Time Lotteries and Stochastic Impatience

We study preferences over lotteries in which both the prize and the payment date are uncertain. In particular, a time lottery is one in which the prize is fixed but the date is random. With Expected Discounted Utility, individuals must be risk seeking over time lotteries (RSTL). In an incentivized experiment, however, we find that almost all subjects violate this property. Our main contributions are theoretical. We first show that within a very broad class of models, which includes many forms of nonexpected utility and time discounting, it is impossible to accommodate even a single violation of RSTL without also violating a property we termed Stochastic Impatience, a risky counterpart of standard Impatience. We then present two positive results. If one wishes to maintain Stochastic Impatience, violations of RSTL can be accommodated by keeping Independence within periods while relaxing it across periods. If, instead, one is willing to forego Stochastic Impatience, violations of RSTL can be accommodated with a simple generalization of Expected Discounted Utility, obtained by imposing only the behavioral postulates of Discounted Utility and Expected Utility.

Probability in Ethics

It is natural to think that the most basic questions in ethical theory do not have much to do with probability. Given answers to these questions, we can try to extend them to cases involving probability, though this job might best be handled by more technical disciplines. This chapter is an argument for the opposite view. The major ethical problems to do with probability involve very little mathematics; many topics which seem to have nothing to do with probability are arguably all about probability; and thinking about various problems to do with probability can help solve analogous problems which do not involve probability, sometimes even revealing that popular positions about such problems are incoherent. Among the topics discussed are: interpretations of probability; expected utility theory; utilitarianism; egalitarianism; fairness; the priority view; population size; incommensurability; continuity; nonexpected utility theory; evaluative measurement; decision theory; act consequentialism; rule consequentialism; contractualism; and deontology.