Modelling the probability distribution of prize winnings in the UK National Lottery: consequences of conscious selection

Summary.  In the statistical and economics literature on lotteries, the problem of designing attractive games has been studied by using models in which sales are a function of the structure of prizes. Recently the prize structure has been proxied by using the moments of the prize distribution. Such modelling is a vital input into the process of designing appealing new lottery games that can generate large revenues for good causes. We show how conscious selection, the process by which lottery players choose numbers non-randomly, complicates the multivariate distribution of prize winners by introducing massive overdispersion of numbers of winners, and large correlations between the numbers of different types of prize winner. Although it is possible intuitively to reach a qualitative understanding of the data, an a priori model does not fit well. We therefore construct an empirical model of the joint distribution of prize winners and use it to calculate the moments of ticket value as a function of sales. The new model gives much higher estimates of ticket value moments, particularly skewness, than previously obtained. Our results will have consequences for policy decisions regarding game design. A spin-off result is that, on the basis of the results of model fitting, lottery players may increase the expected value of their ticket by strategically choosing numbers which are less popular with other lottery players.