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
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