My Casino Background 1: Counting Cards in Las Vegas and Prague
Post 1 of 2 about my background with casino gambling and decision science
This post and the next one are to tell you about my background with casino gambling and decision science, partly so you can decide for yourself whether I’m a reliable source, and partly by way of explaining why I think the topic is interesting and important. This first post is about my early forays into card counting before I started graduate school. The next post will discuss my Ph.D. dissertation research, which provided most of the data that will be described in this Substack.
Newsletter Contents
Vegas Baby, Vegas!
When I turned 21 (no pun intended!), I went on a road trip from Oregon to Colorado via Utah’s canyon country with my then girlfriend. As far back as I can remember, I had liked games and gambling, and so for me one of the highlights of the trip was an inconvenient detour through Las Vegas now that I was finally old enough to gamble. Though I don’t trust my 30+ year old memory of events, I doubt the idea of gambling with the odds against me in the hope of getting lucky appealed to me much, and I doubt I thought I could get an advantage over the casino, but the general idea of checking out the over-the-top Las Vegas Strip and experiencing casino gambling somehow did, despite some part of me also thinking Las Vegas was a gaudy and immoral blemish on a beautiful natural landscape.
On the way we stopped at a used bookstore and I looked for books on how to gamble in casinos. I don’t remember if I already knew about card counting and was looking for books on the topic but, in any event, they had a used copy of the 2nd edition of Beat the Dealer by Edward O. Thorp, which—lucky me—is the first (reliable) book on “Basic Strategy” and “card counting”. It is foundational to all subsequent math-based resources on blackjack strategy.
Card Counting
Card counting is a system for keeping track of the ratio of good to bad cards remaining to be played in casino blackjack, adjusting one’s playing strategy depending on that ratio, and—when that ratio is sufficiently good for the player, increasing one’s bets. Even though the casino usually has an advantage in blackjack, occasionally the ratio of good to bad cards changes enough before the dealer—a casino employee—shuffles the cards, and the player can have an advantage. By betting the minimum during those usual times when the casino has an advantage and betting as much as possible during those uncommon times when the player has an advantage, sufficiently skilled card counters, with the right table conditions, can get a long term positive expected return.
Blackjack Rules and Basic Strategy
Knowing the best way to play each hand when not counting cards—known as Basic Strategy—is essential, not just to successful card counting but also to minimizing the casino’s advantage for non-card counters. Players are competing against the dealer, and the dealer must play by set rules. Those rules often vary depending on the country and region, the particular casino within that region, and even specific tables within a casino, but the gist is that the goal of blackjack is to get more points than the dealer without getting more than 21 points total (getting more than 21 points is called busting and it results in an automatic loss for the player). Points correspond to the card values on a normal 52-card deck, except that face cards are all worth 10 and aces can be worth 1 or 11, depending on what makes a better hand. The dealer must take a card (hit) if their hand total is less than 17 and stop taking cards (stand) once their total reaches 17 or more. If the dealer busts, they must pay all remaining players. An ace and a ten-value card as the first two cards is called a blackjack. It is special in that it beats other types of 21 and it pays out 3 to 2 on the players’ bet (as long as the dealer doesn’t also have a blackjack. Unlike common home versions of blackjack, the dealer does not win ties: if the dealer and player have the same total, including if they both have blackjacks, it is called a push and the original bet is returned to the player.
While the dealer has to play every hand the same way, players do not. They also have several additional options that would seem to give them an advantage over the dealer. Players can hit or stand as they wish and can make other playing choices that the dealer cannot make, such as splitting, which involves separating any two identical-vlaue first cards into two separate hands and matching the original bet for the new second hand, or doubling down, which involves doubling one’s original bet after seeing one’s original two cards and the dealer’s first card, but then being forced to take one and only one additional card. Players make all of their choices after seeing the dealer’s first card, which is dealt face up. They do not see any other dealer cards until after all players have finished their turns, at which point the dealer plays his or her hand. For a more detailed description of blackjack rules, see the blackjack rules page on my companion website.
