The Perfect Bet

by Adam Kucharski

  Von Neumann started by looking at poker at its most basic, where it is a game between two players. To simplify matters further, he assumed that each player was dealt a single card, showing a number somewhere between 0 and 1. After both players put in a dollar to start, the first player—who we’ll call Alice—has three options: fold, and therefore lose one dollar; check (equivalent to betting nothing); or bet one dollar. Her opponent then decides whether to fold, and forfeit the money, or match the bet, in which case the winner depends on whose card has the highest number.

  Obviously, it is pointless for Alice to fold at the start, but should she check or bet? Von Neumann looked at all possible eventualities and worked out the expected profit from each strategy. He found that she should bet if her card shows a very low or very high number, and she should check otherwise. In other words, she should bluff only with her worst hand. This might seem counterintuitive, but it follows logic familiar to all good poker players. If her card shows an average-to-low number, Alice has two options: bluff or check. With a terrible card, Alice has no hope of winning unless her opponent folds. She should therefore bluff. Middling cards are trickier. Bluffing won’t persuade someone with a decent card to fold, and it’s not worth Alice betting on the off chance that her mediocre card will come out on top in a showdown. So, the best option is to check and hope for the best.

  In 1944, von Neumann and economist Oskar Morgenstern published their insights in a book titled Theory of Games and Economic Behavior. Although their version of poker was much simpler than the real thing, the pair had cracked a problem that had long bothered players, namely, whether bluffing was really a necessary part of the game. Thanks to von Neumann and Morgenstern, there was now mathematical proof that it was.

  Despite his fondness for Berlin’s nightlife, von Neumann didn’t use game theory when he visited casinos. He saw poker mainly as an intellectual challenge and eventually moved on to other problems. It would be several decades before players worked out how to use von Neumann’s ideas to win for real.

  BINION’S GAMBLING HALL IS part of the old Las Vegas. Away from the Strip’s shows and fountains, it lies in the thumping downtown heart of the city. While most hotels were built with theaters and concert halls as well as casinos, Binion’s was designed for gambling from the start. When it opened in 1951, betting limits were much higher than at other venues, and in the entrance a giant upturned horseshoe straddled a box displaying a million dollars in cash. Binion’s was also the first casino to give free drinks to all gamblers to keep them (and their money) at the tables. So, it was only natural that when the first World Series of Poker took place in 1970, it was held at Binion’s.

  Over the following decades, players gathered at Binion’s each year to pit their wits—and luck—against each other. Some years were especially tense. Early in the 1982 competition, Jack Straus stumbled onto a losing streak that left him with a single chip. Fighting back, he managed to win enough hands to stay in the game, eventually going on to win the whole tournament. The story goes that when Straus was later asked what a poker player needs for victory, his reply was “a chip and a chair.”

  On May 18, 2000, the thirty-first World Series reached its finale. Two men were left in the competition. On one side of the table was T. J. Cloutier, a poker veteran from Texas. Opposite him sat Chris Ferguson, a long-haired Californian with a penchant for cowboy hats and sunglasses. Ferguson had started the game with far more chips than Cloutier, but his lead was shrinking with each hand that was dealt.

  With the players almost even, the dealer handed out yet another set of cards. They were playing Texas hold’em poker, which meant that Ferguson and Cloutier first received two personal “pocket” cards. After looking at his hand—the ninety-third of the day—Cloutier opened with a bet of almost $200,000. Sensing a chance to retake the advantage, Ferguson raised him half a million dollars. But Cloutier was confident, too. So confident, in fact, that he responded by pushing all his chips into the center of the table. Ferguson looked at his cards again. Did Cloutier really have the better hand? After pondering his options for several moments, Ferguson decided to match Cloutier’s bet of almost $2.5 million.

