## Efficiency of card counting in blackjack (Part 1)

Introduction

In the past, I have discussed some of the mathematics and algorithms involved in analyzing the game of blackjack (see here and here, for example).  Most of that discussion focused on the accuracy and speed of computation.  Here, however, I want to discuss actually playing the game, and in particular, the practice of card counting.  It is possible for an “advantage player” to make money playing blackjack; I propose to consider the question, “How much money can possibly be made, and what is the tradeoff in advantage versus complexity of card counting systems?”  I have conjectured in the past– without any analysis or data to back up the claim– that relatively simple systems are “good enough,” and that the marginal additional advantage of more complicated systems is not worth the greater mental effort needed to practice and execute them.

Trying to answer this question involved writing some new software, some analysis, and a lot of data.  There are several main points in the punchline, so I plan to break this up into a few reasonably-sized pieces.  But first, I want to begin at the beginning, so to speak, with a description of some of the basic concepts involved in card counting, at least as I understand them.  I think this will be useful, partly to collect my thoughts and make this self-contained, but also because I find many such descriptions online and in the literature to be unsatisfying, misleading, or downright inaccurate.

Rules of the game and basic strategy

For consistency, I will assume the following rules throughout: 6 decks dealt to 75% penetration, dealer stands on soft 17, doubling down is allowed on any two cards including after splitting pairs, pairs may be split and re-split up to a maximum of four hands, except that aces may be split only once, with no surrender.

Given these rules, we can describe the simplest “basic” playing strategy with the following table, condensed from the output of the strategy calculator:

```S17, DOA, DAS, SPL3, NRSA, CDZ-

Hard |                Dealer's up card
hand | 2    3    4    5    6    7    8    9    10   A
-----------------------------------------------------------
19  | S    S    S    S    S    S    S    S    S    S
18  | S    S    S    S    S    S    S    s    s    s
17  | s    s    s    s    S    s    s    s    s    s
16  | s    s    s    s    s    h    h    h    h    h
15  | s    s    s    s    s    h    h    h    h    h
14  | s    s    s    s    s    h    h    h    h    h
13  | s    s    s    s    s    h    h    h    h    h
12  | h    h    s    s    s    h    h    h    h    h
11  | DH   DH   DH   DH   DH   DH   DH   DH   DH   H
10  | DH   DH   DH   DH   DH   DH   DH   DH   H    H
9  | H    DH   DH   DH   DH   H    H    h    h    h
8  | h    H    H    H    H    H    h    h    h    h
7  | h    h    h    H    H    h    h    h    h    h
6  | h    h    h    h    h    h    h    h    h    h
5  | h    h    h    h    H    h    h    h    h    h

Soft |                Dealer's up card
hand | 2    3    4    5    6    7    8    9    10   A
-----------------------------------------------------------
20  | S    S    S    S    S    S    S    S    S    S
19  | S    S    S    S    S    S    S    S    S    S
18  | S    DS   DS   DS   DS   S    S    h    h    h
17  | H    DH   DH   DH   DH   H    h    h    h    h
16  | h    H    DH   DH   DH   h    h    h    h    h
15  | h    H    DH   DH   DH   H    h    h    h    h
14  | H    H    H    DH   DH   H    H    h    h    h
13  | H    H    H    DH   DH   H    H    h    h    h

Pair |                Dealer's up card
hand | 2    3    4    5    6    7    8    9    10   A
-----------------------------------------------------------
A- A | PH   PH   PH   PH   PDH  PH   PH   Ph   Ph   Ph
10-10 | S    S    S    S    S    S    S    S    S    S
9- 9 | PS   PS   PS   PS   PS   S    PS   ps   s    s
8- 8 | Ps   Ps   Ps   Ps   Ps   Ph   ph   ph   ph   ph
7- 7 | ps   ps   Ps   Ps   Ps   ph   h    h    h    h
6- 6 | ph   ph   Ps   Ps   Ps   h    h    h    h    h
5- 5 | DH   DH   DH   DH   DH   DH   DH   DH   H    H
4- 4 | h    H    H    PH   PH   H    h    h    h    h
3- 3 | ph   ph   Ph   Ph   Ph   ph   h    h    h    h
2- 2 | ph   ph   Ph   Ph   PH   Ph   h    h    h    h
-----------------------------------------------------------

S = Stand
H = Hit
D = Double down
P = Split

Uppercase indicates action is favorable for the player
Lowercase indicates action is favorable for the house

When more than one option is listed, options are listed
from left to right in order of preference.
```

Note that this strategy is “total-dependent.”  That is, the strategy depends only on the total of the cards in the player’s hand, not on the specific cards that make up that total.  For example, we always hit hard 16 against a dealer 10, whether that 16 is made up of 10-6, 10-3-3, 8-4-4, etc.  Playing this strategy off the top of a full 6-deck shoe yields an expected return of -0.4065%, or a house edge of less than half of one percent of the initial wager.

A first incremental improvement to our strategy might be to make it “composition-dependent.”  For example, although it is indeed optimal to hit 10-6 vs. 10, and to hit 10-3-3, we should stand with 8-4-4.  These are all very close calls; with 8-4-4, the expected return for standing is -53.99%, compared with -54.43% for hitting.  And there are not many of these composition-dependent strategy variations, yielding an overall expected return for the round of -0.4029%, an improvement of less than 4 thousandths of one percent.

Hand composition matters because it affects the distribution of card ranks remaining in the undealt shoe, which in turn affects the probabilities of winning, losing, or pushing the round.  However, let us stick to our total-dependent strategy for now, since as we will see, card counting effectively handles composition-dependence for us.

The effect of not shuffling

Now consider playing multiple rounds from the same shoe before reshuffling.  Given knowledge of the cards dealt in previous rounds, the expected value of using this same fixed total-dependent strategy will in general be different for the next round, possibly higher or lower than before.  Also, the optimal strategy for the next round may also be different.  For example, it may no longer be optimal to hit a particular hard 16 vs. 10, but instead to stand.

Card counting is an attempt to estimate one or both of these two effects: the simplest systems involve only betting strategy, betting more when the expected return is thought to be favorable.  More complex systems also vary playing strategy, e.g., sometimes hitting a hard 16 vs. 10, sometimes standing, depending on which cards have already been dealt in previous rounds.

To see the potential benefit of merely varying your wager, while always playing the same fixed total-dependent strategy above, the following figure shows the distribution of expected return, as a percentage of initial wager, vs. penetration; i.e., how much of the 6-deck shoe is depleted when the round is dealt.  Each of the 4.3 million gray points corresponds to one round in a simulation of 100,000 shoes.  The additional color is an overlaid smoothed histogram to show the density of points in the scatterplot.

Expected return vs. penetration, using a fixed total-dependent strategy.

At the start of each shoe, the expected return is the same -0.4065% every time.  But as we move deeper into the shoe, the range of possible returns increases.  The challenge is to find a way to recognize where we are on this plot, and to bet more when expected return is positive, and bet less– or even sit out– when expected return is negative.

Next time, I will get into how this is done, using card counting systems of varying complexity.  Admittedly, so far none of the above is really anything new.  But my eventual goal is to define and accurately measure the efficiency of these card counting systems; that is, how well do they perform, not only relative to each other, but also relative to the theoretically best possible performance?

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### 5 Responses to Efficiency of card counting in blackjack (Part 1)

1. Bulahka says:

Your basic strategy chart is a little off. Given the rule set you listed, a soft 15 vs 4 is a double, not a hit, and a soft 13 vs 5 is also a double, not a hit.

• You’re right! Thanks for catching my clerical error entering the indicated rules. I have updated the post to reflect the correct strategy table.

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