Given the many applications of expected utility for decision under risk, we dedicate a separate chapter to this topic. Throughout this chapter we make the following assumption, often without further mention. It implies Structural Assumption 2.5.2 (decision under risk and richness), adding the assumption of EU.

Structural Assumption 3.0.1 [Decision under risk and EU]. ≽ is a preference relation over the set of all (probability-contingent) prospects, which is the set of all finite probability distributions over the outcome set ℝ. Expected utility holds with a utility function U that is continuous and strictly increasing. □

The assumption that all finite probability distributions are available in the preference domain entails, in fact, a strong richness restriction, similar to our assumption that all real-valued outcomes are available in the domain. The assumption is, however, commonly made in the literature on decision under risk and it facilitates the analysis, which is why we use it too.

An application from the health domain: decision tree analysis

My experience with applications of decision theory mostly come from the medical domain. Although discussions of medical examples can at times be depressing, dealing with human suffering, the medical domain is one of the most important fields of application for decision theory. Hence, I present a medical application. We consider a simplified decision analysis for patients with laryngeal cancer in stage T3 (a particular medical state of the cancer with no metastases; McNeil et al.1981). In this subsection, as an exception, outcomes are nonmonetary.