The gambler’s fallacy is the belief that, for random events like coin tosses, runs of a particular outcome will be balanced by a tendency for the opposite outcome i.e. longer the streak of heads, higher the probability of tails on the next toss (Ayton & Fischer, 2004). This is a false belief because the probability of both heads and tails is always 50% in a fair coin toss. Previous outcomes have no predictive value in a random process.
The flipside of the gambler's fallacy is the hot hand fallacy which is the belief that the longer a streak is, more likely it is to continue. A well-know example of this is the belief among basketball players, coaches and fans alike, that players have "hot streaks" i.e. they are more likely to make a successful shot if their previous shot was successful. But a study looked at sports data and found no proof of such streaks, other than what would be expected by pure chance. Just like with coin tosses and childbirths, a player's previous shot has no impact on the likelihood of success of their next shot (Gilovich et al., 1985).
COGNITIVE CAUSES
Cognitive causes are the psychological mechanisms that explain these fallacies. It is likely that no one of the multiple explanations can explain every instance of the fallacies, and each explanation is valid in some cases and invalid in others.
Firstly, while trying to answer a difficult question, we often substitute it with a different question that is easier to answer. So, while trying to predict the next outcome in a sequence of events, we often tend to answer the question 'what does a sequence of this type typically look like' (Kahneman & Frederick, 2002).
Now, we expect random sequences like coin tosses to look like this: 'HTHTHT' rather than this: 'HHHTTT'. This is because we expect that the underlying probability i.e. 50% for heads and 50% for tails will be represented even in two coin tosses, giving us one head and one tail. But ignoring the size of the sample set is problematic because rare occurrences are more likely in smaller set of instances. For example, the probability of only heads in 2 or 3 coin tosses is fairly high, but only heads in 100 coin tosses is next to impossible (Sedlmeier & Gigerenzer, 1997).
So, we should expect more variance and rare occurrences like streaks in smaller sample sets. But we expect small samples to look just like large samples where the rare occurrences balance themselves out. This expectation of balancing out leads to the gambler's fallacy. Even when larger samples are available, we tend to focus only on a small set of recent instances because those are more salient in memory and require lesser mental effort to process (Barron & Leider, 2010).
Now, because we expect alternations and reversals in random sequences, seeing streaks in small samples of a completely random process can make us doubt if the process is truly random. We may instead believe that underlying probability leans towards longer streaks and therefore we'd expect streaks to continue, resulting in the hot hand fallacy. Quite interestingly, the belief that longer a streak more likely it is to continue and the belief that longer a streak more likely it is to end, don't only co-occur but also stem from the same phenomena (Suetens et al., 2016).
Indeed, we adjust our expectations of what a sequence looks like based on our beliefs about the process that produces the sequence. We are prone to hot hand fallacy when we believe that the underlying process is based on skills, intentionality or idiosyncrasies, whereas we are subject to the gambler’s fallacy when we believe the process is random (Ayton & Fischer, 2004). Sometimes, when a random process results in streaks, we may start perceiving intentionality and skills, where none exists (Caruso et al., 2010).
And lastly, we have a natural tendency to mentally organize separate events into inter-related meaningful patterns and make predictions based on these patterns. Therefore our natural expectation is for patterns to repeat and for streaks to continue. But while thinking about random events, we deliberately override the repetition expectancy and replace it by an alternation bias i.e. sequence alternating between different equally likely outcomes (Duthoo et al., 2013).
NEURAL CAUSES
Regions involved in reinforcement learning and affective decision-making, like striatum and orbitofrontal cortex, are active during decisions exhibiting hot hand fallacy, and executive control regions such as the dorsolateral prefrontal cortex, are activated during gambler's fallacy. Reinforcement learning creates an expectation of rewards in situations which were rewarding in the past. This is the default learning mechanism of our brain and seems to be a key factor contributing to the hot hand fallacy. Involvement of executive processes in the gambler's fallacy indicates that the default mechanism of reinforcement learning is consciously overridden to produce expectation of reversal rather than continuation (Xue et al., 2011; Huang et al., 2019; Jessup & O'Doherty, 2011).
DECISION ADVANTAGES
For these tendencies to be passed down genetically or culturally to us from our ancestors, they must be beneficial in certain situations.
Pattern detection is one of the most important functions of our brains. Being able to predict outcomes based on observed patterns keeps us one step ahead of the risks and opportunities in our environment, and the human brain is the greatest pattern detector in nature. But this tendency can backfire and lead to fallacies when we encounter events that occur close in time and space, but are completely independent (Xue et al., 2011). Understanding randomness does not improve our predictive capabilities, because randomness is unpredictable by definition. Therefore our cognitive mechanisms may not be well-adapted for such processes.
