Chapter 1

Introduction to Decision

Theory

1.1. Preliminaries

In this chapter we provide a brief introduction to mathematical decision

theory for problems with one decision maker, commonly known as decision

theory. It lays the basis to develop a mathematical decision theory for sit-

uations in which several decision makers interact, i.e., the basis to develop

game theory. We would like to warn the reader that this is an instrumental

chapter and, therefore, it is just a concise introduction. As a consequence,

an unfamiliar reader may miss the kind of motivations, interpretations, and

examples that appear in the rest of the book.1

Binary relations are fundamental elements of decision theory. Given a

set A, every subset

R

of A

×

A is a binary relation over A. For each pair

a, b

∈

A, we denote

(a, b) ∈

R

by a

R

b and, similarly, we denote

(a, b) ∈

R

by a

R

b.

Decision theory deals with decision problems. In a decision problem there

is a decision maker who has to choose one or more alternatives out of a set

A. The decision maker has preferences over A, which are usually modeled

through a binary relation

⊂

A

×

A, referred to as preference relation in this

context. For each pair a, b

∈

A, a b is interpreted as “the decision maker

either prefers a over b or is indifferent between a and b”. Two standard

requirements are normally imposed on : i) is complete, i.e., for each pair

a, b

∈

A, a b or b a (or both) and ii) is transitive, i.e., for each triple

1We

refer the reader to Kreps (1988) and Mas-Colell et al. (1995) for deeper treatments of decision

theory.

1

http://dx.doi.org/10.1090/gsm/115/01