How to use effects of removal

From Stanford Wong's BJ21

Posted by Pete Moss on 13 Nov 1997, at 10:46 p.m., in response to Re: Moss's Algebraic Approximation Formula, posted by Cacarulo on 13 Nov 1997, at 7:38 a.m.

I've posted several versions of this, found a small mistake, deleted it, posted again ... I think this one is right, but no promises. Use at your own risk.

Okay, here's the last piece of the puzzle -- how to use "effects of removal" in conjunction with the best linear predictor. We need two more pieces of info for the algorithm, as you will see.

The effects of removal vector is based on some pack EP (for "effects pack"). It might be a 51 card pack (dealer's up-card removed), or in the case of the insurance EOR vector you posted here, it is a full 52 card deck. It doesn't matter to us what the "effects pack" EP is, so long aswe know how many cards are in it, which we will denote EP.N, and one other bit of intormation.

To calculate the effects vector, you first calculate the difference between the player's expectations for the two plays under consideration when drawing from the pack EP. Call that number EPD, for "effects pack difference". In the Griffin tables, pp 74-85, it is given (in percent) in the next to last column. That is the other extra piece of information we need.

For each card denomination i, the effects of removal number EOR(i) is defined as follows: Remove one card of denomination i from the pack EP, and get a new difference in EV's for the two plays under consideration using that pack of EP.N-1 cards. Call that difference RD(i). The EOR vector is then defined as,

       EOR(i) = RD(i) - EPD            [1]

Reiterating, to use the best linear predictor and an effects of removal vector EOR, we need to know EPD and EP.N. Using that information, we will now form a new vector, and call it E.

       E(i) = (EP.N-1)*EOR(i) - EPD    [2]

We now proceed as described previously, using the E as defined above in the best linear predictor algorithm.

Let's see how it works with the insurance example. We will use the following EOR vector: EOR(ten) = -.0407, EOR(others) = .0181. EP.N is 52. We calculate EPD = 3*(16/52)-1 = -.076923076. That is the difference in EV between betting at 2:1 odds that a ten will be drawn from EP (a full 52 card pack) and the EV of not betting at all (zero).

We calculate E using [2] and come up with this: E(ten) = -1.99878, E(other) = 1.00002. It is very close to the insurance count, but not exact because the EOR vector has roundoff error. So continuing with our calculation as before, we will get an answer that is close to what we got using the insurance count to begin with. We now get index = 1.4342. If you do the same calculation, but for double deck, you get 2.39294. That was calculated using the same effects pack EP, but a different (two deck) pack in the best liner predictor part of the calculation.

Pete