by Nate Silver
The Lakers weren’t even playing that badly, Voulgaris thought. They had won five of their first seven games despite playing a tough schedule, adjusting to a new coach, and working around an injury to Bryant, who had hurt his wrist in the preseason and hadn’t played yet. The media was focused on their patchy 1998–99 season, which had been interrupted by the strike and the coaching changes, while largely ignoring their 61-21 record under more normal circumstances in 1997–98. Voulgaris had watched a lot of Lakers games: he liked what Jackson was doing with the club. So he placed $80,000—his entire life savings less a little he’d left over for food and tuition—on the Lakers to win the NBA championship. If he won his bet, he’d make half a million dollars. If he lost it, it would be back to working double shifts at the airport.
Initially, Voulgaris’s instincts were looking very good. From that point in the season onward, the Lakers won 62 of their remaining 71 contests, including three separate winning streaks of 19, 16, and 11 games. They finished at 67-15, one of the best regular-season records in NBA history. But the playoffs were another matter: the Western Conference was brutally tough in those years, and even with home-court advantage throughout the playoffs—their reward for their outstanding regular season—winning four series in a row would be difficult for the Lakers.
Los Angeles survived a scare against a plucky Sacramento Kings team in the first round of the playoffs, the series going to a decisive fifth game, and then waltzed past Phoenix in the Western Conference Semifinals. But in the next round they drew the Portland Trail Blazers, who had a well-rounded and mature roster led by Michael Jordan’s former sidekick—and Jackson’s former pupil—Scottie Pippen. Portland would be a rough matchup for the Lakers: although they lacked the Lakers’ talent, their plodding, physical style of play often knocked teams out of their rhythm.3
The Lakers won the first game of the best-of-seven series fairly easily, but then the roller-coaster ride began. They played inexplicably poorly in the second game in Los Angeles, conceding twenty consecutive points to Portland in the third quarter4 and losing 106-77, their most lopsided defeat of the season.5
The next two games were played at the Rose Garden in Portland, but in Game 3, the Lakers gathered themselves after falling down by as many as thirteen points in the first half, with Bryant swatting away a shot in the final seconds to preserve a two-point victory.6 They defied gravity again in Game 4, overcoming an eleven-point deficit as O’Neal, a notoriously poor free-throw shooter, made all nine of his attempts.7 Trailing three games to one in the series, the Trail Blazers were “on death’s door,” as Jackson somewhat injudiciously put it.8
But in the fifth game, at the Staples Center in Los Angeles, the Lakers couldn’t shoot the ball straight, making just thirty of their seventy-nine shots in a 96-88 defeat. And in the sixth, back in Portland, they fell out of rhythm early and never caught the tune, as the Blazers marched to a 103-93 win. Suddenly the series was even again, with the deciding Game 7 to be played in Los Angeles.
The prudent thing for a gambler would have been to hedge his bet. For instance, Voulgaris could have put $200,000 on Portland, who were 3-to-2 underdogs, to win Game 7. That would have locked in a profit. If the Blazers won, he would make more than enough from his hedge to cover the loss of his original $80,000 bet, still earning a net profit of $220,000.9 If the Lakers won instead, his original bet would still pay out—he’d lose his hedge, but net $320,000 from both bets combined.* That would be no half-million-dollar score, but still pretty good.
But there was a slight problem: Voulgaris didn’t have $200,000. Nor did he know anybody else who did, at least not anybody he could trust. He was a twenty-three-year-old airport skycap living in his brother’s basement in Winnipeg. It was literally Los Angeles or bust.
