A rational approach to risk? Bayesian decision theory

Abstract

The aim of this chapter is to provide an overview over decision theory, in particular Bayesian decision theory, to explain its main ideas with some emphasis on risk and to flag the most important controversies about the theory. The paper starts out with the St. Petersburg paradox and motivates the idea that rational agents maximize expected utility. Since '‘rationality’’ is ambiguous, the attention is restricted to a core notion of rationality that evaluates an agent’s choices in relation to her desires and beliefs. Accordingly, in Bayesian decision theory, the expected utility arises from utilities and probabilities that measure the strengths of the agent’s desires and beliefs, respectively. This raises two questions: (1) Why do rational agents maximize expected utility? (2) How can the strengths of the agent’s desires and beliefs be measured? Both questions are commonly answered in terms of representation theorems. Such theorems show that the strengths of the agent’s beliefs and desires can be represented in terms of numerical probabilities and utilities if the choices she would take in hypothetical cases fulfill certain conditions. The theorems further entail that an option is preferred to another if and only if it has a higher expected utility than the latter. This chapter covers results by von Neumann-Morgenstern, Ramsey, Savage, and Bolker-Jeffrey. It discusses the consequences for choices that are risky in a nontechnical sense and concludes by pointing out major controversies concerning Bayesian decision theory.