Introduction to Imprecise Probabilities

Chapter 1Desirability

Erik Quaeghebeur

SYSTeMS Research Group, Ghent University, Belgium

1.1 Introduction

There are many ways to model uncertainty. The most widely used type of model in the literature is a function that maps something we are uncertain about to a value that expresses what we know or believe to know about it. Examples are probabilities, which may specify a degree of belief that an event will occur, and previsions, which specify acceptable prices for gambles (cf. Section 1.6 and Chapter 2).

In this chapter, we show that other types of models that are conceptually and intuitively attractive can be built and used as well. The focus lies on the notion of desirability and the theory of sets of desirable gambles. Next to introducing its concepts and structure, we also use it as a nexus for clarifying the relationships between many of the equivalent or almost equivalent models for uncertainty appearing in the imprecise-probability literature: partial preference orders, credal sets, and lower previsions.

We formulate desirability in the context of an abstract betting framework:

We—short for ‘an intentional system’—are uncertain about the outcome of an experiment. A possibility space for the experiment is a finite or infinite set of elementary events— i.e., mutually exclusive outcomes—that is exhaustive in the sense that other outcomes are deemed practically or pragmatically impossible. A bounded real-valued function on a possibility space is called a gamble and interpreted ...