Moss's Algebraic Approximation Formula

From Stanford Wong's BJ21

Posted by Pete Moss on 12 Nov 1997, at 12:11 a.m., in response to Snyder's Algebraic Approximation Formula, posted by Cacarulo on 11 Nov 1997, at 6:22 a.m.

This is supposed to be a follow-up to BRH's message below, but the @#$%^& software won't let me do it. It keeps saying "Duplicate submission". Hope this works.

Moss's algebraic approximation formula (with a lot of help from Modern Probability Theory and its Applications by Parzen, page 387.

I was the one who pointed out that the formula as originally posted was not correct. I'm not sure the above fixes fix it either, but I could be wrong. I haven't studied it.

There is also some question as to whether a best linear predictor is the right tool for generating indices. The fact that the linear combination gives the best least-squares fit overall with the effects of removal does not guarantee it will pass real close to the zero, where the expectation for making one play or the other is the same.

In any case, if I didn't blow it somewhere, here is how to calculate your "mu" and multiplier b.

The best linear predictor of the sum of "effects" E, given a true count calculated with the card tags C

Remove the dealer's up card from the pack. If you are doing composition-dependent stuff, remove the known player's cards also.

Let N be the number of cards remaining in the pack, and for each denomination i, let N(i) be the number of cards of that denomination remaining in the pack.

Let V be either C or E, and define tc(V) to be the true count (per card) of the pack. For example, if you remove an ace (to figure insurance) from a single deck, then tc(HiLo) will be -1/51, the running count divided by the number of cards remaining. 


    define tc(V) = (-1/N) sum_i { N(i)*V(i) }

Let V1 and V2 be effects or tags vectors. Define the dot product of the two vectors over the pack. 

    define dot(V1,V2) = sum_i { N(i)*[tc(V1)-V1(i)]*[tc(V2)-V2(i)] }

The best linear predictor for E, given a C-true count (per card) of index is 


       mu + b*index



where b = dot(C,E)/dot(C,C);

	  mu = tc(E)-b*tc(C);

Notice there is no (N-1)/N factor. I don't know where that came from. This figures true count per card. To get true count per deck, multiply by 52.

For using this formula with an 'effects of removal' vector as defined by Griffin in The Theory of Blackjack see the follow-up article, "How to use effects of removal", below.

Pete

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