Next, streaks have some strange properties. Imagine a streak of 5 events, say 5 heads in a row. Now imagine that this streak continues in the next outcome, so that we have a streak of 6. This streak will be counted as a streak of 6, not a streak of 5, even though the streak of 5 clearly did occur. And so did a streak of 4 and 3 and so on. But only one streak will be counted - the largest one replacing all others that build up to it. This self-overlapping property of streaks leads to undercounting and underrepresentation of streak patterns in our perception. Therefore we expect streaks to be rare and to end sooner than they actually do (Sun, & Wang, 2010).
And lastly, hot hand fallacy is not only a belief about probabilities but a real life strategy. For example, basketball coaches who believe in hot streaks among players are also likely to endorse that players experiencing a streak should be given the shot. Now, studies have found that this strategy can actually be quite successful if also coupled with a consideration of the player's overall strike rate. Individuals may learn about the effectiveness of such strategies purely based on their experiences, without fully understanding why the strategy is successful. So they may form incorrect beliefs to explain their effectiveness, such as the success of a shot being related to the success of previous shots. But as long as the incorrect belief supports the correct behavior, the belief will persist and even grow stronger (Burns, 2004; Raab et al., 2012).
DECISION RISKS
Firstly, as the name suggests, the gambler's fallacy plays a key role in compulsive gambling behavior. The expectation of frequent reversals of luck and chance not only keeps gamblers going after a streak of losses, but even leads to them taking more risks, because they believe that a victory is shortly due because of the streak. So, in an effort to recover their losses, they risk losing much more money (Xue et al., 2011). Indeed these fallacies often accompany and amplify motivated reasoning i.e. we expect that helpful streaks will continue and harmful streaks will end (Alter & Oppenheimer, 2006).
Next, we often underestimate the role of randomness and chance in our lives because we perceive interconnections and relationships between completely random independent events. This may lead to judgement errors like blaming the victims of unfortunate rare tragedies, because of the belief that they must be engaging in a lot of risky behaviors for such a rare event to happen to them. Similarly, rare and lucky successes may be completely attributed to the skills and efforts of the person, and not at all to chance, leading to overestimation of their capabilities (Oppenheimer & Monin, 2009). This is common in options trading where we judge trader's by their past performance which may have been influenced by luck (Huber et al., 2010).
Also, we see belief in hot streaks among investors and traders, despite the awareness about volatility and the role of chance in market. As a result, many investors expect streaks of high earnings to continue, leading to overreaction of stock prices, which further contributes to volatility (Suetens et al., 2016).
Managing Bias
And finally, let’s look at some strategies to manage the undesirable impacts of these fallacies on our decisions and to use the to our advantage.
Firstly, observing and evaluating information sequentially increases the salience of the most recent outcomes, resulting in us making predictions for a small sample. As small samples are less predictable and more variable, we're more prone to fallacies. Instead, by evaluating all the data together at the end of the sequence, we can avoid recency bias and improve predictions (Barron & Leider, 2010).
Second, perceptions of intentionality and control contribute to the hot hand fallacy. So focusing attention on the person who rolls the dice or tosses the coin as the agent for the outcome will amplify the fallacy, and taking attention away from them and towards the random process will diminish the hot hand fallacy (Roney & Trick, 2009).
Next, the gambler's and the hot hand fallacies stem from our tendency to mentally organize separate events into groupings of events that form meaningful patterns. By influencing how people group events together, we can moderate these fallacies. For example, if a series of coin tosses is divided into two groups at an arbitrary point, we tend to not consider streaks from the earlier group to predict outcomes in the latter group (Roney & Trick, 2003).
Next some studies indicate that men are more prone to the gambler's fallacy while women are more prone to the hot hand fallacy (Stöckl et al., 2015; Tyran, 2011).
And lastly, because hot hand fallacy is associated with affective decision-making, strategies that amplify the affective nature of information, like presenting information in emotionally laden language, are likely to amplify the fallacy as well. For example, a study found that when outcome information was presented in their native language, people were more prone to the hot hand fallacy compared to information presented in a second language. The researchers attributed this to the affective or emotional potency of native languages (Gao et al., 2015).
REFERENCES
Ayton, P., & Fischer, I. (2004). The hot hand fallacy and the gambler’s fallacy: Two faces of subjective randomness?. Memory & cognition, 32(8), 1369-1378.
Gilovich, Thomas; Tversky, A.; Vallone, R. (1985). "The Hot Hand in Basketball: On the Misperception of Random Sequences". Cognitive Psychology. 17 (3): 295–314.
Kahneman, D., & Frederick, S. (2002). Representativeness revisited: Attribute substitution in intuitive judgment. Heuristics and biases: The psychology of intuitive judgment, 49, 81.
