Comparative Riskiness of Random Vector Outcomes

Abstract

One lottery over a vector outcome space is said to be riskier than another if every risk averse decision-maker prefers the latter to the former. We consider two other criteria for making such comparisons, one of which is a generalization of second-order stochastic dominance. Our main result is that all these criteria are equivalent for every convex vector outcome space. If the outcome space is also given appropriate topological structure, then lotteries with comparable riskiness have identical weak means; if it is also suitably ordered, then lotteries with comparable riskiness also have comparable sets of risk premia. These results bring a host of new applications within the ambit of the theory of comparative riskiness, most notably models featuring risk embodied in random processes. We illustrate this usefulness with applications to the theories of auctions, utility regulation, inventory control, portfolio choice, public goods and moral hazard in teams.