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Sports fans, announcers, players and coaches all seem to believe in some form of the ``hot hand" phenomenon. Consider the sport of basketball (the source of a number of the articles discussed here). Here the belief is that shooters periodically get the hot hand, and during these periods they hit an unusually high percentage of their shots. Acting on this belief presumably has implications for how players are used by their coaches, how teammates behave in terms of passing the ball to the hot player, or to taking shots when they themselves feel hot. It certainly appears in commentaries on the game. The Kahneman-Tversky school, pioneers in the study of human misperceptions of chance processes, have reached some controversial conclusions about this phenomenon. They contend that the hot hand phenomenon is a ``cognitive illusion."

The Chance (Winter 1989) article by Tversky and Gilovich summarizes research originally reported in Cognitive Psychology (see references below). The article starts with a survey of fan beliefs, with the idea of pinpointing just what it is fans believe they are seeing when they say a player has the hot hand. A majority of fans believe that: (1) a player's chances of making a shot are better after just having made two or three shots in a row than after missing two or three; (2) when shooting pairs of free throws, the chances of making the second shot are better if the first shot is made than if it is missed; and (3) it is important to pass the ball to someone who has just made several shots in a row. The fans' average estimated field goal percentage for a hypothetical 50%shooter for shots taken after just making a shot is 61compared to 42%after just missing.

To study basketball shooting, Tversky, Gilovich and Vallone collected actual game data from the Philadelphia 76-ers entire 1980-81 season. They found no statistical evidence that the number or length of streaks differs from what would be observed in a sequence of independent tosses of a with probability of heads corresponding to a player's shooting percentage. Instead, they concluded that the fan beliefs above derive from biases in their intuitive judgments about probability. In evaluating a string of hits and misses, fans rely on the ``representativeness" heuristic, whereby even short sequences hits and misses are expected to be representative of the process that generates them, in this case shooting process with a (say) 50%hit rate. In evaluating sequences of tosses of a fair coin, people typically regard the sequence H-T-H-T-T-H as more likely than either H-H-H-T-T-T (which appears non-random) or H-H-H-H-T-H (which does not represent equal probabilities of H,T), even though each as a 1/64 probability. In basketball, letting H and T correspond to a shots hit and missed respectively, fans evaluating the last two sequences readily assume that forces other than chance must be at work. In fact, no such assumptions are necessary to explain patterns in the 76-ers shooting data that were collected in the study. Finally, to demonstrate that intuition in predicting the outcomes of shots (based on patterns of successes) is faulty, Tversky et. al.. set up a controlled shooting experiment with the Cornell University men's and women's basketball teams. Both players and spectators were asked to bet on outcomes of shots. Correlations were near zero between shooters' bets and their own performance and between observer's bets and shooter's performance. For both players and spectators, bets were correlated with the outcome of the previous shot.

In a rebuttal to these arguments , Larkey, Smith and Kadane (Chance, Fall 1989) argue that extracting entire sequences of hits and misses from a season-or even from an entire game- is too complicated a process, and that analysis should be restricted to ``cognitively manageable chunks of shooting opportunities." The idea is that player hitting 3 for 3 over the course of a game makes a different impression from one who scores three consecutive baskets in the contest. They propose two measures. The first is ``the hot hand out of context," which computes occurrences to opportunities ratios for streaks (e.g., 5 for 5) and imperfect sequences (e.g. 4 for 5). The second is ``the hot hand in 20-field goal context," which restricts to sequences of 20 baskets (and includes an estimate of the player's probability of taking his team's next shot). In each case, a 5 for 5 streak counts also as two 4 for 4 opportunities, three 3 for 3, etc. Vinnie Johnson, a player with a league-wide reputation as a streak shooter, indeed emerges as truly exceptional by the latter measure.

In their rejoinder in the same issue of Chance, Tversky-Gilovich assert that Larkey, Smith and Kadane have confounded the statistical question of whether the hot hand exists with the psychological question of whether people believe in it. They also find that Vinnie Johnson's exceptional rating hinges on one particular 7 for 7 streak, which, upon examination of game videos, turns out not to have happened (he missed once in the middle)!!

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