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“An alternative betting scheme is the Kelly Criterion. The Kelly Criterion basically states: Determine the percentage advantage over the house and bet that percentage of the total bank.”
Ken Uston, Million Dollar Blackjack
D.A. and I were getting more passionate about the game every day. We decided to meet for lunch to talk about blackjack theory and swap information on the latest books we were reading. I finished up some morning trades at work and headed back into the city to meet up with D.A., who was able to carve out an hour between classes. I sent him a text with the address. Boylston Street. It was close enough that he could take the green Line there, and I wanted to try this place out for myself.
I met D.A. at the hostess station and he immediately asked, “Why the hell did you pick this place? I love sushi but I’m not so much a fan of taking off my shoes.”
“Trust me. I don’t like the idea of you taking your shoes off, either. But think about it—the MIT guys used to meet to discuss team strategies at a basement sushi place on Boylston. This has got to be that place.”
D.A.’s eyes lit up. Neither of us was sure if this was actually the restaurant referenced in Bringing Down the House, but it seemed to make sense. And the placebo effect helped enhance the feeling that we might be sitting at the same table that some of the legendary players from MIT sat at when they were at their peak. Maybe some of their good fortune would rub off on us.
We read and re-read books, from technical, to instructional, to pure entertainment. Mezrich’s books had made blackjack seem like a simple game—bet big when the player has the advantage. But there was so much more to it than that. After all, what did betting big actually mean?
The true count was the basis for bet variation, or optimal betting. In other words, the amount wagered has to be specifically based on the team’s bankroll and the current playing advantage. The concept of “betting unit” became a big part of our discussions. In an effort to minimize risk of ruin (going broke) we used a system that hinged on a probability formulation developed by John Kelly in the 1956 issue of the Bell System Technical Journal, a publication for Bell Laboratories. Later, Ed Thorp demonstrated it in a 1961 address to the American Mathematical Society, but more notably in his books Beat the Dealer (for blackjack) and Beat the Market (for investing). More recently, William Poundstone’s Fortune’s Formula describes the work of Kelly and Thorp in more readable terms. We appreciated the irony that both investors and gamblers alike used the formula. My two worlds, finance and blackjack, were overlapping.
The idea of the formula is to determine the optimal sizes for a series of bets. It’s been claimed that even legendary investors like Warren Buffet and Bill gross employ Kelly-based strategies, and I’d be surprised if they didn’t. The theory states that wagers (or investments) should be sized to maximize the expected value of the outcome. With that in mind, even many proponents of Kelly and his theory advise using “fractional Kelly” to reduce volatility. In blackjack terms, the Kelly Criterion is a formula for determining how much to bet based on the size of a player’s bankroll and his advantage.
D.A. and I learned that the amount we would wager was based on the true count, which was a gauge of our playing advantage. After considering risk factors, the optimal bet was equal to three-fourths of the advantage multiplied by the bankroll. Our goal was to scrape together a bankroll of $16,000 and play it optimally using Kelly betting. We knew that at a true count of 2, we’d first reach an advantageous situation.
A true count of 2, less 1 for the offset, equals 1. And 1 x 0.50% = an advantage of +0.50%. So if our advantage was 0.50%, we could multiply that by three-fourths and then by our $16,000: 0.50% x ¾ x $16,000 = $60.
For every increase of 1 in the true count, our bet would increase by $60. We could start there and work on building our bankroll. The point was to play as close to optimal as possible, so it was critical that we thought in terms of “betting unit.” In our case, with a $16,000 bankroll it would be $60.
An easier way to calculate bet size was to divide the bankroll by 266. That would be considered “full Kelly.” But we knew from our research that most investment strategies called for “fractional Kelly.” Without playing more conservatively, the risk of ruin was exaggerated. Certainly, full Kelly was optimal, but fractional Kelly was much safer. Playing at one-half Kelly meant dividing our bankroll by 532 units. The MIT blackjack team was able to play a conservative unit because their bankroll was so large. They played at about one-third Kelly, dividing their bankroll by 887 to derive their betting unit.
We decided to keep things simple and divide our bankroll by 500, slightly more than half-Kelly. By limiting our betting unit to 1/500th of our entire bankroll, we could safely assume that we would have little chance of losing everything. With a $16,000 bankroll, the betting unit would be $30, meaning we would target the lowest tables at Foxwoods, which was the closest place to apply our craft whenever the time was right. This approach to bankroll management would allow us to evaluate our bankroll after each session and adjust the betting unit according to increases or decreases in the bankroll.
Of course, that didn’t mean that every bet was $30. That was just our unit. As the true count rose, so would the number of betting units wagered. By using the Kelly Criterion we could work within the parameters of sound money management, a key component to playing blackjack professionally. Unfortunately, many proficient card counters go bust because they don’t properly account for the seemingly implausible swings that come with the game. Since casinos have a never-ending supply of money when compared to the players at the tables, they can handle the swings. Players, on the other hand, can not. So they need to prepare for the fact that losing streaks are a common part of the game.
The measurement of that volatility is done through the concept of standard deviation, something I discussed frequently with my clients about their portfolios. A portfolio that may have averaged, say, 8% per year over a certain span of years most likely would have yielded returns higher and lower than 8% from year to year. The 8% return is just the average, or the mean. The years with higher or lower returns represented a deviation from that mean.
