A Bridge from Monty Hall to the (Anti-)Hot Hand: Restricted Choice, Selection Bias, and Empirical Practice

Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers

“…21 While GVT's analysis of conditional probabilities provides their only measure of the magnitude of the hot hand, they also analyze the number of runs, serial correlation, and variation of shooting percentage in 4-shot windows. Miller and Sanjurjo (2014) show that the runs and serial correlation tests, along with the conditional probability test for k = 1, all amount to roughly the same test, and moreover, that they are not sufficiently powered to identify hot hand shooting. The reason why is due to measurement error: the act of hitting a single shot is only a weak signal of a change in a player's underlying probability of success, which leads to an attenuation bias in the estimate of the increase in the probability of success associated with entering the hot state (see Appendix B and Stone (2012)'s work on measurement error when estimating autocorrelation in ability).…”

“…Outline of Proof : In the proof contained in Appendix A we begin by showing that the conditional expectation E[P k (X)|I k (X) = ∅] is equal to the conditional probability P(X τ = 1|I k (X) = ∅), (Friedman 1998;Nalebuff 1987;Selvin 1975;Vos Savant 1990). See Miller and Sanjurjo (2015a) for more details on the connection between the selection bias, the Monty Hall problem, and other conditional probability puzzles. 11 In particular, Miller and Sanjurjo (2016) show that the bias introduced here, in conjunction with a quasi-Bayesian model of decision making under sample size neglect (Benjamin, Rabin, and Raymond 2014;Griffin and Tversky 1992;Kahneman and Tversky 1972), provides a novel structural candidate explanation for the persistence of gambler's fallacy beliefs.…”

“…The reason why is due to measurement error: the act of hitting a single shot is only a weak signal of a change in a player's underlying probability of success, which leads to an attenuation bias in the estimate of the increase in the probability of success associated with entering the hot state (see Appendix B and Stone (2012)'s work on measurement error when estimating autocorrelation in ability). The test of variation in 4-shot windows is even less powered than the aforementioned tests (Miller and Sanjurjo 2014;Wardrop 1999).…”

“…23 The in-game free throw data that GVT analyze (Study 3: Celtics, 1980-81, 1981 seasons: 9 players), while arguably controlled, is not ideal for the study of hot hand shooting for a number of reasons: (i) hitting the first shot in a pair of isolated shots is not typically regarded by fans and players as hot hand shooting (Koehler and Conley 2003), presumably due to the high prior probability of success (≈ .75), (ii) hitting a single shot is a weak signal of a player's underlying state, which can lead to severe measurement error (Arkes 2013;Stone 2012), (iii) it is vulnerable to an omitted variable bias, as free throw pairs are relatively rare, and shots must be aggregated across games and seasons in order to have sufficient sample size (Miller and Sanjurjo 2014). In any event, subsequent studies of free throw data have found evidence that is inconsistent with the conclusions that GVT drew from the Celtics' data (Aharoni and Sarig 2011;Arkes 2010;Goldman and Rao 2012;Miller and Sanjurjo 2014;Wardrop 1995;Yaari and Eisenmann 2011). Table 2: Columns 4 and 5 reproduce the shooting percentages and number of shots that appear in Table 4, Columns 2 and 8, from Gilovich et al (1985) (note: 3 hits (misses) includes streaks of 3, 4, 5, etc.).…”

Dissecting the Monty Hall Anomaly

“…Much is at stake in evaluating these competing explanations. If the main cause is improper Bayesian updating following restricted choice, then the Monty Hall problem is a decision‐theoretic manifestation of longstanding statistical puzzles in education, psychology, and sports science (Miller and Sanjurjo ) and in medical decision‐making (Cox et al ). If it is a more general failure of hypothetical thinking, then we get new insight into economically important phenomena such as the “winner's curse” in common value auctions and a variety of other “cursed equilibria” (Esponda and Vespa ; Eyster and Rabin ).…”