Small samples and evolution: did the law of small numbers arise as an adaptation to environmental challenges?

Abstract

In the context of casino gambling, only a minority (~15%) of players presented with a streak of at least length 6 in roulette disregard recent events in deciding their next move, which is the normatively optimal approach to such a decision (Croson and Sundali, 2005). The majority of people would instead subscribe to a belief in a recency effect. This intriguing pattern of reasoning is categorized as either the gambler's fallacy, when the subject perceives negative recency (GF; Laplace, 1951; Tune, 1964; Tversky and Kahneman, 1971), or as the hot hand fallacy, when positive recency is perceived (HH; Gilovich et al., 1985). Such tendencies demonstrate, among a variety of things, that magical thinking is not exclusive to astrologists and tarot fanatics. Both the GF and HH refer to instances of the subject projecting a relationship between prior and present events, albeit in opposing directions. For example, subsequent to observing a run of 6 “heads,” a subject committing the GF would expect “tails” on the next coin toss. Alternatively, a subject committing the HH, following a similar streak of, say, successful basketball throws, would expect another “hit” on the next throw. Both fallacies have been posited as consequences of our immanent adherence to the law of small numbers—a distorted conception of chance, according to which short random sequences are considered highly representative of their underlying generating process (Tversky and Kahneman, 1971; Gilovich et al., 1985); But, counterintuitively, when dealing with sequences governed by chance, the short sub-sequences that we mistake as essentially representative of the overall generating process, actually deviate systematically from sequential properties on the global level; such small sub-sequences, on the basis of which we draw predictive inferences, are rather misrepresentative, containing excessive alternations and lacking sufficient long runs (Gilovich et al., 1985).