Given this wide variety of playing choices relative to the limited options for the dealer, the bonus payout for blackjacks, and the fact that the dealer does not win ties, why does the casino have an advantage at all, even against non-card counters? Even if the player just copied the dealer’s strategy (hitting values less than 17 and standing with 17 or more), shouldn’t the 3:2 payout for blackjacks give the player an advantage? The source of the casino’s advantage is one tie that is not a push: when players bust, the casino takes that bet and removes the player’s cards immediately, before the dealer plays. Given the blackjack rules where I played, dealer’s will bus more than 28% of the time, but they’ll win all of those times if the player busts first. That advantage for mutual busts is high enough that players who mimic the dealer have a disadvantage of about 6%. That’s far better than the approximate 28% advantage the casino would have if blackjack players chose randomly when making each type of possible decision1, but it’s far worse than the advantage the casino would have if players used perfect Basic Strategy, optimally varying when they hit, stand, split, or double down, depending on their cards and the dealer’s exposed card. Perfect basic strategy in casinos where I played generally reduced the casino’s advantage to 0.4-0.5% without any card counting at all. That is, for each $10 bet, the casino would only win 4 or 5 cents on average, over the long run.
Card Counting is Simpler Than You Think
Contrary to common depictions of card counting (such as Dustan Hoffman’s character in Rain Man), successful card counting does not rely on high-level math or exceptional memory. Instead, card counters do simple arithmetic, adding and subtracting small numbers, often just ones, to get a “running count” that is either positive or negative, and sometimes adjusting that count by multiplying by some fraction to adjust for the number of desk remaining (arriving at the “true count”). When that true count gets sufficiently high, card counters have a positive expected return and increase their bets. The main demands on memory are limited to memorization of the basic strategy given the particular casino rules, and of how to deviate from that strategy, depending on the count. But that memory comes from hours of training and practice that moves those rules into long-term memory before stepping foot in the casino, something that anyone able to get a college degree could do with enough practice. Indeed, the most brilliant mathematician on the planet could not calculate the basic strategy on the fly, much less adjust that basic strategy given which cards have been removed from the deck and evaluate the impact of those changes on expected return. The first statistically accurate Basic Strategy was published in 1956 and was limited to a specific set of casino rules, and it still had minor errors that were only identified after Monte Carlo simulations—repeatedly sampling particular card combinations millions of times to assess outcome probabilities—made it possible to check the math. This is where Thorpe’s 2nd Edition of Beat the Dealer, and his collaboration with IBM employee Julian Braun, were so important. Card counters do not calculate probabilities at all. They just use simple heuristics (rules of thumb, such as adding or subtracting ones based on whether the card is good or bad for the player). Perfect assessment of the impact of all cards on outcome probabilities and optimal playing strategy still requires a computer.
Card counting strategies vary widely in terms of how sophisticated or complicated they are, balancing the limitations of human cognitive ability with the potential gain from using more complicated systems, a theme which has turned out to be central to the history of behavioral decision theory as well. The early trend was to focus on maximizing ideal expected return, and therefore the complexity of the card counting system, but later methods have focused on maximizing simplicity so as to limit mistakes, trying to limit any required real-time calculations and choices so players can spend more time faking it (looking like they’re not counting cards).
Card Counting is a Dumb Way to Make a Living
Part of the reason simple strategies are so important is that it is important not to look like a card counter, because once a casino suspects a player of counting cards, it’s easy to put a stop to it. In Nevada and the Czech Republic, casinos can simply kick players they suspect of counting cards out of the casino. In Atlantic City, they can’t kick players out, but they can change how they manage the table. For example, they can shuffle more often or limit the betting spread between maximum and minimum best or require all players to enter the table only immediately after the decks have been shuffled. All of these changes minimize the potential benefits of counting cards.
As such, card counters put a lot of effort into looking like they’re not counting cards, including doing things like occasionally drinking alcohol, sometimes betting large off the top of the deck, sometimes violating the Basic Strategy before any cards have been removed from play, and trying to make keeping track of the count sufficiently automatic that they can carry on conversations and look away from the table even while the cards are being dealt. Along with the potential for human error, all of those efforts masking card counting reduces the theoretical expected return from mathematical models, calling into question how readily practiced card counters—if not using other methods—actually play with a long-term advantage.
On top of that, as with casino gambling more generally, there is large degree in variability in outcomes just due to the vagaries of chance. Just as many gamblers can spend a week in Las Vegas and leave with a big win, many good card counters can spend a week counting cards and still lose a lot of money. A simulated example taken from Stanford Wong’s wonderful book on card counting, Professional Blackjack (1994), wonderfully quantifies the impact of variance. A card counter using one widely used system (“hi-lo”), without making any errors and spreading bets between $10 and $100, can expect to win $16 per hour on average but with a standard deviation of $415 per hour (meaning that about one third of the time, players will lose more than $399 per hour or win more than $431 in an hour). As such, many skilled but naïve card counters would lose an entire betting bankroll of thousands of dollars despite using a system with a positive long-term expected return.
