The entire Hot Hand Fallacy is wrong!

Originally published in The Reasoner Volume 10, Number 9– Semptember 2016

The Rio 2016 Olympic Games have been, as usual, a great illustration of the Laplacian dictum according to which probability is partly due to our ignorance and partly to our knowledge. For the certainty that the best athlete(s) will win or, at the other end of the spectrum, the impossibility to figure out who is more likely to win, would turn the greatest sporting event on the planet into a very dull couple of weeks.

Laplacian romanticism notwithstanding, sport generates data, lots of it. Not only does this allow for high-tech betting, it also feeds uncertain reasoning research. In particular SportVU, a body-tracking system which builds on Israeli military technology, played an unexpected role in addressing a long-standing question: Is the Hot Hand phenomenon real, or is it just one of the many ways in which we tend to see patterns where there aren’t any?

In the lingo popularised by arcade games, a basketball player is “heating up” when he makes two hits in a row. After that he is believed, primarily by himself, to be more likely to score again, until the lucky streak ends. Everyone in the business appears to believe in it, and after three decades of controversy it may turn out that the popular belief is right. Yes, after three successful shots, players are more likely to score again. This conclusion is supported by data collected with SportVU technology which has been (re)interpreted in the light of a recent subtle and quite surprising finding by theoretical economists Joshua Benjamin Miller (Bocconi University) and Adam Sanjurio (University of Alicante). Intuitively, they identify a surprisingly subtle property of randomness which had so far managed to escape statistical analysis. The idea is that in finite series of coin tosses the probability of getting alternating results is strictly less than one half. This indeed proves the existence of a “cold hand” against which players have always been fighting, unknowingly so far. This result is fundamental, for previous SportVU data did not highlight a significant variation in the success rate of NBA players as a consequence of the Hot Hand. With the adjustments provided by the cold-hand effect, increases of up to 12% in scoring rates become observable. As Woody Allen would put it, the entire Hot Hand Fallacy is wrong. The details are of course rather subtle. The “user-friendly” explanation by J.B. Miller and A. Sanjurio (2016: A Primer and Frequently Asked Questions for `Surprised by the Gamblers and Hot Hand Fallacies? A Truth in the Law of Small Numbers‘, Available at SSRN: is indeed quite useful.

As reported on Andrew Gelman’s blog about a year ago this result generated some controversy among specialists. Those include T. Gilovich, whom had contributed with A, Tverski and R. Vallone to guiding the probability and statistics community towards the belief the Hot Hand should be thought of as fallacious. Back in 1985 they were convinced of the contrary, and sought to establish this experimentally, without success. Whilst players and experts believed the phenomenon was real, the data eventually pointed in the opposite direction. The Hot Hand was then relegated to a “misperception of random sequences”. Not so for basketball players, who, as it turned out, knew better.

Two quick observations. First, however big, data is of little help in the absence of an adequate context for its interpretation. The reason as to why the SportVU data alone confirmed, wrongly, the fallacy is interesting in its own right, and is nicely summed up in this recent piece by Jesse Singal on the New York Magazine. Second, when modelling social behaviour, mathematics must be sometimes bended to accommodate common sense. Daniel’s Bernoulli dissolution of the St. Petersburg Paradox is an early, spectacular, example of that.