  Once two initial pocket cards have been dealt in Texas hold’em, there are up to three additional rounds of betting. The first of these is known as the “flop.” Three more cards are dealt, this time placed face up on the table. If betting continues, another card—the “turn”—is revealed. Another round of betting means that the game reaches the “river,” where a fifth card is shown. The winner is the player who has the best five-card hand when the two pocket cards are combined with the five communal cards.

  Because Cloutier and Ferguson had both gone all in at the start, there would be no additional betting. Instead, they would have to show their pocket cards and watch as the dealer turned over each of the five additional cards. When the players showed their hands, the crowd surrounding the table knew Ferguson was in trouble. Cloutier had an ace and a queen; Ferguson had only an ace and a nine. First, the dealer turned over the flop cards: a king, a two, and a four. Cloutier still had the better hand. Next came the turn, and another king. The game would therefore be settled on the river. As the final card was revealed, Ferguson leapt from his seat. It was a nine. He’d won the game, and the tournament. “You didn’t think it would be that tough to beat me, did you?” Cloutier asked Ferguson after he’d netted the $1.5 million prize money. “Yes,” Ferguson replied, “I did.”

  UNTIL CHRIS FERGUSON’S TRIUMPHANT performance in Las Vegas, no poker player had won more than $1 million in tournament prizes. But unlike many competitors, Ferguson’s extraordinary success did not rely solely on intuition or instinct. When he played in the World Series, he was using game theory.

  The year before he beat Cloutier, Ferguson had completed a doctorate in computer science at UCLA. During that time, he worked as a consultant for the California State Lottery, picking apart existing games and coming up with new ones. His family members have mathematical backgrounds, too: both parents have PhDs in the subject and his father, Thomas, is a professor of mathematics at UCLA.

  While studying for his doctorate, Chris Ferguson would compete for play money in some of the early Internet chat rooms. He saw poker as a challenge, and it was one he was rather good at. The chat room games didn’t lead to any profit, but they did give Ferguson access to large amounts of data. Combined with improvements in computing power, this enabled him to study vast numbers of different hands, evaluating how much to bet and when to bluff.

  Like von Neumann, Ferguson soon realized that poker was too complicated to study properly without making a few simplifications. Building on von Neumann’s ideas, Ferguson decided to look at what happens when two players have more options. Of course, he would have more than one opponent at the start of a real poker game, but it was still useful to analyze the simple two-player scenario. Players may fold as the betting rounds progress, so by the time the endgame arrives, there are often only a couple of players left.

  Yet there are still a number of things the two players might do at this point. The first player, Alice, had three simple choices in von Neumann’s game—bet one dollar, check, or fold—but in a real game she might do something else, like change her bet. And the second player might not respond by matching the bet or folding. The second player might be confident like Cloutier was and raise the betting.

  As more options creep into the game, picking the best one becomes more complicated. In a simple setup, von Neumann showed that players should employ “pure strategies,” in which they follow fixed rules such as “if this happens, always do A” and “if that happens, always do B.” But pure strategies are not always a good approach to use. Take a game of rock-paper-scissors. Picking the same option every time is admirably consistent, but the strategy is easy to beat if your opponent works out what you’re doing. A better idea is to use a “mixed strategy.” Rather than always going with the same approach, you should switch between one of the pure strategies
—rock, paper, or scissors—with a certain probability. Ideally, you will play each of the three options in a balance that makes it impossible for your opponent to guess what you’re going to do. For rock-paper-scissors, the optimal strategy against a new opponent is to choose randomly, playing each option one-third of the time.

  Mixed strategies also make an appearance in poker. Analysis of the endgame suggests that you should balance the number of times you are honest and the number of times you bluff so that your opponent is indifferent to calling or folding. Like rock-paper-scissors, you don’t want the other person to work out what you are likely to do. “You always want to make your opponents’ decisions as difficult as possible,” Ferguson said.

  Sifting through the data from the chat room games, Ferguson spotted other areas for improvement. When experienced players had good hands, they would raise heavily to encourage their opponents to fold. This removed the risk of a weak hand turning into a winning hand when the communal cards were revealed. But Ferguson’s research showed that the raises were too high: sometimes it was worth betting less and allowing people to remain in the game. As well as winning more money with strong hands, it meant that if a hand did lose, it wouldn’t lose as much.

