Dual representations of cardinal preferences when outcomes are vectors

Given a set of vector outcomes and the set of lotteries over it, we define sets of (a) von Neumann-Morgenstern representations of cardinal preferences over the lotteries, (b) mappings that yield the certainty equivalent outcomes corresponding to a lottery, (c) mappings that yield the risk premia corresponding to a lottery, (d) mappings that yield the acceptance set of lotteries corresponding to an outcome, and (e) vector-valued functions that yield generalized Arrow-Pratt coefficients corresponding to an outcome. Our main results establish bijections between these sets of mappings for very general specifications of outcome spaces, lotteries and preferences. As corollaries of these results, we derive analogous dual representations of risk averse preferences. Some applications to financial theory illustrate the potential uses of our results. Finally, we provide criteria for comparing the risk aversion of preferences in terms of the dual representations. JEL classification: C02, D01, D81, G12