Adam Smith on lotteries: an interpretation and formal restatement

The few pages that Adam Smith devoted to lotteries, mainly in the Wealth of Nations (1776) did not receive much attention. They nonetheless constituted an opportunity to introduce a sophisticated analysis of individual decision under risk. Through various examples, Smith pointed out a risk-seeking attitude, figured out in the paper in terms of inverse stochastic dominance. However, it is well-known that a contradiction occurs between such an attitude and the principle of an asymmetric sensitivity to favorable and unfavorable events, expressed by a concave function, introduced in the Theory of Moral Sentiments (1759). We argue that an appropriate solution to this difficulty should rest on Smith's emphasis on a universal tendency to overestimate the chance of gain, which leads to favor a rank-dependent utility approach within which optimism toward risk can compensate asymmetric sensibility in order to produce some kind of risk-seeking. The question raised by the coexistence of various attitudes toward risk illustrated by the figures of the entrepreneur (typically, the "projector" and the "sober man") gives rise to an extensive analysis, which aims at explaining, on moral grounds, how an initial attitude of risk-seeking can generate prudence before being transformed into risk-aversion.