In terms of blackjack, we could easily calculate the expected return based on the number of hands we played, the amounts we were wagering, and our playing advantage. Realistically, it was probable that our actual returns would vary from that expected return. The amount of variation could be defined by the applicable standard deviation. But we didn’t have to be math majors to understand what it all meant. We just had to believe in the law of large numbers.
Although we had mastered each piece of the card counting system individually, putting it all together seemed like an overwhelming task. We knew that we’d have to process the information in our minds while sandwiched between other players at the table and amid the distractions of cocktail waitresses in provocative outfits offering free drinks, sounds of slot machine trays filling up with nickels, and all of the lights and sounds casinos use to attack the senses.
Intimidated as we were, we also knew that we could do it. It just took focus and determination. And a lot of practice.
We decided to buy an important piece of software—a powerful blackjack simulator, Casino Verite, that could be customized for dealer speed, payoff mistakes, rules variations, and counting systems. It was a valuable tool to provide playing experience at incredible speeds and difficulties. Coupled with our in-person practices, D.A. and I played through hundreds of virtual shoes on Casino Verite, even incorporating many of its drills into our checkout procedures.
We were playing so much blackjack that it was nearly all we could think about. We continuously approached hands with the same step-by-step process:
Focus.
Running count.
Decks played.
Decks remaining.
True-count conversion.
Offset.
Proper betting.
Basic strategy.
Or, for example:
A running count of 17.
3.5 decks played.
2.5 decks remaining.
True count conversion: 17/2.5 = 6.8.
Offset: 6.8 – 1 = 5.8 (rounded to the nearest half = 6).
Bet: $180 (6 x $30).
I’d created three piles of flash cards to carry with me for times when I wasn’t near a computer. One had running counts listed from 1 to 30. Another listed decks played from ¼ to 5¾. And the last had betting units ranging from $25 to $500, including arbitrary units, such as $70 or $315.
First, I would turn over a running count card. Next, I would turn over a decks-played card. Then, I would turn over a betting-unit card. Lastly, I would process the math, and then I’d do it all over again.
I was processing basic numbers all around me. The Saturday matinee starts at 4:15 (or, 4¼) and I’m 33 years old. 33/4.25. The date is April 26. 26/4. In some ways, I realized that I was being obsessive, but deep down I knew that I was pushing myself as hard as I could. It was all beginning to come together.
There was still one more piece to the puzzle. In blackjack, as in life, some difficult decisions are made using an alternate rulebook.
Basic strategy was the rule and we knew it perfectly. It was the mathematically correct way of playing blackjack assuming no additional information. However, as the count rises or falls, so does the player’s advantage or disadvantage. As such, adjustments in the playing strategy need to be made, as well. Basic strategy is still the foundation for all decisions, but making the correct play based on the count sometimes means veering from basic strategy.
We had one more chart to learn: index numbers. These were the count numbers at which it was correct for our playing decisions to deviate from basic strategy.
The standard decision based on basic strategy, for instance, when a player has a total of 8 with a dealer’s upcard of 6 is to hit. The chart of index numbers, however, indicated that at a certain true count or higher, the proper playing decision was not the one dictated by basic strategy.
I followed my finger across the top of the page where the header indicated “Dealer’s Hand” until I reached 6. Then I dragged my finger down until it intersected with the “Player’s Hand” of 8. Sitting inside the little square in the large grid of every possible hand was the number 2. That meant that if the true count was +2 or higher, the correct decision with an 8 versus a 6 was no longer to hit. It was to double down.
The chart represented more than a hundred different situations where the correct play would be to veer from basic strategy. A 9 versus a dealer 2, normally a hit, was a double at a true count of +1 or higher. A 12 versus 4, normally called for a decision to stand, but was instead a hit when the true count dropped to any negative number. It was a new world of information and one that would require the greatest test of brainpower.
A few days earlier, we’d acknowledged that index numbers were our next hurdle, and we’d agreed to prepare for our next practice. The moderate effort we both put into memorizing this new code was not settling in very well and we were both a little discouraged.
“I found a solution,” I assured D.A. “Blackjack Attack outlines something called the ‘Illustrious eighteen.’”
“That’s right, the Illustrious eighteen.” he added, “I read about that in Blackjack Blueprint, too. I forgot about it.”
“Right,” I went on, holding up the bent-back pages of Don Schlesinger’s book, my latest reference. “It says here that about ninety percent of the real value in index numbers comes from these eighteen decisions alone. All we need to do is work on the Illustrious eighteen and not worry about learning all of the basic strategy variations. Just these.”
“That works for me.”
I continued, as if to convince D.A. further. In retrospect, I was rationalizing the decision to myself. “Not to mention, learning more than these eighteen comes with added risk of playing error, too. Why take the added risk? If ninety percent of the value of index plays comes from these eighteen numbers alone, I’m good with just learning them.”
“Me too,” D.A. agreed.
We knew that learning the Illustrious 18 could be done in no time at all. We found a shortcut in the system but for some reason, deep down I think we both felt like we were cheating ourselves in some way. Not in terms of playing edge, but in terms of the perfection we originally sought. We’d worked hard to get to the place we were. Shortcutting on the final skill set seemed counter to the approach we’d taken all along.
It was like a cook forgetting to add the spices. It just didn’t taste right.