Surprised by the Gambler’s and Hot Hand Fallacies? A Truth in the Law of Small Numbers
نویسندگان
چکیده
We find a subtle but substantial bias in a standard measure of the conditional dependence of present outcomes on streaks of past outcomes in sequential data. The mechanism is driven by a form of selection bias, which leads to an underestimate of the true conditional probability of a given outcome when conditioning on prior outcomes of the same kind. The biased measure has been used prominently in the literature that investigates incorrect beliefs in sequential decision making—most notably the Gambler’s Fallacy and the Hot Hand Fallacy. Upon correcting for the bias, the conclusions of some prominent studies in the literature are reversed. The bias also provides a structural explanation of why the belief in the law of small numbers persists, as repeated experience with finite sequences can only reinforce these beliefs, on average. JEL Classification Numbers: C12; C14; C18;C19; C91; D03; G02.
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Miller and Sanjurjo (2015) suggest that many analyses of the hot hand and the gambler’s fallacies are subject to a bias. The purpose of this note is to describe our understanding of their main point in terms we hope are simpler and more accessible to non-mathematicians than is the original.
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