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I study endogenous learning dynamics for people expecting systematic reversals from random sequences - the gambler's fallacy. Biased agents face an optimal-stopping problem. They are uncertain about the underlying distribution and learn its parameters from predecessors. Agents stop when early draws are good enough, so predecessors' experience contain negative streaks but not positive streaks. Since biased agents understate the likelihood of consecutive below-average draws, society converges to over-pessimistic beliefs about the distribution's mean and stops too early. Agents uncertain about the distribution's variance overestimate it to an extent that depends on predecessors' stopping thresholds. Subsidizing search partially mitigates long-run belief distortions.
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