Early on in the game his chances didn’t look good. The Blazers went after O’Neal at every opportunity, figuring they’d either force him to the free-throw line, where every shot was an adventure, or get him into foul trouble instead as he retaliated. Halfway through the second quarter, the strategy was working to a tee, as O’Neal had picked up three fouls and hadn’t yet scored from the field. Then Portland went on a ferocious run midway through the third quarter, capped off by a Pippen three-pointer that gave them a sixteen-point lead as boos echoed throughout the Staples Center.10
Voulgaris’s odds at that point were very long. Rarely did a team11 that found itself in the Lakers’ predicament—down sixteen points with two minutes left to play in the third quarter—come back to win the game; it can be calculated that the odds were about 15-to-1 against their doing so.12 His bet—his ticket out of Winnipeg—looked all but lost.13
But early in the fourth quarter, the downside to Portland’s brutally physical style of play suddenly became clear. Their players were beaten-up and fatigued, running on fumes and adrenaline. The Lakers were playing before their home crowd, which physiologists have shown provides athletes with an extra burst of testosterone when they need it most.14 And the Lakers were the younger team, with a more resilient supply of energy.
Portland, suddenly, couldn’t hit a shot, going more than six minutes without scoring early in the fourth quarter, right as the Lakers were quickening their pace. L.A. brought their deficit down to single digits, then five points, then three, until Brian Shaw hit a three-pointer to even the score with four minutes left, and Bryant knotted two free-throws a couple of possessions later to give them the lead. Although Portland’s shooting improved in the last few minutes, it was too late, as the Lakers made clear with a thunderous alley-oop between their two superstars, Bryant and O’Neal, to clinch the game.
Two weeks later, the Lakers disposed of the Indiana Pacers in efficient fashion to win their first NBA title since the Magic Johnson era. And Bob the skycap was halfway to becoming a millionaire.
How Good Gamblers Think
How did Voulgaris know that his Lakers bet would come through? He didn’t. Successful gamblers—and successful forecasters of any kind—do not think of the future in terms of no-lose bets, unimpeachable theories, and infinitely precise measurements. These are the illusions of the sucker, the sirens of his overconfidence. Successful gamblers, instead, think of the future as speckles of probability, flickering upward and downward like a stock market ticker to every new jolt of information. When their estimates of these probabilities diverge by a sufficient margin from the odds on offer, they may place a bet.
The Vegas line on the Lakers at the time that Voulgaris placed his bet, for instance, implied that they had a 13 percent chance of winning the NBA title. Voulgaris did not think the Lakers’ chances were 100 percent or even 50 percent—but he was confident they were quite a bit higher than 13 percent. Perhaps more like 25 percent, he thought. If Voulgaris’s calculation was right, the bet had a theoretical profit of $70,000.
FIGURE 8-1: HOW VOULGARIS SAW HIS LAKERS BET
Outcome
Probability
Net Profit
Lakers win championship
25%
+$520,000
Lakers do not win championship
75%
–$80,000
Expected profit
+$70,000
If the future exists in shades of probabilistic gray to the forecaster, however, the present arrives in black and white. Bob’s theoretical profit of $70,000 consisted of a 25 percent chance of winning $520,000 and a 75 percent chance of losing $80,000 averaged together. Over the long term, the wins and losses will average out: the past and the future, to a good forecaster, can resemble one another more than either does the present since both can be expressed in terms of long-run probabilities. But this was a one-shot bet. Voulgaris needed to have a pretty big edge (the half dozen different reasons he thought the bookies undervalued the Lakers), and a pretty big head on his shoulders, in order to make it.
FIGURE 8-2: THE WORLD THROUGH THE EYES OF A SUCCESSFUL GAMBLER
Now that Voulgaris has built up a bankroll for himself, he can afford to push sm
aller edges. He might place three or four bets on a typical night of NBA action. While the bets are enormous by any normal standard they are small compared with his net worth, small enough that he can seem glumly indifferent about them. On the night that I visited, he barely blinked an eye when, on one of the flat screens, the Utah Jazz inserted a seven-foot-two Ukrainian stiff named Kyrylo Fesenko into the lineup, a sure sign that they were conceding the game and that Voulgaris would lose his $30,000 bet on it.
