The cross-sectional “Gambler's Fallacy”: Set representativeness in lottery number choices

• We study the cross-sectional consequences of Representative Bias using lottery number choices.
• Number sets highly representative of the underlying uniform distribution are over-selected.
• This incurs an expected cost to gamblers under the pari-mutuel lottery structure.

Traditionally, the Gambler's Fallacy is described as the belief that a sequence of independent outcomes over time should exhibit short-run reversals. The underlying psychological bias thought to drive this fallacy is Representativeness Bias: the idea that even a small sample of outcomes should closely reflect the theoretical probability distribution (Tversky and Kahneman, 1971). Yet representativeness also has less commonly explored consequences in the cross-sectional dimension. We find strong evidence for this in lottery play where probabilities are well-defined and transparent, using a dataset of over 1.6 million lottery tickets purchased by over 28,000 players. Specifically, individuals prefer number combinations that are cross-sectionally representative of the uniform distribution from which they are drawn. We test two possible approaches to implementing representativeness; a heuristic 3-bin approach which is promoted in some gambling advice literature, and a direct optimization approach in which gamblers try to spread the numbers in the chosen set as evenly as possible across the lottery number range. By both measures, gamblers over-gravitated to highly representative lottery number sets and over-avoided less representative sets, compared to the proportions that the true lottery odds would suggest. In this pari-mutuel lottery setting, a cost is incurred by gamblers with this type of bias, by reducing their expected winnings.