Two years ago, while researching an article on sports, I came upon a conundrum that resisted any attempt to confine it to the language of the conventional sports page. It concerns a cherished gospel of the playing field that athletes and their fans call the “Hot Hand.” Heat in this case refers to transcendence, an inexplicable escalation of energy and skill. A golfer with a Hot Hand will send his drives twenty or thirty yards beyond his ordinary range; an archer will see her arrows graze each other as they strike the bull’s eye; a basketball player will hit a string of shots so acrobatic and indifferent to defense that he seems linked by invisible channels to the basket. Subjective accounts of these experiences, which athletes sometimes call “playing in the zone,” include perceptual changes, euphoric sensations, and other alterations of consciousness, and their reality is so universally accepted that they commonly dictate strategy. In baseball, every pitcher in the league knows which hitters, bringing streaks to town, are not to be offered hittable pitches, and when a basketball player gets “hot,” his teammates pass up open shots to get him the ball.
It never occurred to me that anyone could question the Hot Hand until, in the course of my research, I encountered Dr. Amos Tversky. A MacArthur Fellow, professor of psychology at Stanford, Tverksy made a statistical study of basketball streaks which brought him to the brutal, sacrilegious, and seemingly inarguable conclusion that they do not exist. Building on the assumption that the Hot Hand is primarily a function of self-confidence, that a player who makes a couple of shots will grow more confident and therefore more likely to make his next, Tversky reasoned that, if Hot Hand occurs, a player’s shooting average would tend to be slightly higher on shots following successful shots than on shots following misses. Shot-by-shot studies of several NBA teams indicated, however, that no such deviation occurs. Tversky found that a player’s overall average will remain approximately the same after one, two, or three hits, or one, two, or three misses. Such figures, he says, demonstrate that even a string of ten or eleven successful shots is no more significant, statistically, than a string of ten or eleven heads in a run of coin-tossing. Streaks, he says, are “epiphenomena,” which is to say, patterns imposed, after the fact, on random events. He traces their universal acceptance to a cognitive limitation in the human mind that makes it intolerant of randomness. “Basketball players don’t make their shots because they’re hot. They feel hot because they make them. And that feeling disappears as soon as they miss.”
Horrified, as any sports fan would be, at the implications of Tversky’s work, I conducted an informal study of the responses it evoked among people from various disciplines. Athletes and fans were too emotional to be helpful. Shocked that anyone would compare a basketball player’s shooting skill with a chance process like coin-tossing, they ignored the fact that Tversky had compensated for this by factoring in his players’ shooting percentages. Most mathematicians and statisticians approved of Tversky’s method as well as his conclusion. The only viable objections came from Berkeley philosophy professor, Michael Scriven, an expert on Probability Theory, and Robert Jahn, an engineering professor who directs Princeton University’s Anomalies Institute (which studies “the relationship between man and machines”). Scriven called Tversky naive and simplistic. “If a player has streaks,” he said, “they wouldn’t show up as statistical spikes but in overall scoring average.” As for Jahn, his objection was metaphysical. “Tversky’s work is part of a holy war between those who believe that man can influence his reality and those who believe that everything is chance.” Confronted with these objections, Tversky was unshaken. “I have presented this work all over the country for the last four years, and I’ve never seen it successfully challenged.”
It seems to me that more is at stake in this debate than a bit of sports mythology. Statistics is a bastion of materialism—dualism if you like—a means by which patterns of phenomena are measured objectively, and retrospectively. What we have here is a situation in which such measurement contradicts intuitive experience on the one hand and doesn’t touch it on the other. No sports fan, hearing of Tversky’s work, will cease to believe in streaks. Even Tversky, an avid basketball fan himself, admits that he remains among the faithful. “My mind is after all no less subject to illusion than anyone else’s.” The question then is not about sport and performance but illusion and reality. Does the fact that the Hot Hand is contradicted by statistics prove that it does not exist or does it prove that statistics themselves are limited? Statistics depend on probability curves. Streaks by definition are anomalous, extraordinary occurrences which fall beyond such curves. But any such curve can be sufficiently enlarged so that it includes more and more variables heretofore considered extraordinary. Thus, it is not impossible that the point of view offered by the window of statistics, by the very fact of fitting peaks and valleys of performance into a curve, distorts them the way landscape is distorted when viewed from an airplane. If one were to conduct a statistical study which measured the frequency with which individuals in a large group of people fell in love or wrote beautiful poems or had enlightenment experiences, would he discover that such behavioral anomalies are in fact as “predictable” as a run of heads in coin-tossing? If this could be demonstrated, would such phenomena be confined to the realm of illusion?
Consider that Tversky is really measuring a breakthrough in performance that involves the entire organism. Speaking of Hot Hands, athletes describe mental and emotional changes that sound like optimum brain-states. Basketball players say the basket seems “bigger” for them or that they feel an almost mystical connection to it. Ted Williams has said that when he was hitting well he could see the seams on a pitched ball, and more than one gymnast has reported that, on good days, the balance beam becomes “wider” so that any thought of falling off disappears. Again, scientific observers are skeptical. In the course of interviews I’ve conducted, several neuroscientists called such reports “hallucinations” or “illusions.” As one put it, “Williams didn’t see the seams. He thought he saw them. What he was probably experiencing was a cue his brain had developed to aid him in anticipating where the ball would be when he was ready to swing.”
Does Tversky’s data support this view? If so, does it contradict the athlete’s experience, misread it, or simply measure it inappropriately? Which data do we honor? If the player perceives an event, if his brain registers it, if it leads to verifiable success in his performance, what sort of definition of reality would permit us to conclude that it wasn’t really true? What does the word “really” mean in this context?
All such reasoning of course amounts to a sports fan’s not unpredictable answer to Tversky, but it may be that he offers us more than is at first apparent. Consider that what he is actually doing is defining the extraordinary as ordinary. The Hot Hand, he says, is absolutely within the scope of ordinary human behavior. And he arrives at that conclusion, not by demeaning breakthrough performance, but by enlarging the probability curve through which it is seen. A basketball player who makes a great number of consecutive shots during a single game may seem extraordinary when viewed only in terms of that particular game; but if viewed historically, his performance becomes—to use a favorite Zen expression—”nothing special.”
Is it possible that Tversky, far from denigrating the Hot Hand, is simply placing it in a spiritual context? Think of Shakyamuni Buddha’s observation that between any two moments in time an infinite number of “mind-moments” occur. What does such temporal expansion amount to, but a vast enlargement of the probability curve which any moment contains? In the context of that curve, any behavior—blissful or painful, enlightened or deluded—is indeed “nothing special.” Great teachers, of course, have been reminding their students of this fact for centuries. It may well be that enlightenment is the spiritual version of the Hot Hand and that Amos Tversky’s work—this great celebration of dualism—is as useful to us as anything in the sutras.
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