WHEN KLAY THOMPSON struck the match on perhaps the biggest inferno the game had ever seen, the world was still debating whether he deserved to be an All-Star.
It was Jan. 23, 2015, against the Sacramento Kings, and Thompson was ice cold coming out of halftime. The then-24-year-old had missed his previous five shots, including an uncontested layup from point-blank range, and a pair of 3-pointers without a defender in the vicinity. No one could have foreseen what happened next. With the score tied with 9 minutes, 45 seconds left in the third quarter, Thompson’s masterpiece began.
A midrange pull-up. Splash. A slam-on-the-brakes transition 3-pointer at the top of the key. Boom. Another transition 3-pointer. A one-handed alley-oop dunk on the fast break. A 3-pointer off the dribble in the face of rookie Nik Stauskas. Another 3 on Stauskas, top of the key, as the decibels in Oracle Arena began distorting the audio feed from the CSN broadcast.
He was just getting started. A 3, left wing. A 3, right wing. A finger-roll over three defenders. Another transition 3. A pull-up jumper at the free throw line. A corner 3-pointer with Stauskas blanketing him so fully that it prompted an incredulous “No way!” from Golden State Warriors broadcaster Bob Fitzgerald before the shot went down.
And then, finally, the signature elevator-doors play. After letting their shooter squeak through at the top of the key, David Lee and Draymond Green sealed off Thompson’s chasing defenders. Open look. Bucket.
The place went bonkers.
When the flames had abated, when the smoke had cleared, the Warriors were up 95-71. And it was on the back of 13 makes in a row, all from one player, in under 10 minutes of action. Thompson finished the quarter with a record-smashing 37 points.
After the game, a stunned Green was asked whether someone could possibly do that in the “NBA 2K” video game. His response? “Nah, you don’t get that hot in 2K.”
In that moment, Thompson claimed his status as a bona fide NBA star. But Thompson’s father, Mychal Thompson, a two-time champion with the Los Angeles Lakers, wasn’t there to see it. Turns out he watched the whole episode unfold on his iPhone while flying on the Lakers’ team plane. Klay later joked about his father, “I guarantee you first thing he says to me is, ‘You probably should have had 60.'”
And then, this past December, almost three years later, Thompson did score 60. In 29 minutes. No one in the shot-clock era had scored 60 points in less than 30 minutes, according to Elias Sports Bureau research. Thompson had gone en fuego again.
Or had he? In his 2013 New York Times best-seller, “Thinking, Fast and Slow,” Nobel Prize-winning economist Daniel Kahneman wrote: “The hot hand is a massive and widespread cognitive illusion.” No less an authority than Michael Lewis, author of “Moneyball,” has declared: “The streaks observed by fans and announcers and the players themselves [are] illusions.”
For 30 years, the consensus in the statistical and scientific community has been resounding: Our eyes are deceiving us. The hot hand is little more than a mirage.
But what if they were all wrong?
JULIUS ERVING, DARYL DAWKINS AND ANDREW TONEY were dripping sweat when the scrawny researchers shuffled onto the Philadelphia Spectrum floor. It was 1982, just before the team reached the NBA Finals that season, and Cornell professor Tom Gilovich had surprised his staff with word that they’d received permission to interview the 76ers for part of his groundbreaking study of the hot hand in sports.
“As you can imagine, the research assistants were beside themselves,” Gilovich remembers. “I never had my RAs be as thankful again.”
The task was to ask players whether they believed in the hot hand — the notion that a player can be more successful at an act after recent success. As expected, most of the 76ers they spoke with that day believed in the hot hand, just like the fans Gilovich and his crew surveyed. The players felt it. The coaches observed it. It was a thing. Everyone knew it.
At the time, Toney — known as the Boston Strangler for his penchant for tormenting the Celtics — was considered the Klay Thompson of his era, the streak shooter of all streak shooters. But Gilovich, along with researchers Robert Vallone and Amos Tversky (a trio known colloquially in the research community as “GVT”), analyzed shot charts provided by the team statistician and did not, in fact, find that Toney’s hot shooting was statistically significant. “Despite this widespread belief that Toney runs hot and cold,” the paper reads, “his runs of hits and misses did not depart from chance expectations.”