Given the bankroll requirements, the unreliability of income, the amount of practice and time it takes to become a sufficient card counter, the unknown cost to expected return because of both human error and the need to not look like a card counter, and the somewhat limited expected return ($16/hour) even with perfect play and an impressive betting spread of 10-1 (which would not be allowed in most casinos remotely suspicious that someone is a skilled card counter), card counting is a lousy way to make a living. The vast majority of card counters do it because they’re naïve or because they’re motivated by other things besides earning a good living. I had a lot of fun counting cards, imagining myself up against the powerful casino with their mob roots, having to put on a show, and facing both financial risk and the fear of being found out and perhaps roughed up. The adrenaline, dopamine, and serotonin mixing around in my brain alongside large wins and losses no doubt added to that fun (and to the pain). But it’s clear to me now that making a good living was never in the cards.
Casinos make it possible to count cards by offering single-deck games and by not shuffling after every hand even though institutionalizing those changes and making card counting impossible would be easy. They do that because they make a heck of a lot of money from the prospect of profitable card counting which brings so many people to the table, along with the reality that very few of the people attracted by that prospect actually know Basic Strategy, much less how to count cards.
This is putting aside the non-trivial fact of problem and pathological gambling. Casinos are unhealthy environments, crafted in large part for their ability keep as much of the money from their patrons wallets as possible, while also keeping those patrons coming back repeatedly. The long-term impact on impulse control and well-being that comes with repeated wins and losses with games designed over generations of experimentation and refinement to condition players to stay as long as possible and come back repeatedly is not something to take lightly, and it has ruined many lives.
But I Didn’t Know That at the Time
I bought Beat the Dealer within a day or so of reaching Las Vegas. With limited time to learn the Basic Strategy and no time to practice, I learned the simplest and weakest card counting strategy in the book, which essentially amounted to (a) learning Basic Strategy for the tables I would play at by heart (though most Las Vegas casinos are happy to let players use Basic Strategy cards at the table, and many sell credit-card sized Basic Strategy cards in their casino bookstores or even give them away at the table), (b) finding a casino with a low minimum bet and single-deck blackjack (at the time I was able to find a single-deck game with $1 minimum bets on the Strip, but such low minimums with single-deck games may not be available anywhere but in downtown Las Vegas today, and they may not be available anywhere), (c) betting the minimum amount until all four 5s were dealt out of the deck before the next shuffle (a frustratingly uncommon event), and (d), betting as much as I could bear to risk when that happened (which was probably about $10 per hand). I ended up getting lucky and winning $50 during the couple of hours we spent in Las Vegas. I didn’t know it at the time, but I just as well might have lost that much or a lot more using the strategy I was using, and I might have easily won as much or more than that without counting cards at all. But winning $50 was exciting for me, I attributed it to my card-counting prowess, and I was hooked.
Over the next few years I studied increasingly advanced card counting strategies and went back to Vegas many times, culminating in the Uston Advanced Point Count system from Ken Uston’s influential, practical, and well-written book, Million Dollar Blackjack. That strategy assigns different values depending on the cards impact on the player’s expected return (each 10 removed from play is counted as minus 3, each 5 is counted as plus 3, etc.), it keeps a separate count of aces using the feet for making decisions about blackjack likelihoods, it adjusts the running count based on the number of half-decks remaining (to get the true count), and it requires memorization of 159 different deviations from the Basic Strategy depending on the player’s hand, the dealer’s upcard, and the count. Uston had a wonderful section on training, and I spent hundreds of hours practicing and memorizing so I could count through multiple decks in his prescribed times and make playing decisions based on counts without conscious cognitive effort.
I found “investors” (friends of my parents who were intrigued and convinced by my understanding of the math of the game and my ability to convey the ideas behind card counting, or maybe who just wanted to show support to their friend’s naive kid). They invested $1000, what I thought at the time was a meaningful bankroll along with a few hundred of my own dollars, and I went to back to Vegas and made what I remember to be a 50% (or 150%?) return on their investment. I didn’t know it then, but this was again primarily a result of luck. Given my bet size and range, I had been more likely to lose my entire bankroll over the course of the few days I was there than not. I also got banned from my first casino, which felt like a victory. I continued to read books and practice and visit Las Vegas, but as I learned more, I also realized I would need a bigger bankroll than I was comfortable risking for a lower expected return than the work could warrant, unless I wanted to join a card-counting team, which came with its own risks (see Uston’s book for more on teams and why they can be so effective; or for a romanticized Hollywood version about a famous team from MIT, see the (IMO unwatchable) film 21.
