Card Counting

Abstract

The analysis in Chap. 7 of Optimal Basic Strategy considered only the round immediately following a shuffle, so that the first card drawn had value j with likelihood d ₀(j). But for subsequent rounds, deeper into the pack, the distribution of the remaining cards varies and so do the likelihoods d(j).

1 Appendix 1 Asymptotic Distribution of Card Likelihoods

Equation (8.1), exhibited at the start of this chapter, should be recognizable to those familiar with discrete probabilities; but its asymptotic limit (8.2) may be less familiar, so the derivation is worth outlining. For compactness we use d _j ⁰ and d _j here rather than d ₀(j) and d(j) as in the text body; and we set ${d}_{j} \equiv {d}_{j}^{0}(1 + {\varepsilon }_{j})$.

If the number of cards of each value is considered large, ν _j > > 1, then each factorial in (8.1) can be replaced by its asymptotic Stirling approximation, $n! \approx \sqrt{2\pi n}\,{(n\left /e\right.)}^{n}$; Stirling is within 2% of correct even for n as small as 4, and the error diminishes rapidly as n increases. After considerable manipulation, (8.1) can then be written as the distribution of likelihoods d,

$$\begin{array}{rcl} p(\hat{M}\mathbf{d})& & = \delta ( \sum _{j}{d}_{j}^{0}{\varepsilon }_{ j},0)\sqrt{ \frac{2\pi (52D )f\tilde{f}} {\prod \limits_{j}\left [2\pi {\nu }_{j}(\tilde{f} +\tilde{ f}{\varepsilon }_{j})(f -\tilde{ f}{\varepsilon }_{j})\right ]}} \\ & & \qquad \times \exp \left [ \sum _{j}{\nu }_{j}\Phi ({\varepsilon }_{j})\right ], \end{array}$$

(8.34)

with the further notation $\tilde{f} \equiv 1 - f$ and

$$\Phi (\varepsilon ) \equiv \ln \left [ \frac{{f}^{f}{(\tilde{f})}^{\tilde{f}}} {{(\tilde{f} +\tilde{ f}\varepsilon )}^{\tilde{f}+\tilde{f}\varepsilon }{(f -\tilde{ f}\varepsilon )}^{f-\tilde{f}\varepsilon }}\right ].$$

(8.35)

In the asymptotic limit, the otherwise discrete variables d become quasi-continuous: the expected value of a function of d, which in the discrete case sums the function over the distribution $p(\hat{M}\mathbf{d})$, becomes an integration over d. Also, the Kronecker delta converts to a Dirac delta. Furthermore, within that integration the exponential part of (8.34) is mostly very small, since ν _j is very large and Φ can be shown to be nonpositive. Thus, it is asymptotically valid for each d _j integral to focus on the integration range near ɛ _j = 0, where Φ(0) = 0: we replace Φ and the square root in (8.34) by the lowest nonvanishing term in their Taylor series about ɛ _j = 0. In particular, $\Phi ({\varepsilon }_{j}) \approx -\tilde{f}\,{\varepsilon }_{j}^{2}/2f$. The process described here for taking the asymptotic limit is typically called the Method of Stationary Phase.

To complete the derivation, rescaling is needed for the conversion of the ten discrete sums to continuous integrations (over the range extended to − ∞ < d < + ∞), and from the Kronecker to Dirac delta functions. Also, bring back the parameter $\Delta \equiv \sqrt{f/52D\tilde{f}}$ and restore the original variables via the reverse substitution to ${\varepsilon }_{j} = ({d}_{j} - {d}_{j}^{0})/{d}_{j}^{0}.$. Then the asymptotic limit of (8.34) becomes just the form (8.2), QED.

2 Appendix 2 Eigenmodes

In matrix algebra, an n ×n dimensional real symmetric matrix M _{i, j} = M _{j, i} is said to have eigenvectors e _i ^μ, where 1 ≤ μ ≤ n, if ∑ _j M _{i, j} e _j ^μ = m _μ e _i ^μ is satisfied for each μ; m _μ is termed the corresponding eigenvalue. If the eigenvectors are also chosen to be orthonormal, $\sum _{j=1}^{n}{e}_{j}^{\mu }{e}_{j}^{\nu } = \delta (\mu ,\nu )$, then the matrix M can be represented in terms of its eigenmodes as ${M}_{i,j} = \sum _{\mu =1}^{n}{m}_{\mu }{e}_{i}^{\mu }{e}_{j}^{\mu }$. For a fuller exposition see Shores (2007).