be break-even when the insurance count is zero, so you just need to solve for the Hi/Lo index HLI: 


        a + b*HLI/52 = 0

        HLI = -52*a/b

        HLI = 1.41667

We use HLI/52, because the predictor estimates the per-card true count.

Is that index, 1.41667, any good? For single deck, we can actually calculate the index precisely, by cycling the computer through all possible deck configurations, weight them by likelihood, and tablulate expectations by running count. It turns out that with a running count of +1 and 38 cards remaining (true count 1.36842) insurance is barely "off", and with running count +1 and 37 cards remaining (true count 1.405405), insurance is barely "on", so any number in between 1.368 and change and 1.405 minus change is as good as any other. The index we estimated will only give the wrong answer when the running count is +1 and exactly 37 cards remain unseen; It will say not to take insurance, although insurance has a positive EV of .0927385 percent.

The error is slight, but it is there. The best linear predictor missed the zero. Another approximate method I have uses Gaussian distributions to approximate running count distributions. It gives an index of 1.38338, which will never indicate a wrong decision. The difference is that that method "goes for the goal" -- the index where the decision is break-even, whereas the best linear predictor finds a general solution that minimizes the sum of squared prediction errors across all indices, and then we use that general solution to solve for the zero.

As I said, I'll get back to you on using "effects" vectors. I think it's probably just a bit more algebra.

Pete

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