In fact, GVT found no evidence of the hot hand in any of the 76ers they studied during the 1980-81 season, including Dr. J. When Julius Erving — a 52 percent shooter — made three baskets in a row, he shot 48 percent on the fourth.
Maybe, they thought, they might find the elusive hot hand in free throw sequences, which would be unaffected by the shot selection that could muddy up the picture. Nope. The Boston Celtics’ team statistician forked over the team’s free throw logs, and GVT found nothing there, either. Sure, some players shot better after making one, but the difference wasn’t statistically significant. In other words, any fluctuation around the mean was seen as purely random variation, not evidence of being “in the zone.”
Perhaps they’d find it in a controlled experiment? The Cornell researchers designed a workout with 14 members of the men’s varsity and junior varsity teams, along with 12 members of the women’s varsity team. The players all took 100 shots from locations where they were expected to shoot roughly 50 percent, to mimic coin flips. After analyzing the data, the conclusion was clear: again, no hot hand.
“One has to realize that coin-tossing produces just as many runs,” GVT wrote.
And so the Fallacy of The Hot Hand was born. For 30 years, GVT’s findings would be held sacred in the scientific community. In the basketball community, not so much. Tversky once said: “I’ve been in a thousand arguments over this topic. I’ve won them all, and I’ve convinced no one.” When asked about the hot-hand fallacy, Red Auerbach wasn’t too keen on Tversky’s study, “Who is this guy? So he makes a study. I couldn’t care less.” The legendary Bobby Knight wasn’t a fan of the so-called fallacy, either.
Nonetheless, lessons gleaned from the 1985 “Hot Hand in Basketball” paper were repeated from lecture halls to business conferences to casinos across the globe. Beware the stockbroker who has beaten the S&P five years in a row. Don’t be tricked by the roulette table that seems to be running “hot” on red. And more recently, Don’t buy into Klay Thompson’s hot hand.
The fallacy, in truth, had become more well-known than the very notion of the hot hand itself. Type the phrase “hot hand” into Google’s search bar today and the first webpage returned is a Wikipedia entry. The title? “Hot-hand fallacy.”
IT’S SATURDAY AFTERNOON at the Warriors’ practice facility in downtown Oakland and Thompson’s gray shirt is drenched after a long shooting session with player development coach Chris DeMarco. Thompson is cooling down and talking to reporters at the NBA Finals, occasionally pulling up his shirt to blot the sweat falling from his forehead.
Thompson hasn’t been his best. The 27-year-old is shooting 36.6 percent from the floor in the postseason. In Game 1 of the Finals, he shot 3-of-16, the worst in a decade among players with at least 15 attempts in the Finals. Everyone wants to know: When is this guy going to get hot again?
As he sits on a stool in front of a tangled mess of voice recorders, he’s reminded that he once missed five shots in a row before making 13 in a row. The notion gives him hope. “Yeah, it definitely helps,” Thompson says. “I always think about the times I’ve gotten hot in this league. It’s been a lot, thank God.”
Thompson says he felt the hot hand for the first time in sixth grade, about a decade before he became perhaps the most flammable player in the NBA. The first time he felt the magic as a pro? He recalled it almost immediately. It came in Denver. He scored 19 points in 18 minutes off the bench just before the All-Star break. “It really gave me confidence I can score at this level at a high clip,” Thompson says. That feeling proved prophetic. Thompson averaged 17 points per game after the break — up from just 7.2 prior to it.
And the rest has been hot-hand history — or it would be, if anyone believed in the hot hand: There was Thompson’s 60 points in Indiana, and the quarter-long blaze in Sacramento. In fact, a study of 26,425 shots over the past three seasons reveals that Thompson has the longest streak of makes when compared to a Finals crew of Curry, Kevin Durant, LeBron James, Kyrie Irving and Kevin Love. Only Love had a lower field goal percentage than Thompson over that time, but Thompson was the only player of the bunch with a streak of 11, 12 or 13 makes (all coming from that Sacramento game).
And Thompson’s streakiness leaps off the page when he’s compared with Curry, considered by many the best shooter ever. Over the past three seasons, Curry displayed superior shooting percentages (48.2 percent versus Thompson’s 46.1). Still, the longest shooting streak for Curry? Eight makes in a row, which he did only once. Thompson? He did that not once, not twice, but six times.