Voulgaris’s big secret is that he doesn’t have a big secret. Instead, he has a thousand little secrets, quanta of information that he puts together one vector at a time. He has a program to simulate the outcome of each game, for instance. But he relies on it only if it suggests he has a very clear edge or it is supplemented by other information. He watches almost every NBA game—some live, some on tape—and develops his own opinions about which teams are playing up to their talent and which aren’t. He runs what is essentially his own scouting service, hiring assistants to chart every player’s defensive positioning on every play, giving him an advantage that even many NBA teams don’t have. He follows the Twitter feeds of dozens of NBA players, scrutinizing every 140-character nugget for relevance: a player who tweets about the club he’s going out to later that night might not have his head in the game. He pays a lot of attention to what the coaches say in a press conference and the code that they use: if the coach says he wants his team to “learn the offense” or “play good fundamental basketball,” for instance, that might suggest he wants to slow down the pace of the game.
To most people, the sort of things that Voulgaris observes might seem trivial. And in a sense, they are: the big and obvious edges will have been noticed by other gamblers, and will be reflected in the betting line. So he needs to dig a little deeper.
Late in the 2002 season, for instance, Voulgaris noticed that games involving the Cleveland Cavaliers were particularly likely to go “over” the total for the game. (There are two major types of sports bets, one being the point spread and the other being the over-under line or total—how many points both teams will score together.) After watching a couple of games closely, he quickly detected the reason: Ricky Davis, the team’s point guard and a notoriously selfish player, would be a free agent at the end of the year and was doing everything he could to improve his statistics and make himself a more marketable commodity. This meant running the Cavaliers’ offense at a breakneck clip in an effort to create as many opportunities as possible to accumulate points and assists. Whether or not this was good basketball didn’t much matter: the Cavaliers were far out of playoff contention.15 As often as not, the Cavaliers’ opponents would be out of contention as well and would be happy to return the favor, engaging them in an unspoken pact to play loose defense and trade baskets in an attempt to improve one another’s stats.16 Games featuring the Cavaliers suddenly went from 192 points per game to 207 in the last three weeks of the season.17 A bet on the over was not quite a sure thing—there are no sure things—but it was going to be highly profitable.
Patterns like these can sometimes seem obvious in retrospect: of course Cavaliers games were going to be higher-scoring if they had nothing left to play for but to improve their offensive statistics. But they can escape bettors who take too narrow-minded a view of the statistics without considering the context that produce them. If a team has a couple of high-scoring games in a row, or even three or four, it usually doesn’t mean anything. Indeed, because the NBA has a long season—thirty teams playing eighty-two games each—little streaks like these will occur all the time.18 Most of them are suckers’ bets: they will have occurred for reasons having purely to do with chance. In fact, because the bookmakers will usually have noticed these trends as well, and may have overcompensated for them when setting the line, it will sometimes be smart to bet the other way.
So Voulgaris is not just looking for patterns. Finding patterns is easy in any kind of data-rich environment; that’s what mediocre gamblers do. The key is in determining whether the patterns represent noise or signal.
But although there isn’t any one particular key to why Voulgaris might or might not bet on a given game, there is a particular type of thought process that helps govern his decisions. It is called Bayesian reasoning.
The Improbable Legacy of Thomas Bayes
Thomas Bayes was an English minister who was probably born in 1701—although it may have been 1702. Very little is certain about Bayes’s life, even though he lent his name to an entire branch of statistics and perhaps its most famous theorem. It is not even clear that anybody knows what Bayes looked like; the portrait of him that is commonly used in encyclopedia articles may have been misattributed.19
What is in relatively little dispute is that Bayes was born into a wealthy family, possibly in the southeastern English county of Hertfordshire. He traveled far away to the University of Edinburgh to go to school, because Bayes was a member of a Nonconformist church rather than the Church of England, and was banned from institutions like Oxford and Cambridge.20
Bayes was nevertheless elected as a Fellow of the Royal Society despite a relatively paltry record of publication, where he may have served as a sort of in-house critic or mediator of intellectual debates. One work that most scholars attribute to Bayes—although it was published under the pseudonym John Noon21—is a tract entitled “Divine Benevolence.”22 In the essay, Bayes considered the age-old theological question of how there could be suffering and evil in the world if God was truly benevolent. Bayes’s answer, in essence, was that we should not mistake our human imperfections for imperfections on the part of God, whose designs for the universe we might not fully understand. “Strange therefore . . . because he only sees the lowest part of this scale, [he] should from hence infer a defeat of happiness in the whole,” Bayes wrote in response to another theologian.23
Bayes’s much more famous work, “An Essay toward Solving a Problem in the Doctrine of Chances,”24 was not published until after his death, when it was brought to the Royal Society’s attention in 1763 by a friend of his named Richard Price. It concerned how we formulate probabilistic beliefs about the world when we encounter new data.
