Posted by Pete Moss on June 06, 1997 at 00:02:41:

Here is what I came up with for calculating optimal betting strategies and for rating counting systems. It is based on the same idea as the schemes described by Winston and BRH, namely optimizing the growth rate of bankroll, but the implementation is different. I describe a score function that can be used to rate a counting system in the same way the price of an annuity or mutual fund might be determined -- by expected future value. A system with twice the score of another is worth twice as much in future value of bankroll. Scores for five counting systems will be given.

First we define for each count i, a random variable Ri, namely the per-unit outcome or "result" of a round of blackjack. The count i may be either a true count for a system like Hi/Lo or HiOpt II, or it may be a running count for a system like K-O. By way of example, the random variable will take on the value +1 when the player wins the original wager (only), -2 when he doubles and loses, +3/2 when he gets an untied blackjack and does not take insurance, -3 when he splits, doubles one hand, and loses all, et cetera.

The expected net return for a round played at count i, expressed as a fraction of the player's bankroll, is then

E[B*Ri] = B*E[Ri]

where B is the fraction of bankroll wagered initially on that round. ("E" is the expectation operator, read "expected value of".)

An estimate of E[Ri] can be read directly from the output of a blackjack simulator such as SBA (Statistical Blackjack Analyzer).

We shall also be interested in the second moment, or expected square of the per unit return,

E[(B*Ri)^2] = B^2 E[Ri^2]

The value E[Ri^2] can be obtained from the output of SBA, as a function of the per unit variance and the per unit expectation:

E[Ri^2] = Var(Ri) + E[Ri]^2

The second term in this expression is small compared to the first, and is frequently ignored, making the variance or "second central moment" a stand-in for the second moment.

Now we define a betting strategy S as a function that takes two arguments, the size of a "unit" (as a fraction of bankroll), and a count i, and returns a number of units to bet. A strategy is called "fixed" if the number it returns does not actually depend on the size of a unit. One strategy which will be of interest to us is the "constrained Kelly strategy with spread Q", which will be described later. We may also examine "fixed" strategies that are set forth in books about counting systems.

OPTIMUM BETTING UNIT SIZE FOR A GIVEN STRATEGY S(u,i)

The certain-equivalent growth rate of bankroll is the expected value of the random variable 1+B*R, where R is the per-round, per-unit expectation, and B is the fraction of bankroll wagered. To obtain the best unit size (as a fraction of bankroll) for a given betting strategy, if thus suffices to maximize the expected value of the logrithm of that random variable. That is done by finding the unit size (fraction of bankroll) u that maximizes the expected value of the expression

Sum_i P(i) log(1+S(u,i)*u*Ri)

where "Sum_i" means to take the sum over all counts i, and P(i) is the frequency of the count i. Let B(i,u) denote the fraction of bankroll wagered, u*S(i,u). We may use the first two terms in the Taylor's series expansion of the log function to obtain the following approximation to the expected value of the log rate of growth of bankroll, given the unit size u and betting strategy S. The function is expressed in terms of the first two moments of the random variables Ri.

"Score function", approximates expected value of log bankroll growth rate:

Sum_i P(i) { B(i,u)*E[Ri] - .5 * B(i,u)^2 E[Ri^2] }

P(i) and E[Ri] are taken directly from the output of SBA, and E[Ri^2] is calculated as previously described from E[Ri] and Var[Ri].

The function can be maximized using any one-dimensional function optimizer. I use Brent's method, because I had the code for it already written.

THE (UNCONSTRAINED) KELLY CRITERION K(i)

Suppose there is no constraint on how much we are allowed to wager. We would choose not to bet at all when the expectation E[Ri] is negative. When it is positive, we would wish to maximize the longterm growth of bankroll.

For every i, find the fraction of bankroll K(i) that maximizes the expected value of

log(1+K(i)Ri)

Again using the first two terms in the Taylor's series expansion, we wish to maximize the approximation,

K(i)E[Ri] - .5 K(i)^2 E[Ri^2]

That expression is at a maximum when its derivative with respect to K(i) is zero:

0 = E[Ri] - K(i) E[Ri^2]

Thus,

K(i) = E[Ri]/E[Ri^2]

when K(i) maximizes longterm growth. That is the familiar approximation to the Kelly criterion generalized for arbitrary results Ri.

THE CONSTRAINED KELLY CRITERION WITH BET SPREAD Q

Because of casino "heat", we may wish to constrain our betting "spread", the ratio of our largest to smallest bet.

For each positive real number q (representing a "1 to q spread") let Kq(u,i) be the betting strategy that returns the value in the range 1 to q that is closest to the number of units the unconstrained Kelly criterion would call for, K(i)/u.

It can be shown that the optimal betting strategy that is constrained to bet from 1 to q units is Kq(u,i) with a unit size u that optimizes the score function described above.

RATING COUNTING SYSTEMS

Systems are often rated by B*E[Ri], the product of overall expected return and average wager size, after B has been adjusted in some way to "account for risk". That method does not measure the system's capacity to make a bankroll grow, because it does not adequately account for bankroll fluctuations that make money unavailable for (optimal) wagering. It uses only the first moment of the random variable Ri. It should be clear now that the score function

score = sum_i P(i) { B(i)*E[Ri] - .5 B(i)^2 E[Ri^2]}

should be the prefered method. Using that score function, a counting system may be rated by how quickly it can be expected to grow a bankroll. We may estimate the expected number of rounds required to double a bankroll using the score function:

E[doubling_time] = log(2)/score(system)

The relative worth of one system as compared to another may be calculated as the ratio of their scores (or doubling times). A system with half the doubling time is worth twice as much profit out the door, given the same initial bankroll and the same number of rounds played.

Using the output of SBA, and assuming a 1 to 5 spread at single deck, one player, head's up, H17, DOA, 65% penetration, face-down game, some expected doubling times were estimated:

(1) Omega II with Ace side count, optimal betting ................... 7437
(1) HiOpt II with Ace side count, optimal betting ................... 7486
(2) Uston Advanced Plus/Minus with Ace side count, optimal betting .. 7915
(3) K-O, running count mode, optimal betting ....................... 12370
(4) K-O, running count mode, published betting strategy ............ 12831


Notes:

(1) The Omega II indices that come with SBA were used for both Omega II and HiOpt II.

(2) Carefully risk-adjusted indices for doubling and splitting were used, which may account for the score being better in relation to the two-level counts than might be expected.

(3) The "rounded" indices in the K-O book were used with the optimal constrained Kelly betting strategy: 1 unit through pivot-2, approximately 2.5 units at pivot-1, 4 units at pivot, 5 units at pivot+1. Optimal unit size for that strategy was used.

(4) The "rounded" indices in the K-O book were used, with the "fixed" betting strategy indicated in book. Optimal unit size for that strategy was used.