And so, on this Saturday afternoon, Thompson has faith that he’ll snap out of the funk soon. He draws upon the Denver game, the Sacramento game, the Indiana game. To catch fire, he retraces his steps. “It’ll come,” Thompson says. “I think about those moments leading up to the game, in the game that got me to that point. I tell myself this whole postseason that it’s gonna happen. I really believe it.”
JOSHUA B. MILLER, A LIFELONG WARRIORS FAN and Bay Area native, has flown into San Francisco from Milan just in time for the NBA Finals. But the professor at Italy’s Bocconi University is not here for hoops. This is a business trip, a visit to nearby Stanford University, where he’ll represent the joint work he has done with research partner Adam Sanjurjo as he attends talks by world-renowned economists.
And now, at a Ferry Building coffee shop overlooking the San Francisco Bay, Miller is attempting to prove to my audience of one that everything I’ve ever believed about streakiness in sports is wrong. He’s drawing diagrams on a notepad, trying to explain the thinking behind an explosive assertion that he and Sanjurjo discovered: The hot hand is real — we’ve just been using faulty math all along to identify it.
“Say Steph Curry has taken 100 shots in a row …” he begins.
Miller, with piercing emerald eyes and long, professorial wavy brown hair, looks down as he draws X’s and O’s in a long sequence. In this hypothetical, Curry is positioned on the court where he’d make half his shots like flipping a coin. “Put him at half court or something,” Miller says with a laugh.
He draws brackets underneath strings of three X’s, signifying a trio of makes, then draws an arrow on the fourth shot of the sequence. The fourth shot is important here, it seems.
“You select each shot in which Curry hit the previous three shots and write it down on a piece of paper — whether it was a make or miss — and put each piece of paper in a bucket,” Miller says. “He makes three in a row anywhere in the sequence, you put that fourth shot in a bucket.”
After then, after drawing a series of brackets and arrows on the line of X’s and O’s, he looks up and poses a simple question: “If you reached into the bucket and pulled out a shot at random, what are the chances it’s a make?” Miller asks.
I begin to panic. The answer seems obvious. The hot hand is a fallacy, right? It’s 50-50, no different from a coin flip. Still, I have a creeping sense that my answer is wrong. Or why would he be doing this?
“Fifty-fifty,” I say.
It is wrong. According to Miller, in this hypothetical, with Curry shooting from a 50-50 spot, if you study the fourth shots that follow three makes in a row, the probability that those fourth shots are makes is … lower. In fact, he says, it’s 46 percent, in the case of 100 shots, for example.
And now I am Neo, downing the red pill. Everything I believe is suddenly wrong. I ask Miller: Why isn’t it 50-50? He replies that a subtle but substantial selection bias exists when picking out shots to look at that are conditional on the outcomes of other shots within the same sequence. In this case, the condition “given three hits in a row” actually changes the game and suppresses the likelihood that the fourth shot turns up as a hit.
Could you wrap your head around that? Neither could I. But Miller and Sanjurjo get this response so often that he shared one of the go-to analogies they’ve designed to enlighten the masses. Miller asks me to consider the case of the mansion and the broken clock. Imagine you own a mansion with two rooms that are identical except for one key detail. One room has a fully functioning clock on the wall that reads the correct time. The other room has the same clock on the wall, except it is broken and stuck at 7:42. And you know this fact.
So, you have a long night of drinking. You stumble home. You pass out in one of the two rooms. Later, you wake up in a daze. You look on the wall and see the clock reads 7:42. Remember, there was a 50-50 chance you woke up in the broken-clock room and a 50-50 chance you woke up in the correct-clock room. But given that you know one clock is stuck at 7:42, what room would you guess you’re in?
(The jerk’s answer is to say that you’d simply wait one minute and find out soon enough. But put that aside for now.) “The broken-clock room,” I say.
Miller smiles. “Exactly,” he says. “This is the correct assumption. I mean, What are the odds?”
Miller delivers this last line rhetorically, as if to say, “Duh.” But then he works through the probabilities on paper and writes it down to show that there’s a 1-in-720 (given 720 minutes every 12 hours) chance that you happened to wake up in the correct-clock room — a long shot. But there are much stronger odds that you fell into the broken-clock room, where it’s always 7:42. Suddenly, your underlying 50-50 odds change with knowledge of the broken clock.
