Card Counting

Thinking and Games

Card counting is a common casino strategy used primarily in the blackjack family of games to help the player decide whether the next hand is likely to give advantage to the player or the dealer. The most basic variation of card counting in blackjack is rooted in the idea that high cards, most notably aces and 10s, benefit the player more than the dealer, whereas the low cards, particularly 5s, but also 3s, 4s, and 6s, benefit the dealer more than the player. The strategy takes advantage of basic statistics: A high concentration of aces and 10s tends to diminish the inherent house advantage by increasing the player’s chances of hitting a natural blackjack, whereas a high concentration of 5s and other low cards further compounds the house advantage. Those who are able to count cards, typically characterized as skilled players, can therefore modulate their betting by altering bet sizes based on the composition of remaining cards. Thus even though, on average, a blackjack player can expect to win only about 48% of the hands dealt and lose the remaining 52% (ignoring ties which can be expected in about 9% of the hands), varying the size of bets based on card count-adjusted outcome probability – that is, placing higher bets when counts are advantageous to the player and placing lower bets otherwise – can result in a player beating the house, as measured by monetary outcomes. As famously portrayed in the 2008 movie ‘21’,1 card counting blackjack teams have been known to win big – millions of dollars big. Although legal in all major gaming jurisdictions, card counting with the mind 2 is, not surprisingly, frown upon by casinos, with many investing considerable resources in technological and human countermeasures.

At its core, counting cards offers blackjack players the ability to reduce decision ambiguity by relying on objective, in this case statistical, evidence. An unskilled player is likely to make betting choices based on his intuition alone – as such, that player is depending almost entirely on chance, which as noted earlier favors the house. A skilled player, on the other hand, enhances his intuition by taking into account empirical evidence which chips away at the house advantage. We certainly cannot dismiss the possibility of an unskilled player winning, and possibly even winning big – after all, virtually all lottery jackpot winners are just lucky pickers of essentially random numbers. However, there is a considerable difference between picking a favorable outcome in a single-trial event, such as a lottery drawing, and winning a game comprised of series of somewhat sequentially dependent decisions. The truly interesting point here is that decision-guiding precision of card counting-derived information is relatively low, ultimately just enabling players to develop more refined expectations regarding the likely composition of the mix of cards in the shoe 3 – in essence, it just tightens the estimated probability ranges. And yet that seemingly small amount of information is enough to deliver impressive and repeated player wins, and certainly enough to compel casinos to invest in a variety of countermeasures including decreasing deck penetration, preferential shuffling, and large wager increase-triggered shuffling, to name just a few. In short, a skilled player’s mind can translate relatively small infusions of objective insights into disproportionately large benefits.

Although the exact mechanics of how that happens are still shrouded in mystery, some possible clues might be offered by considering a different game, but this time not a game of chance, but what might be considered the ultimate game of strategy – chess. This centuries-old game (it is believed to have originated in India around 6th century AD) is a true test of cerebral fitness – to prevail, a player needs to consider a wide range of available strategies and tactical moves, all while recognizing and adapting to the opponent’s moves. Given its highly analytical, zero-sum (one player’s gain is the other player’s loss), perfect information (all positions are perfectly visible to both players), and perhaps most importantly combinatorial (each successive move generates a typically large set of possibilities) nature, chess naturally lends itself to machine learning in the form of computer-based chess-playing systems. Thus not surprisingly, the history of those systems roughly parallels the history of what is known as ‘artificial intelligence’ (AI).

AI is a scientific endeavor of growing interest (and controversy) that aims to answer the basic question: Can a machine be made to think like a person? Almost from the start, that question was tied to the question of whether a machine could be made to play chess, as the strategic nature of that game embodies, in many regards, what we tend to view as uniquely human combination of reason and creativity. Building on the work of Alan Turing,4 John Von Neumann, Claude Shannon, and other early 20th-century information theory pioneers, computerized chess-playing system designers forecasted that machines would come to dominate humans as early as the 1960s,5 but it took three more decades of algorithmic design and computing power advances for that forecast to come true (now computerized chess systems routinely beat the best human players, so much so that those systems play other such systems for the ‘best of the best’ bragging rights). That tipping point was reached in the famous 1997 match, which pitted the then reining chess world champion Garry Kasparov against IBM’s supercomputer known as Deep Blue. That event marked the first time a computer defeated a human chess champion,6 under regular time controls (on average 3 min per move). While the news of Deep Blue’s victory stirred worldwide sensation, looking back at those events we should have been more astounded that it took that long for a computer to better the best human player. After all, the game of chess is combinatorial in nature 7; moreover, given the ever-increasing computer processing power and advances in evaluation algorithms, the eventual dominance of computerized chess-playing systems was an inescapable consequence of technological and scientific progress – the only true question was ‘when’.

Thus, the truly fascinating aspect of Kasparov’s duel with Deep Blue was the astounding evidentiary asymmetry: Capable of processing more than 200 million instructions per second, in a timed match, on turn-by-turn basis, Deep Blue was able to evaluate millions of possible move sequences, whereas its human opponent was only able to carefully consider a small handful, likely fewer than ten, sequences. Further adding to that disparity was the fact that although the designers of Deep Blue had access to hundreds of Kasparov’s games, Kasparov himself was denied access to recent Deep Blue’s games. Thus, the contest ultimately pitted a human chess player relying primarily on ends-and-means heuristic to intuitively determine optimal outcomes of few move sequences, against a computer programmed with the best strategies human chess players – as a group – devised, lightning fast access to dizzying number of alternatives, and an equally lightning fast decision engine powered by the most advanced algorithms devised and programmed by leading scientists. And let’s not lose sight of the fact that Deep Blue, like all machines, was not hindered by factors such as fatigue or recall decay (forgetting). In view of the enormous informational and computational disparity, it is nothing short of amazing that the human champion convincingly won the initial (1996) bout, 4-2, and was only narrowly edged out in the (1997) rematch, where in six games there were three draws, two games were won by Deep Blue and one by Kasparov. Or at least that is how it would appear to a casual observer, one curious enough to ponder those matters, but not necessarily knowledgeable enough to grasp the true essence of the answer. Let us take a closer look at the storage and processing speed aspects of human thinking, as those two considerations are at the core of perceived machine processing superiority.