We use natural experiments embedded in state-run lotteries and a new nationally representative survey to provide reduced-form and structural estimates of risk preferences and behavioural biases in lottery demand, and to explore the implications for optimal lottery design. We find that sales respond more to the expected value of the jackpot than to price but are unresponsive to variation in the second prize—a pattern that is consistent with probability weighting but is inconsistent with standard parameterizations. In the survey, we find that lottery spending decreases modestly with income and is strongly associated with measures of innumeracy, poor statistical reasoning, and other proxies for behavioural bias, which also decline with income. Regression predictions suggest that Americans would spend 43% less on lotteries if they were unbiased, while the remaining lottery demand is due to other factors such as anticipatory utility or entertainment value. We use these empirical moments to estimate a model of socially optimal lottery design. In the model, current multi-state lottery designs increase welfare but may harm heavy spenders.

We use public finance sufficient statistic approaches to characterize the welfare effects of “nudges,” such as simplified information and warning labels, in markets with taxes and endogenous prices. While many studies focus on average effects, we show that welfare also depends on how the nudge affects the variance of choice distortions, and average effects become irrelevant with zero pass-through or optimal taxes. We implement the framework with experiments evaluating automotive fuel economy labels and sugary drink health labels. Labels decrease purchases of low-fuel economy cars and sugary drinks but may decrease welfare because they increase the variance of choice distortions.