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We prove that a subtle but substantial bias exists in a common measure of the conditional dependence of present outcomes on streaks of past outcomes in sequential data. The magnitude of this streak selection bias generally decreases as the sequence gets longer, but increases in streak length, and remains substantial for a range of sequence lengths often used in empirical work. We observe that the canonical study in the influential hot hand fallacy literature, along with replications, are vulnerable to the bias. Upon correcting for the bias, we find that the longstanding conclusions of the canonical study are reversed.

Oppenheimer and Monin (2009) recently found that subjectively rare events are taken to indicate a longer preceding sequence of unobserved trials than subjectively common events, an effect which they refer to as the retrospective gambler's fallacy. The current paper extends this idea to the situation where participants judge the likelihood of streak continuation. Participants were told about a streak produced by a random process (coin flips) or human performance (basketball shots), and either predicted the next outcome or inferred the immediately preceding outcome. For the coin scenarios, participants tended to expect streak termination -- the gambler's fallacy --- and this effect was the same for predictions and retrospective inferences. In the basketball scenarios, no overall bias was found in either prospective or retrospective judgments. The results support Oppenheimer and Monin's suggestion that reconstruction of the past entails the same heuristics as prediction of the future; they also support the idea that the nature of the data-generating process is a key determinant of whether people fall into the gambler's fallacy. It is suggested that the term retrospective gambler's fallacy be used to describe situations where a streak is taken to indicate that the preceding unobserved outcome was of the opposite type, and that the phenomenon discovered by Oppenheimer and Monin be referred to as retrospective representativeness, or a retrospective belief in the law of small numbers.

We demonstrate that lottery markets can exhibit the “hot-hand” phenomenon, in which past winning numbers tend to have a greater share of the betting proportion in future draws even though past and future events are independent. This is surprising as previous works have instead documented the presence of an opposite effect, the “gambler’s fallacy” in the U.S. lottery market. The current literature also suggests that the gambler’s fallacy prevails when random numbers are generated by mechanical devices, such as in lottery games. We use two sets of naturally occurring data to show that both the gambler’s fallacy and the hot-hand fallacy can exist in different types of lottery games. We then run online experimental studies that mimic lottery games with one, two, or three winning numbers. Our experimental results show that the number of winning prizes impacts behavior. In particular, whereas a single-prize game leads to a strong presence of the gambler’s fallacy, we observe a significant increase in hot-hand behavior in multiple-prize games with two or three winning numbers. This paper was accepted by David Simchi-Levi, behavioral economics.

We suggest a productive categorization of positive and negative recency biases – such as the gambler’s fallacy and the hot hand fallacies – in terms of bottom-up and top-down processes. In this manner, the biases may be treated similarly to perceptual illusions. We provide a brief review of this framework, and in two experiments suggest a novel methodology that utilizes a nonbinary outcome design. In Experiment 1, we showed that a negative recency belief can generate either a positive or negative recency behavioral bias. In Experiment 2, we performed a more granular examination of bottom-up processes that might underpin probability judgments by manipulating reward variance and a sequence window, showing a list of recent winning positions First, we found that the gain/loss variance and the size of the sequence window affected the frequency of subjects showing either positive or negative recency biases. Second, we found a main effect of sequence window on a subject’s positive recency bias after streaks of either rewards or outcomes. Finally, we found data qualitatively supporting an inverse spatial proximity bias for subjects displaying behavior consistent with a negative recency bias.

This research demonstrates a new effect of consumer similarity in a chance context. Six studies show how similarity with previous winners can positively or negatively affect potential participants’ perceived likelihood of winning the subsequent independent sweepstakes draw. Attributions of winning outcomes to a personal cause or randomness change potential participants’ expectations regarding the sequence of more or less similar winners. When personal attribution is prevalent, exposure to more (vs. less) similar winners causes potential participants to feel they are more likely to win and, as a consequence, judge the sweepstakes as more attractive. This positive effect of similarity is mediated by the expectation of more repetitions of similar winners consistent with the belief that luck can be transferred among similar people. When randomness is presented as the salient cause for winning, though, people’s subjective conception of randomness leads them to expect more alternations in the sequence of more and less similar winners, thus prompting a reversal of the similarity effect. That is, they feel less likely to win when the sweepstakes features a more compared to a less similar winner.

