The Myth of Card Counting

There are lots of articles info on how to count cards but v little on the returns. Ever wonder why there is so little, or why casinos are relatively unconcerned with card counting – there are card counting books in casino gift shops and a few innocuous rule changes would probably wipe it out altogether. The underlying reason is that far from being a ticket to wealth, the returns from card counting are atrocious.

Background

First why does card counting work at all? In Blackjack the dealer has the advantage of collecting a player’s bet whenever the player busts regardless of the dealers outcome. The player has two main advantages, a payout of 3/2 for Blackjack (an Ace / Ten pair) and the ability to stand on any card combination whereas the dealer must hit up to 17.

The result of this is that a deck with a heavy concentration of high value cards greatly favours the player. High cards increase the likelihood of busts, and a player can avoid these by standing on low values.  For example, a player with a card combination equaling 12 facing a dealer’s 6 show card should stand as the next card is likely to bust the player, the dealer however, will have a high likelihood of busting as he will be required to take at least two cards from a deck loaded with high value cards. Thus the effect of high cards is more to bust the dealer than win the player hands. High cars also increase the chances of blackjack for the player.

It would therefore be advantageous to know when there is a high proportion of high value cards in a deck so a player can increase bets and stand on lower values.

Basic Strategy

The starting point for any blackjack counting system is basic strategy – a set of rules determining when to hit, stand, split and double-down based on the player’s cards and the dealer’s show card. This can be represented as a grid or listing of rules. This shouldn’t be a daunting task although you need to be almost flawless at this – only one error per twenty shoes is permissible.

Counting

Next is learning a counting system. Hi-Lo is the most popular system in which a value of one is added for a cards of value 6 and under, one is deducted for 10 value cards and aces, and 7,8,9 are ignored. Thus for a sequence of Jack, 8, 3, Ace , the count would be -1 (ie -1, 0, +1, -1).

This appears extremely simple but requires a great amount of effort since it needs to be executed almost flawlessly (only one or two counting errors are permitted over a six deck shoe). This typically takes several hours per day for two to three months.

This count (‘running count’) only gives the excess number of high cards over low cards, however what we need is the proportion of high cards relative the to the remaining cards. So the ‘running count’ needs to be divided by the number of remaining decks in the shoe to arrive at the ‘true count’. To do this the counter also needs to keep a count of the number of cards played (a rough approximation is usually sufficient).

Once you have mastered card counting, you need to learn the modifications to basic strategy. Since the strategy will be very different in situations where the true count is high. That’s a solid three to six months work, spending of several hours per day.

Calculating Returns

What advantage does all this effort give a card counter over the dealer? A shade over 1%. Hardly juicy, but lets work through the returns.

We will look at this as an investment and so work backwards from the bankroll. Say you have $100,000 to invest in the bankroll. Your betting unit (i.e. the amount your bet is increased for every +0.5 in the true count) should be $200 – this isn’t an exact number but to avoid the fatal blow of wiping out the betting unit 0.1% to 0.5% of the bankroll, in this example I went with 0.2%.

With perfect basic strategy and perfect card counting the expected returns will be the betting advantage multiplied by the betting unit. Thus, in this case the expected return on a hand would be 0.01 x $200 = $2.

Assuming you can play at a rate of 50 hands per hour that gives you $100 per per hour. Its possible to play more hands per hour, especially playing one-on-one with the dealer, but this increases the chance of being detected and also leads to counting errors due to the speed of play.

Next, to set up a confidence interval to estimate the distribution of returns over time. The standard deviation of a bet in blackjack is 1.1 (slightly larger than the bet size due to the increased payout in the event of blackjack). Assuming a normal distribution we can therefore say that the earnings will be the expected return (ie mean) plus or minus three standard deviations with 99.7% certainty.

The real issue for blackjack card counters is that randomness dominates until you play a very large number of hands.

Take the scenario after 100 bets :
The expected return would be $2 x 100 = 200. The standard error increases with the square root of trials therefore the standard error after 100 trials is 10 x (1.1 x 200) = 2200. Thus we can say with 99.7% certainty that after 100 hands of blackjack the return should be $200 plus or minus $6600 (or between -$6400 and +$6800).

Even 100,000 hands doesn’t provide a guaranteed return.  The expected return after 100,000 hands  is $200,000 plus or minus $208,560 (SQRT(100,000) x 220 x 3).

You really need to be approaching half a million hands of blackjack to be deep into positive returns. After 500,000 hands you would be 99.7% sure of having a return of $1,000,000 plus or minus $466,620. Alas 500,000 hands would take about 5000 hours or 625 days of playing 8 hours per day.

Now To The The Real World

Unfortunately the returns only get worse once you start adjusting for the real world. Card counting is actually very obvious, even an inexperienced dealer can quickly identify a counter. The reason is that the betting profile of a counter is totally different to any other player. Dealers will all have a working knowledge of basic strategy and will know a player playing decent basic strategy – which a card counter will do.

The problem is that a card counter will suddenly deviate from playing perfect basic strategy with a low bet to a high bet with major deviations from basic strategy. Some situations in particular are a major tell. Never splitting tens is not just basic strategy but also common sense, however, if when the true count and the dealer show card is a 5 or 6 then a counter must split tens.

The best way to avoid detection is to play as a team with one player as the counter who plays basic strategy but no more. Once the count is high the counter will signal the high-roller who plays a larger bet and does not follow basic strategy. This, however, vastly dilutes returns since the 1% advantage is divided between two players and one player will be playing at approximately a 0.5% disadvantage (although on a lower bet).

Given the large number of hands required and the necessity of playing with more than  one player, card counters usually play in large teams. However, the meager 1% advantage over the house means there is very little return to be shared around.

Hence, most card counting teams have a hierarchy – investors/team leaders, senior player and junior players. With the players being paid a very small return for their work.

In practice the real trick of card counting is finding smart, energetic workers willing to put in six months of training (probably unpaid) and then work back-breaking shifts of 8-12 hours a day at the tables for a shade above minimum wage.