  Through his research, Ferguson discovered that finding a successful approach to poker doesn’t necessarily mean chasing profits at all costs. As he once told The New Yorker, the optimal strategy isn’t a case of “How do I win the most?” but one of “How do I lose the least?” Novice players usually confuse the two and don’t fold often enough as a result. True, it’s impossible to win anything by folding, but sitting out a hand allows players to avoid costly betting rounds. Collecting together his results into detailed tables, Ferguson memorized the strategies—including when to bluff, when to bet, how much to raise—and started playing for real money. He entered his first World Series in 1995; five years later he was champion.

  Ferguson has always been fond of picking up new skills. He once taught himself to throw a playing card so fast from a distance of ten feet that it could slice a carrot in two. In 2006, he decided to take on a new challenge. Starting with nothing, he would work his way up to $10,000. His aim was to show the importance of bankroll management in poker. Just as the Kelly criterion helped gamblers adjust their bet size in blackjack and sports betting, Ferguson knew it was essential to adjust his playing style to balance profit and risk.

  Because he was starting with zero dollars, Ferguson’s first task was to get ahold of some cash. Fortunately, some poker websites ran daily “freeroll tournaments.” Hundreds of players could enter for free, with the top dozen or so receiving cash prizes. It’s not often that a big-name player enters a freeroll tournament, let alone takes it seriously. When other online players found out who they were playing against, most thought it was a joke. Why was a world champion like Chris Ferguson plying his trade on the free tables?

  After a few attempts, Ferguson eventually netted some all-important cash. “I remember winning my first $2 a couple of weeks into the challenge,” he later wrote, “and I strategized for three days, deliberating over what game to play with it.” He settled on the lowest-stakes game possible, but within one round he’d lost it all. Finding himself back to zero, he returned to the freeroll tournaments and started over again. It was clear that he would have to be extremely disciplined if he was going to reach his target.

  Playing around ten hours per week, it took Ferguson nine months to get to $100 (he’d expected it to take around six). He kept going, sticking to a strict set of rules. For instance, he would only ever risk 5 percent of his bankroll in a particular game. It meant that if he lost a few rounds, he would have to go back to lower-stakes tables. Psychologically, he found it difficult to drop down a level. Ferguson was used to the excitement of high-stakes games and the profits they brought. After moving down, he would lose focus and struggle to keep to his rules. Rather than take more risks, he stepped back; it was pointless playing the game until he’d regained his concentration. The self-restraint paid off. After another nine months of careful play, Ferguson finally reached his $10,000 total.

  The bankroll challenge, along with his earlier World Series victory, cemented Ferguson’s reputation as a virtuoso of poker theory. Much of his success came from working on optimal strategies, but do such strategies always exist in games like poker? The question was actually one of the first that von Neumann asked when he started working on two-player games at the University of Berlin. As well as laying the foundations for the entire field, the answer would go on to cause a bitter dispute about who was the true inventor of game theory.

  GAMES LIKE POKER ARE “zero-sum,” with winning players’ profits equal to other players’ losses. When two players are involved, this means one person is always trying to minimize the opponent’s payoff—a quantity the opponent will be trying to maximize. Von Neumann called it the “minimax” problem and wanted to prove that both players could find an optimal strategy in this tug-of-war. To do this, he needed to show that each player could always find a way to minimize the maximum amount that could potentially be lost, regardless of what their opponent did.

  One of the most prominent examples of a zero-sum game with two players is a soccer penalty. This ends either in a goal, with the kicker winning and the goalkeeper losing, or a miss, in which case the payoffs are reversed. Keepers have very little time to react after a penalty is taken, so generally make their decision about which way to dive before the kicker strikes the ball.