We test for a “hot hand” (i.e., short-term predictability in performance) in Major League Baseball using panel data. We find strong evidence for its existence in all 10 statistical categories we consider. The magnitudes are significant; being “hot” corresponds to between one-half and one standard deviation in the distribution of player abilities. Our results are in notable contrast to the majority of the hot-hand literature, which has generally found either no hot hand or a very weak hot hand in sports, often employing basketball shooting data. We argue that this difference is attributable to endogenous defensive responses: basketball presents sufficient opportunity for transferring defensive resources to equate shooting probabilities across players, whereas baseball does not. We then develop a method to test whether baseball teams do respond appropriately to hot opponents. Our results suggest teams respond in a manner consistent with drawing correct inference about the magnitude of the hot hand except for a tendency to overreact to very recent performance (i.e., the last five attempts). The online appendix is available at https://doi.org/10.1287/mnsc.2017.2804 . This paper was accepted by Amit Seru, finance.

We study a class of permutation tests of the randomness of a collection of Bernoulli sequences and their application to analyses of the human tendency to perceive streaks of consecutive successes as overly representative of positive dependence—the hot hand fallacy. In particular, we study permutation tests of the null hypothesis of randomness (i.e. that trials are i.i.d.) based on test statistics that compare the proportion of successes that directly follow k consecutive successes with either the overall proportion of successes or the proportion of successes that directly follow k consecutive failures. We characterize the asymptotic distributions of these test statistics and their permutation distributions under randomness, under a set of general stationary processes, and under a class of Markov chain alternatives, which allow us to derive their local asymptotic power. The results are applied to evaluate the empirical support for the hot hand fallacy provided by four controlled basketball shooting experiments. We establish that substantially larger data sets are required to derive an informative measurement of the deviation from randomness in basketball shooting. In one experiment, for which we were able to obtain data, multiple testing procedures reveal that one shooter exhibits a shooting pattern significantly inconsistent with randomness—supplying strong evidence that basketball shooting is not random for all shooters all of the time. However, we find that the evidence against randomness in this experiment is limited to this shooter. Our results provide a mathematical and statistical foundation for the design and validation of experiments that directly compare deviations from randomness with human beliefs about deviations from randomness and thereby constitute a direct test of the hot hand fallacy.

We use natural experiments in Haiti and Denmark to test recent theoretical predictions about how agents react to random events. Using player-level administrative data, we find that the average lottery player avoids numbers that recently won (the gambler’s fallacy). A small subset of players in each country exhibit the hot hand fallacy, and bet recent winners. We find no evidence of ‘streak switching,’ in which beliefs switch from the gambler’s fallacy to the hot hand fallacy as winning streaks grow. Follow-up survey data in Haiti indicate that almost all lottery players believe that some numbers are more likely to win than others, and that recent winning history is an important factor in subjective beliefs about numbers’ win probabilities.

A critical component of both economic and perceptual decision making under uncertainty is the belief-formation process. However, most research has studied belief formation in economic and perceptual decision making in isolation. One reason for this separate treatment may be the assumption that there are distinct psychological mechanisms that underlie belief formation in economic and perceptual decisions. An alternative theory is that there exists a common mechanism that governs belief formation in both domains. Here, we test this alternative theory by combining a novel computational modeling technique with two well-known experimental paradigms. We estimate a drift-diffusion model (DDM) and provide an analytical method to decode prior beliefs from DDM parameters. Subjects in our experiment exhibit strong extrapolative beliefs in both paradigms. In line with the common mechanism hypothesis, we find that a single computational model explains belief formation in both tasks and that individual differences in belief formation are correlated across tasks. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2016.2453 . This paper was accepted by Yuval Rottenstreich, judgment and decision making .

How does risk aversion change in wealth? To answer this question, we implemented a field experiment in the form of a free-to-play mobile game. Players made lottery choices at various points in the game and at different levels of in-game wealth. Since the game was designed as a closed economic system, that is, wealth could not be transferred into or out of the game, only in-game wealth was relevant for players' choices. Analyzing the choices of over 2000 players, we find evidence for decreasing absolute risk aversion and decreasing relative risk aversion. We also find evidence of an \"always safe\" heuristic in a subgroup of decisions and observe a tendency of players to act according to the \"hot hand fallacy\". Our research design allows us to exclude inertia and lets us analyze lottery stakes of significant size relative to in-game wealth. Our results render implications for theoretical research, empirical studies, and for the optimal design of financial products.