Barron, G., & Leider, S. (2010). The role of experience in the Gambler's Fallacy. Journal of Behavioral Decision Making, 23(1), 117-129.
Sedlmeier, P., & Gigerenzer, G. (1997). Intuitions about sample size: The empirical law of large numbers. Journal of Behavioral Decision Making, 10(1), 33-51.
Gennaioli, N., & Shleifer, A. (2010). What comes to mind. The Quarterly journal of economics, 125(4), 1399-1433.
Caruso, E. M., Waytz, A., & Epley, N. (2010). The intentional mind and the hot hand: Perceiving intentions makes streaks seem likely to continue. Cognition, 116(1), 149-153.
Duthoo, W., Wühr, P. & Notebaert, W. The hot-hand fallacy in cognitive control: Repetition expectancy modulates the congruency sequence effect. Psychon Bull Rev 20, 798–805 (2013).
Xue, G., Lu, Z., Levin, I. P., & Bechara, A. (2011). An fMRI study of risk-taking following wins and losses: implications for the gambler's fallacy. Human brain mapping, 32(2), 271–281.
Alter, A. L., & Oppenheimer, D. M. (2006). From a fixation on sports to an exploration of mechanism: The past, present, and future of hot hand research. Thinking & Reasoning, 12(4), 431-444.
Xu, J., & Harvey, N. (2014). Carry on winning: The gamblers’ fallacy creates hot hand effects in online gambling. Cognition, 131(2), 173-180.
Oppenheimer, D. M., & Monin, B. (2009). The retrospective gambler's fallacy: Unlikely events, constructing the past, and multiple universes. Judgment and Decision Making, 4(5), 326.
Huber, J., Kirchler, M., & Stöckl, T. (2010). The hot hand belief and the gambler’s fallacy in investment decisions under risk. Theory and decision, 68(4), 445-462.
Suetens, S., Galbo-Jørgensen, C. B., & Tyran, J. R. (2016). Predicting lotto numbers: a natural experiment on the gambler's fallacy and the hot-hand fallacy. Journal of the European Economic Association, 14(3), 584-607.
Jessup, R. K., & O'Doherty, J. P. (2011). Human dorsal striatal activity during choice discriminates reinforcement learning behavior from the gambler's fallacy. Journal of Neuroscience, 31(17), 6296-6304.
Huang, X., Zhang, H., Chen, C., Xue, G., & He, Q. (2019). The neuroanatomical basis of the Gambler's fallacy: A univariate and multivariate morphometric study. Human brain mapping, 40(3), 967–975.
Sun, Y., & Wang, H. (2010). Gambler's fallacy, hot hand belief, and the time of patterns. Judgment and Decision Making, 5(2), 124.
Miller, J. B., & Sanjurjo, A. (2018). Surprised by the hot hand fallacy? A truth in the law of small numbers. Econometrica, 86(6), 2019-2047.
Burns, B. D. (2004). Heuristics as beliefs and as behaviors: The adaptiveness of the “hot hand”. Cognitive psychology, 48(3), 295-331.
Raab, M., Gula, B., & Gigerenzer, G. (2012). The hot hand exists in volleyball and is used for allocation decisions. Journal of Experimental Psychology: Applied, 18(1), 81.
Roney, C. J., & Trick, L. M. (2009). Sympathetic magic and perceptions of randomness: The hot hand versus the gambler's fallacy. Thinking & reasoning, 15(2), 197-210.
Roney, C. J., & Trick, L. M. (2003). Grouping and gambling: a Gestalt approach to understanding the gambler's fallacy. Canadian Journal of Experimental Psychology/Revue canadienne de psychologie expérimentale, 57(2), 69.
Tyran, J.-robert. (2011). The Gambler's Fallacy and Gender. CentER Discussion Paper No. 2011-011.
Stöckl, T., Huber, J., Kirchler, M., & Lindner, F. (2015). Hot hand and gambler's fallacy in teams: Evidence from investment experiments. Journal of Economic Behavior & Organization, 117, 327-339.
Gao, S., Zika, O., Rogers, R. D., & Thierry, G. (2015). Second language feedback abolishes the “hot hand” effect during even-probability gambling. Journal of Neuroscience, 35(15), 5983-5989.
Sundali, J., & Croson, R. (2006). Biases in Casino Betting: The Hot Hand and the Gambler’s Fallacy. Judgement and Decision Making, 1 (1), 1-12.
Gold, E., & Hester, G. (2008). The gambler's fallacy and the coin's memory. In J. I. Krueger (Ed.), Rationality and social responsibility: Essays in honor of Robyn Mason Dawes (pp. 21–46). Psychology Press.