This, Miller explains, is what math-heads call the principle of restricted choice. Because the room with the broken clock is restricted to “choose” just one time, whereas the room with the fully functioning clock can “choose” any time (including 7:42), knowing that the “chosen” time is 7:42 makes it relatively more likely that you are in the room with the broken clock.
(At this point, I was with him … and you might be, too. Who hasn’t woken up drunk in a house with matching rooms and mismatched clocks?)
Fair warning: The following seven paragraphs traffic in math that can give even mathematicians migraines. If you’d like to skip this part and reconvene at the start of the next section, we won’t blame you. If you give it a whirl but find yourself feeling confused, just know you’re not alone.
And now Miller brings it back to coin flips and the subtle selection bias of hit streaks. The broken-clock example, Miller says, is an extreme illustration of what’s happening in a long line of coin flips. Let’s say that in analyzing the results of 100 coin flips you see a string of three heads (HHH_) and “_” is, say, Flip 42 in the sequence — which was selected because it was preceded by HHH. What is the _ in the sequence? Of course, you don’t know which kind of sequence you are in. it could be HHHT or it could be HHHH. You think it’s a 50-50 chance that Flip 42 is H (or T).
But here’s where it changes, and it is sneaky. In sequence HHHH, you could have selected Flip 43 because it continues a run of three heads, and you haven’t excluded Flip 44 or 45, like you would have in sequence HHHT (it ends the run of heads). The excluded flips in the HHHT world mean it is more likely that Flip 42 will turn up T rather than H, given the condition that the flip was selected because it was preceded by three heads in a row. The choice has been restricted. To tie it back to the big ol’ mansion, the broken clock excludes more times that are different from the one you are seeing, and the kind of sequence with HHHT excludes more flips that are different from the one you selected.
“So, that was interesting,” Miller says.
Miller brings it home.
“So if we tell you to look at Steph Curry’s shooting records for the 3-point contest and pick out one of his shots that is preceded by three hits in a row, that shot is more likely to be a miss than his ability would suggest.”
The implication is huge: If Curry is statistically less likely to make a shot after a string of makes (46 percent), then even shooting 50 percent is evidence of the hot hand.
Interesting indeed. This coin-flip discovery, upon its publishing in 2015, flipped the statistical community upside down. One prominent statistician called it “the zeitgeist jackpot.” In a world of people who study probabilities, the unlikeliest of things had happened. And it had all begun with Craig Hodges.
MILLER AND SANJURJO COULDN’T GET it out of their heads. At the 1991 All-Star Weekend Three-Point Shootout, Hodges had made 19 3-pointers in a row. It might have been the single-most glaring piece of evidence that the hot hand exists. Miller and Sanjurjo had watched it all as kids, and it stuck with them. And yet, in 2003, when researchers had analyzed 3-point shooting data in a follow-up study to the 1985 GVT paper, it was the same outcome: zero evidence of the hot hand.
In summer of 2013, Miller and Sanjurjo, friends from their undergraduate studies at UC Santa Barbara, were convinced something was off. Sanjurjo, a die-hard Celtics fan and professor at University of Alicante in Spain, shared Miller’s belief that the hot hand existed but couldn’t figure out why no one had found it in the data. So Miller and Sanjurjo called up a connection who helped run a Spanish league B-level pro team and asked whether they could set up an experiment in their gym to test the hot hand for themselves.
Eight players on the team showed up. Miller and Sanjurjo asked each to shoot 300 shots in this session from a spot on the floor where they’d make roughly 50 percent of their shots. After shooting, each player ranked his teammates from most likely to least likely to have the hot hand in the task, without having seen them shoot.
The shots went up. Thousands of them. But it took only a few minutes before Miller and Sanjurjo suspected they were onto something. One particular shooter, referred to only as “R.C.,” showed what appeared to be an extremely hot hand. They went back to the survey.
“He was at the top of everyone’s ranking,” Miller says. “He’s the hot-hand guy.”
Clearly, R.C. was the Klay Thompson of this Spanish league team. But it was more than that. From best to worst, the predictions were predictive: “The amazing thing,” Miller says, “was that their ranking was very much correlated with their performance.” In other words, players might be able to pick up on hot-hand tendencies without even looking at the math.
