Decision Theory

12Dynamic programming

In this chapter we begin our discussion of multistage decision problems, where multiple decisions have to be made over time and with varying degrees of information. The salient aspect of multistage decision making is that, like in a chess game, decisions made now affect the worthiness of options available in the future and sometimes also the information available when making future decisions. In statistical practice, multistage problems can be used to provide a decision-theoretic foundation to the design of experiments, in which early decisions are concerned with which data to collect, and later decisions with how to use the information obtained. Chapters 13, 14, and 15 consider multistage statistical decisions in some detail. In this chapter we will be concerned with the general principles underlying multistage decisions for expected utility maximizers.

Finite multistage problems can be represented by decision trees. A decision tree is a graphical representation that allows us to visualize a large and complex decision problem by breaking it into smaller and simpler decision problems. In this chapter, we illustrate the use of decision trees in the travel insurance example of Section 7.3 and then present a general solution approach to two-stage (Section 12.3.1) and multistage (Section 12.3.2) decision trees. Examples are provided in Sections 12.4.1 through 12.5.2. While conceptually general and powerful, the techniques we will present are not of easy implementation: ...