Price, in framing Bayes’s essay, gives the example of a person who emerges into the world (perhaps he is Adam, or perhaps he came from Plato’s cave) and sees the sun rise for the first time. At first, he does not know whether this is typical or some sort of freak occurrence. However, each day that he survives and the sun rises again, his confidence increases that it is a permanent feature of nature. Gradually, through this purely statistical form of inference, the probability he assigns to his prediction that the sun will rise again tomorrow approaches (although never exactly reaches) 100 percent.
The argument made by Bayes and Price is not that the world is intrinsically probabilistic or uncertain. Bayes was a believer in divine perfection; he was also an advocate of Isaac Newton’s work, which had seemed to suggest that nature follows regular and predictable laws. It is, rather, a statement—expressed both mathematically and philosophically—about how we learn about the universe: that we learn about it through approximation, getting closer and closer to the truth as we gather more evidence.
This contrasted25 with the more skeptical viewpoint of the Scottish philosopher David Hume, who argued that since we could not be certain that the sun would rise again, a prediction that it would was inherently no more rational than one that it wouldn’t.26 The Bayesian viewpoint, instead, regards rationality as a probabilistic matter. In essence, Bayes and Price are telling Hume, don’t blame nature because you are too daft to understand it: if you step out of your skeptical shell and make some predictions about its behavior, perhaps you will get a little closer to the truth.
Probability and Progress
We might notice how similar this claim is to the one that Bayes made in “Divine Benevolence,” in which he argued that we should not confuse our own fallibility for the failures of God. Admitting to our own imperfections is a necessary ste
p on the way to redemption.
However, there is nothing intrinsically religious about Bayes’s philosophy.27 Instead, the most common mathematical expression of what is today recognized as Bayes’s theorem was developed by a man who was very likely an atheist,28 the French mathematician and astronomer Pierre-Simon Laplace.
Laplace, as you may remember from chapter 4, was the poster boy for scientific determinism. He argued that we could predict the universe perfectly—given, of course, that we knew the position of every particle within it and were quick enough to compute their movement. So why is Laplace involved with a theory based on probabilism instead?
The reason has to do with the disconnect between the perfection of nature and our very human imperfections in measuring and understanding it. Laplace was frustrated at the time by astronomical observations that appeared to show anomalies in the orbits of Jupiter and Saturn—they seemed to predict that Jupiter would crash into the sun while Saturn would drift off into outer space.29 These predictions were, of course, quite wrong, and Laplace devoted much of his life to developing much more accurate measurements of these planets’ orbits.30 The improvements that Laplace made relied on probabilistic inferences31 in lieu of exacting measurements, since instruments like the telescope were still very crude at the time. Laplace came to view probability as a waypoint between ignorance and knowledge. It seemed obvious to him that a more thorough understanding of probability was essential to scientific progress.32
The intimate connection between probability, prediction, and scientific progress was thus well understood by Bayes and Laplace in the eighteenth century—the period when human societies were beginning to take the explosion of information that had become available with the invention of the printing press several centuries earlier, and finally translate it into sustained scientific, technological, and economic progress. The connection is essential—equally to predicting the orbits of the planets and the winner of the Lakers’ game. As we will see, science may have stumbled later when a different statistical paradigm, which deemphasized the role of prediction and tried to recast uncertainty as resulting from the errors of our measurements rather than the imperfections in our judgments, came to dominate in the twentieth century.