After three more sessions of 300 shots over the course of a year, Miller and Sanjurjo ran the numbers. And there it was: evidence of the hot hand. And not just with R.C. Others, too, demonstrated statistical evidence of a higher likelihood of making a shot after having made a string of shots. So they decided to take their analysis and apply it to GVT and other similar studies.
“Maybe if we analyze their data our way, we might find something in their data,” Miller says.
Then they re-examined the underlying data in the 2003 study and realized those researchers had looked at only four years of data, from 1994 to 1997. Why not look at all 3-point shooting contests? Miller and Sanjurjo called up a guy in Switzerland who kept DVDs of every NBA contest. They bought each recording.
And they watched. And watched and watched. … And watched. After reviewing the film, they tallied up the data and studied the 33 shooters with at least 100 shots in the contest. Eight of the shooters showed significant evidence of the hot hand, including Hodges, Curry, Ray Allen and Kyle Korver.
“Nah, you don’t get that hot in 2K.”
Draymond Green, when asked about Klay Thompson’s 37-point, 13-makes-in-a-row quarter
Miller and Sanjurjo were dumbfounded. Why had no one discovered this before?
And that’s when they found it: When Miller and Sanjurjo dug into the metrics of the prior studies, they found that they were tainted by a downward bias in their math that hid the hot-hand effect. Put simply, their detection meter wasn’t properly calibrated. Remember the coin flip exercise? The bias demonstrated in that example was the basis for their new insights. In 2015, they published their findings in a working paper with the following thunderous conclusion:
“In contrast with previous studies, we find substantial evidence of hot-hand shooting in the NBA Three-Point Contest. This leaves little doubt that the hot hand not only exists, but actually occurs regularly.”
The basketball world — or at least the portion of the basketball world that cared deeply about statistical studies — rejoiced. Among academics, the debate raged on.
IT WAS AROUND GILOVICH’S 60TH BIRTHDAY when he realized his days of playing pickup basketball at Cornell campus gymnasiums were over. As it happens, the pounding on the body forced Gilovich to move on to the tennis court, playing regularly with Cornell mathematicians and economists.
When Miller and Sanjurjo’s paper hit Gilovich’s desk in 2015, Gilovich knew it was big. The paper was buzzing. It was the top downloaded academic paper in the world for a few weeks and landed in the top 10 for 2015 despite not hitting the press until late in the year. But Gilovich also sought the help of his mathematicians at the tennis court — “I’m not a statistician; I’m a psychologist,” he concedes to me — to help him understand the nitty-gritty models.
Gilovich wasn’t the only one trying to wrap his head around the math. You can lump former NBA commissioner David Stern in there, as well. Speaking at a Columbia University panel in 2015, Stern held a copy of a Wall Street Journal article by Ben Cohen detailing Miller-Sanjurjo’s groundbreaking findings.
“You talk about sports being important,” a retired Stern said while sitting on the Columbia panel. “Nobel laureates spending their time debating the hot hand. … Help!”
Gilovich printed out copies of the study and handed them to his tennis friends, some of them giants in their fields. Was the Hot-Hand selection bias that Miller-Sanjurjo discovered real? Did the problem of restricted choice taint the GVT paper and suppress a hidden hot-hand effect? Essentially, was GVT playing checkers instead of chess? After review, Gilovich contends that the “jury is still out” on Miller-Sanjurjo’s technique.
“People with tremendous math skills are all over the map on this one,” Gilovich says over the phone. “I simply say that because the mathematicians I talk to are befuddled about the proper statistical analysis. Time will have to tell on that one.”
There is, though, a contingent of mathematicians in Miller-Sanjurjo’s corner. Among them, Maya Bar-Hillel, a cognitive psychologist at Hebrew University who also worked with Tversky, who passed away in 1996; and prominent statistician Andrew Gelman, who wrote a blog post titled, “Hey guess what, there really is a hot hand!” in which he called Miller-Sanjurjo’s findings “big news” in the field. Three papers by Miller-Sanjurjo on the hot hand appear in the Harvard syllabus of Matthew Rabin’s Behavioral Economics course. Additionally, a popular mathematician’s blog run by two professors reviewed Miller-Sanjurjo’s work and called it “a simple, yet striking insight” before adding:
“It uses the most elementary of arguments from probability theory, yet seems to shed new light on a well-studied, much-argued problem. This achievement is wonderful, and gives me hope that there are still simple insights out there, simple insights that we all have missed — yet insights that could dramatically change the way we look at some of our most important open problems.”
