Introduction to Imprecise Probabilities

Chapter 8Decision making

Nathan Huntley¹, Robert Hable², and Matthias C. M. Troffaes¹

¹Department of Mathematical Sciences, Durham University, UK

²Department of Mathematics, University of Bayreuth, Germany

8.1 Non-sequential decision problems

In this section, we consider a single choice under uncertainty: the subject must choose one of a number of possible actions, each of which leads to an uncertain reward. The reward depends on the act and the true state of nature. How does a subject use her beliefs about the state of nature and her relative preferences over rewards to choose her act?

We use the following terminology. An act is something the subject can choose to do. A choice or decision refers to the choosing of a single act from a set. A reward is whatever the subject can receive once the state of nature is revealed.

Example 8.1

Suppose we can bet £1 that a six-sided die will score or higher. The possible acts are {bet, do not bet}. The possibility space consists of the scores shown on the die: . Three rewards are possible: losing £1, gaining £1, and receiving nothing—so rewards can also include negative results and also the status quo.

We want to employ coherent lower previsions (Section 2.2) for decision making. To do so, we need to express the decision problem in terms ...