Decision Theory

2Coherence

Decision making under uncertainty is about making choices whose consequences are not completely predictable, because events will happen in the future that will affect the consequences of actions taken now. For example, when deciding whether to play a lottery, the consequences of the decision will depend on the number drawn, which is unknown at the time when the decision is made. When deciding treatment for a patient, consequences may depend on future events, such as the patient’s response to that treatment. Political decisions may depend on whether a war will begin or end within the next month. In this chapter we discuss de Finetti’s justification, the first of its kind, for using the calculus of probability as a quantification of uncertainty in decision making.

In the lottery example, uncertainty can be captured simply by the chance of a win, thought of, at least approximately, as the long-term frequency of wins over many identical replications of the same type of draw. This definition of probability is generally referred to as frequentist. When making a prognosis for a medical patient, chances based on relative frequencies are still useful: for example, we would probably be interested in knowing the frequency of response to therapy within a population of similar patients. However, in the lottery example, we could work out properties of the relevant frequencies on the basis of plausible approximations of the physical properties of the draw, such as independence and ...