Ruin Again: Part 1 - Feedback

In my June 2011 article I addressed a ruin question about Blackjack from one of my readers.  In it I proposed a simple scheme that used a one-dimensional random walk model that had the same expected value as Blackjack.  Well, did I get feedback!  

Several readers, including my good friend and colleague Stewart Ethier from the University of Utah, wrote and told me that there was a better way to approximate Blackjack using a one-dimensional random walk.  Indeed there is.  One should take into account not only the expected value but the variance as well.  So in this and the two following articles I am going to explain how this is done.

If X represents any random variable, E(X) represents the expected value (or mean) of X where E is the expectation operator.  Let us set

M₀ = E(X)                 (1)

The variance of X is given by

V₀ = E((X – M₀)²)      (2)

The reason for the 0 subscripts will become clear as we proceed.  Using (2) we can write

V₀ = E(X² – 2M₀X + M₀²)

and using the linearity of E this can be rewritten as

V₀ = E(X²) – 2M_oE(X) + E(M₀²)

Recalling (1) we finally have

V₀ = E(X²) – M₀²           (3)

Next I am going to define a one dimensional random walk that has a step size b (for what will be interpreted later as bet size) given by

b = sqrt(V₀ + M₀²) = sqrt(E(X²)         (4)

where sqrt is the square root function and the second equality is given by (3).  The winning and losing probabilities, respectively p and q, are given by

p = ½ + M₀/2b                (5)

and

q = ½ - M₀/2b              (6)

Let’s calculate the mean M of this random walk:

M = pb + q(-b) = (p – q)b = (2M₀/2b)b = M₀

The random walk has the same mean as our original random variable X so we can now ignore the subscript 0 on M.  To calculate the variance V of our random walk, we use formula (3) as follows:

V = pb²  + q(-b)² – M²

so since p + q = 1

V = b² – M² = V₀

The second equality follows from (4).  Hence the variance of our random walk is exactly the same as the variance of our original random variable X ( no more subscripts).

Next month I’ll derive the ruin formula for a one-dimensional random walk and put it in a form that will be useful for our Blackjack question and the following month I’ll put it all together for you.  Thanks to all the folks who wrote to me.  See you next month.

Don Catlin can be reached at 711cat@comcast.net