Today, Gilovich and Miller-Sanjurjo have corresponded over email and have even discussed working together on further research. “It’s very common for different sides to fire missiles at each other from a distance, but I’m not interested in that,” Gilovich says, before offering an olive branch. “There’s a certain way in which [Miller and Sanjurjo]’s findings are not inconsistent with our broader point, which is ‘people exaggerate how much streakiness there is.'”
To be clear, Miller-Sanjurjo’s findings are squarely at odds with GVT’s 1985 study, which clearly stated that there is no evidence of a hot-hand effect in basketball. Miller-Sanjurjo’s work also followed a 2015 Harvard study presented at the MIT Sloan Sports Analytics Conference that found small, but significant evidence of the hot-hand, once controlling for the defense using SportVU data.
In fact, Miller-Sanjurjo found an effect that amounted to, on average, 8 percentage points in the 3-point shooting contest. In laymen’s terms, a contestant’s percentages would rise from, say, 40 percent following three straight misses (cold state) to 48 percent on shots following three straight makes (hot state). An effect of 8 percentage points is no small thing. Consider that only about 9 percentage points separated Kyle Korver’s leading 3-point percentage (45.1 percent) this season and the league average (35.8 percent).
Still, despite many of his academic peers siding with Miller-Sanjurjo’s work, Gilovich isn’t willing to say that his 1985 study doesn’t stand up. Meanwhile, he’ll continue hitting the tennis courts with his friends, where even he admits that he sometimes falls prey to the hot-hand fallacy. “There are days when the tennis racket feels like a magic wand in your hand and you go, ‘God this a great sport, I’ve figured it out!’ And you show up next time and it teaches you otherwise and nope, you haven’t figured it out.”
AND NOW THE $60,000 QUESTION: This is landscape-changing news in science departments across the world, but does it matter on the court? After all, if the majority of coaches and players already believed in the hot hand without even hearing about a nerdy paper from the other side of the Atlantic, then what’s the big deal?
Consider the rise of 3-point shooting in the NBA. Even though everyone in the sport knew for decades that 3s are worth more than 2s, it wasn’t until analytics became a fixture in the sport that teams began to systematically leverage the power of 3-point shooters. The math created a surge. And even though coaches knew that fatigue and injuries might be related, it wasn’t until sports scientists flooded the sport before strategic rest recently became the norm. The science can be influential.
And then there’s this: If the “science” changes on hot-hand fallacy, it can convert the holdouts. And yes, there are the holdouts. Kyle Korver, for instance, views each of his shots as independent of each other and thinks heat checks can be corrosive to a healthy offense. When asked if he believes in the hot hand, he hardly counts himself as a devotee. “I believe a big part of shooting is confidence,” Korver says. “I also believe there’s overconfidence.”
Korver’s teammate James Jones, a 3-point shooting champion with a finance degree from University of Miami, characterizes himself as “a math guy” who thinks streak shooting is largely a mirage. Jones feels locked in sometimes even after shots that barely miss. “I can believe I’m hot,” Jones says, “but the streak has nothing to do with it.”
But then there’s Thompson, who is absolutely a believer. And in Game 2 of the NBA Finals, the belief that things would turn around and turn around quickly may well have helped him climb out of his slump. Thompson shot 8-of-12 from the floor in the Warriors’ 19-point win over the Cavaliers, making four 3-pointers in a game for the first time in more than a month. Then he followed it up with 21 points in the first half of Game 3, on 7-of-11 from the floor, and 5-of-7 from 3-point range. One might even have called him “hot.”
In Thompson’s eyes, the Warriors are not just championship contenders because they boast a host of MVP candidates. Their power comes from having great shooters who can get hot at any time. And when I asked him at a weekend practice before Game 2, which teammate did Thompson think gets the hottest on the team — not counting himself, of course — he let out a “Ah, man …” to buy some time.
After a few seconds of thought, Thompson finally named the two former MVPs on the squad, Curry and Durant. But which one gets hotter, if he could pick only one? After hemming and hawing for a few seconds, he gave up.
Said Thompson: “Flip a coin.”