  Because players are either right- or left-footed, choosing the right- or left-hand side of the goal can alter their chances of scoring. When Ignacio Palacios-Heurta, an economist at Brown University, looked at all the penalties taken in European leagues between 1995 and 2000, he found that the probability of a goal varies depending on whether the kicker chooses the “natural” half of the goal. (For a right-footed player, this would be the left-hand side of the goal; for a left-footed kicker, it would be the right side.)

  The penalty data showed that if the kicker picked the natural side and the keeper chose the correct direction, the kicker scored about 70 percent of the time; if the keeper got it wrong, around 90 percent of shots went in. In contrast, kickers who went for the nonnatural side scored 60 percent of shots if the keeper picked correctly and 95 percent if they didn’t. These probabilities are summarized in Table 6.1.

  If kickers want to minimize their maximum loss, they should therefore pick the natural side: even if the goalkeeper gets the correct direction, the player has at least a 70 percent chance of scoring. In contrast, the keeper should dive to the kicker’s nonnatural side. At worst it will result in the player scoring 90 percent of the time rather than 95 percent.

  If these strategies were optimal, the worse-case probabilities for kicker and keeper would be equal. This is because a penalty shootout is zero-sum: each person is trying to minimize the potential loss, which means if each plays the perfect strategy, it should minimize the maximum payoff for the opponent. Yet this is clearly not the case, because the worst outcome for the player results in scoring 70 percent of the kicks, whereas the worst result for the goalkeeper leads to letting in 90 percent of shots.

  TABLE 6.1. The Probability of Scoring a Penalty Depends on Which Side the Kicker and Goalkeeper Choose

  The fact that the values are not equal implies that each person can adjust tactics to improve the chances of success. As in rock-paper-scissors, switching between options might be better than relying on a simple pure strategy. For example, if the kicker always chooses the natural side, the goalkeeper should occasionally pick that option, too, which would bring the 90 percent worst-case scenario down closer to 70 percent. In response, the kicker could counter this tactic by also opting for a mixed strategy.

  When Palacios-Heurta calculated the best approach for the kicker and goalkeeper, he found that both should choose the natural half of the goal with 60 percent probability, and the other side the rest of the time. Like effective bluffing in poker, this would hav
e the effect of making the other person indifferent to what is going to happen: opponents would be unable to boost their chances by changing their strategy. Both the goalkeeper and the kicker would therefore successfully limit their loss as well as minimize the other person’s gain. Remarkably, the recommended 60 percent value is within a few percent of the real proportion of times players choose each side, suggesting that—whether aware of it or not—kickers and goalkeepers have already figured out the optimal strategy for penalties.

  VON NEUMANN COMPLETED HIS solution to the minimax problem in 1928, publishing the work in an article titled “Theory of Parlour Games.” Proving that these optimal strategies always existed was a crucial breakthrough. He later said that without the result, there would have been no point continuing his work on game theory.

  The method von Neumann used to attack the minimax problem was far from simple. Lengthy and elaborate, it has been described as a mathematical “tour de force.” But not everyone was impressed. Maurice Fréchet, a French mathematician, argued that the mathematics behind von Neumann’s minimax work had already been in place (though von Neumann had apparently been unaware of it). By applying the techniques to game theory, he said that von Neumann had “simply entered an open door.”

  The approaches Fréchet was referring to were the brainchild of his colleague Émile Borel, who had developed them a few years before von Neumann. When Borel’s papers were eventually published in English in the early 1950s, Fréchet wrote an introduction crediting him with the invention of game theory. Von Neumann was furious, and the pair exchanged barbed comments in the economics journal Econometrica.

  The dispute raised two important issues about applying mathematics to real-world problems. First, it can be hard to pin down the initiator of a theory. Should credit go to the researcher who crafts the mathematical bricks or to the person who assembles them into a useful structure? Fréchet clearly thought that brick maker Borel deserved the accolades, whereas history has given the credit to von Neumann for using mathematics to construct a theory for games.