Estimating the Unknown by the Hot Hand Belief

Estimating the Unknown by the Hot Hand Belief Yanlong Sun (Yanlong.Sun@uth.tmc.edu) Hongbin Wang (Hongbin.Wang@uth.tmc.edu) School of Biomedical Informatics, University of Texas Health Science Center Houston, TX 77030 USA Abstract Compared to the gambler’s fallacy in which one makes predictions negatively dependent on the past information, in the hot hand belief, one makes predictions positively dependent on the past information. Both phenomena have been attributed to people’s misperception of randomness. The present study examines an alternative explanation that the positive dependency in the hot hand belief may be due to people’s effort to reduce uncertainty by estimating the unknown probability (common probability estimation), a result known as the Laplace’s rule of succession. We report an experiment to demonstrate that the dependency on the history can be reversed from negative to positive by manipulating the participants’ assumptions about the unknown probability. Keywords: hot hand belief; gambler’s fallacy; common probability estimation; probability matching. Introduction When faced with a series of events, people often attempt to predict what is to occur next based on the history of the previous outcomes, even when the underlying process governing those events is independent and stationary (or, statistically indistinguishable, for example, the same fair or biased coin is tossed repeatedly). As the independence and stationarity assumptions are usually characteristics of a random process, such tendency has often been labeled as misperception of randomness (for a recent review see, Oskarsson, Van Boven, McClelland, & Hastie, 2009). Among those documented, the gambler’s fallacy has the longest history, even older than the history of experimental psychology (see, Ayton & Fischer, 2004). When a fair coin is tossed repeatedly, a person with the gambler’s fallacy will predict a tail after a streak of heads. On the other hand, the same kind of past information can sometimes invoke an opposite prediction. A person with the hot hand belief will predict that a basketball player who has just scored several shots in a row is more likely to score again. In the actual shooting sequences, however, little statistical evidence has been found to reject the independence and stationarity hypotheses (Gilovich, Vallone, & Tversky, 1985; Tversky & Gilovich, 1989). (For a comprehensive review on the hot hand studies, see Bar-Eli, Avugos, & Raab, 2006.) The contrast between the gambler’s fallacy and the hot hand belief has received much attention (Ayton & Fischer, 2004; Burns & Corpus, 2004; Caruso, Waytz, & Epley, 2010; Croson & Sundali, 2005; Rabin, 2002; Sundali & Croson, 2006). Most notably, the same representativeness heuristic has been used to account for both the gambler’s fallacy and the hot hand belief (Gilovich, et al., 1985; Tversky & Kahneman, 1971). By this account, people’s perception of random events are governed by a “law of small numbers” such that a local sample should resemble the underlying population and chance is perceived as “a self-correcting process in which a deviation in one direction induces a deviation in the opposite direction to restore the equilibrium” (Tversky & Kahneman, 1974, p. 1125). Thus, in the gambler’s fallacy, a tail is “due” to reverse a streak of heads. In the hot hand belief, a streak of successes would make the observer to reject the randomness of the process and believe that a “hot hand” will make another shot (see also Tversky & Gilovich, 1989). Nevertheless, the representativeness account has been criticized for its incompleteness. Ayton and Fischer (2004) suggest that the gambler’s fallacy arises from the experience of negative recency in sequences of natural events such as sampling without replacement and the hot hand belief arises from the experience of positive recency in serial fluctuations in human performance such as in sports. Burns and Corpus (2004) show that subjects assume negative recency for scenarios they rated as “random” and positive recency for forecasting scenarios they rated as “nonrandom.” Moreover, it has been proposed that the hot hand belief may arise as an inference to the properties of other processes based on the outcomes of a random process. For example, people may infer the ability of a mutual fund manager from the fluctuations of the portfolio performance (Rabin, 2002), or, infer a person’s luck from the outcomes of a roulette game (Croson & Sundali, 2005; Sundali & Croson, 2006). More recently, Caruso, et al. (2010) report that when people perceive an intentional mind in the underlying process, they are more likely to show the hot hand belief than the gambler’s fallacy. Whereas the perception of randomness has often been implicated in explaining the gambler’s fallacy and the hot hand belief, the notion of randomness is highly debated even among mathematicians and philosophers. It has been suggested that the concept has to be broken down into more fundamental properties in order to be of any practical use (e.g., Blinder & Oppenheimer, 2008; Lopes & Oden, 1987). In particular, the gambler’s fallacy and the hot hand belief have been considered as misperceptions of randomness since they violate either one or both of the independence and the stationarity assumptions (see, Bar-Eli, et al., 2006; Gilovich, et al., 1985). In the present paper, we examine an alternative factor that may contribute to the contrast between the gambler’s fallacy and the hot hand belief,