Prague, Czech Republic
I no longer remember how much I had won or lost playing blackjack by the time I decided it was not worth the time and effort, but my opportunities to play were few and far between and the lack of regular practice meant that I lost the ability to count cards well enough to hope for an advantage. Until I moved to the Czech Republic after college, that is. I moved there in December 1993, not long after graduating from college and about four years after the fall of the Berlin Wall and the Velvet Revolution that marked the peaceful end of communism in Czechoslovakia, and about a year and a half after the Velvet Divorce (the peaceful separation of the Czech and Slovak Republics into two separate countries). I knew there were opportunities to teach English in Prague and I was excited by the prospect of spending an extended period in a different culture and especially by this region where the changes were so new and so clearly transient and in a city that was beautiful, vibrant, friendly, and cheap.
Casinos offering blackjack were plentiful in Prague and the rules were similar to the best rules in Las Vegas (for 6-deck shoes). I decided to get back in to card counting, recognizing by then that it wouldn’t be a reliable or good source of income, but also enjoying the fun of it and believing the small advantage I had at least meant I was more likely to win than lose.
By then, I had learned enough to know that less is sometimes more in blackjack. There were newer systems that emphasized simplicity and minimizing errors over systems that flaunted their complexity and emphasized maximizing expected returns under idealized conditions. During the first several months in Prague, I played a lot of casino blackjack. But I also lost more than I won, more than I could afford, and enough to make me lose interest in the game (and doubt whether the casinos in Prague, with few tables, few patrons, and known to be run by organized crime) had the same incentives to run honest games as the casinos in Las Vegas.
Culture and Experience Promote Systematic Deviations From Basic Strategy
Several interesting observations (to me!) about casino blackjack players follow. First, despite the widely available and well-established mathematical credibility of the basic strategy, nearly none of the experienced casino blackjack players I met actually played according to Basic Strategy. This was true even on the Las Vegas Strip where Basic Strategy is available in casino bookstores, and despite the fact that it would probably take fewer than 10 minutes for most experienced blackjack players to learn how it differed from their existing strategy.
Second, deviations from Basic Strategy were not random: experienced blackjack players on the Las Vegas Strip and in Prague violated basic strategy in similar ways to one another and with shared conviction that those violations were correct, suggesting that through some unclear process of learning from the game or from one another they had developed similar false beliefs about how best to play.
Third, this wasn’t due to lack of awareness that there was a Basic Strategy. Experienced blackjack players know about card counting and the basic strategy. Many of them claim to play according to Basic Strategy and claim to count cards, but they do not play according to the actual Basic Strategy and they do not count cards in a way that could potentially give them a positive expected return (or even know enough about the math or theory behind it to understand that). Others knew that their playing strategy was different from Basic Strategy but believed Basic Strategy was simply wrong.
Fourth, experienced blackjack players in Prague and on the Las Vegas Strip deviated from basic strategy in systematically different ways. Players in Prague were far more concerned about how other players at the table played, and they were noticeably less likely to risk busting than players on the Las Vegas strip. In both locations, players would get sufficiently annoyed at me for how I was playing—even when I playing according to Basic Strategy without counting cards—that they would voice their annoyance and sometimes leave the table in frustration (despite my knowledge that the impact of other players’ choices on my own success was random), but they did so far more in the Czech Republic than in Las Vegas, and they did so for different reasons in Las Vegas than in Prague.
For what to me seemed to be a purely mathematical game with a statistically determinable best way to play each hand (assuming the goal is to maximize expected value, which eventually became obvious is not the goal for most blackjack players), it seemed extremely weird that experience playing would lead to shared false convictions about how to play that would differ from the widely available theory and compelling math that contradicted those shared beliefs. The fact that these shared false convictions differed across cultures seemed a clear indication that the learned suboptimal play was more than just a result of built-in characteristics of the human cognitive system (heuristics and biases) and that they reflected aspects of culture and social learning.
Back to School
Those observations, along with a variety of other experiences with casino gamblers and their strategies and beliefs about winning, contributed to a broader interest in the topic. It eventually led me to spend two years in casinos in northwestern Indiana (riverboats), in the Las Vegas area (as both a blackjack dealer and a player), and in Prague, researching the topic for a PhD in psychology. And it will provide most of the backbone for this Substack. The next newsletter will describe that dissertation research, along with its relevance to psychological science.
Keren, G., & Wagenaar, W. A. (1985). On the psychology of playing blackjack: Normative and descriptive considerations with implications for decision theory. Journal of Experimental Psychology: General, 114(2), 133-158.