Gambler's fallacy and Law of Small Numbers
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Gambler’s Fallacy and The Law of Small Numbers
Analytical Summarization of Matthew Rabin’s paper: Inference by Believers in the Law of Small Numbers
The Law of Small Numbers
- Tversky and Kahneman started to explore the psychology behind false assumptions made during economic situations of risk and uncertainty in 1971. What they dubbed as, The Law of Small Numbers, explains how financial analysts can misperceive a small sample to be indicative of the entire population.
- Take for example, a study used by Tversky and Kahneman: they told a sample of university students that “the mean IQ of the population of eighth graders in a city is known to be 100”. After telling the sample group of students this, they went on to say they selected a random sample of 50 eighth graders from the population. The first child who they tested scored tremendously with an IQ of 150. The question asked of the students was “What do you expect the mean IQ to be for the whole sample?
Rational Answer
- It is impossible to say for sure what the mean average would be for the students. If the sample population was chosen randomly, then, through rational, we must assume it is indicative of the population as a whole. However, since one student scored 150, our mystery sample is now only 49 students. Assuming these students have a mean score of 100, then once we factor back in the student who scored 150 we get:
4900 is the total points scored on the IQ tests by the 49 sample students (under our assumptions)
4900+150 (the score of student #50) = 5050
5050/50 = 101: the average IQ of the sampleResults from the Experiment
Conclusion
Tversky and Kahneman realized that those people who thought the samples IQ would be 100 had put to much faith in “The Law of Small Numbers”. They still believed the sample would be an exact replication of the population, even after they knew one of the fifty was 50 points higher than average. These people figured that the other 49 must somehow make up for the genius of student #50 by collectively being 50 pts. below the average. Those who answered with an average of 101 used a Bayesian school of thinking, a theory developed on the basis of rational probability.
Gambler’s Fallacy
- The Gambler’s Fallacy is closely related to the Law of Small Numbers. It shows another situation where humans ignore rational probability. In the Gambler’s Fallacy, people expect that the second outcome of a given situation must be negatively correlated to the first. Follow the example below:
- Assume an investor is certain that a financial analyst invests successfully have of the time. If in year 1 the analyst is successful than, using the gambler’s fallacy, the investor assumes he will not be successful in year two. Rationally, this does not make sense because year 2 is independent of year 1. If the analyst is truly successful 50% of the time, then he has a 50% chance of being successful in year two. Someone using the gambler’s fallacy will underestimate these odds.
- This is an important concept to understand when evaluating financial analysts. If an investor is thinking under the gambler’s fallacy then they would shy away from using an analyst who was successful the previous year. Consequently, they would be more likely to put faith in an unsuccessful analyst from the year before.
- This person using the gambler’s fallacy is also a believer in small numbers. So following their logic, two years of investing would be enough to judge whether an analyst was good, average, or bad. Having two straight successful years would make you good, one good and one bad would be average, and two bad years would make someone a bad analyst. However, rational probability shows that this is too extreme of an assumption. In the long run, the numbers of successful years will get closer to 50% for the analysts who were 2 for 2 (100% successful) and 0 for 2 (0% successful).
