Chapter 33 Utility theory with uncertainty

Abstract

Utility theory with uncertainty describes a class of models designed to formalize the manner in which a decision maker chooses among alternative courses of action when the consequences of each course of action are not known at the time the choice is made. The distinguishing characteristic of the subject matter is that each course of action results in one of several consequences. The chapter discusses that the problem is conveniently formalized with the use of the notions of consequences, states of nature, and acts. The first primitive of the theory is a nonempty set of consequences, denoted by C. The empirical counterpart of a consequence is anything that has to do with the welfare of the decision maker. The second primitive of the theory is a nonempty set of feasible acts, denoted by A 0 , whose elements are courses of action from which the decision maker may choose. Decision problems in which the set of states is a singleton, or in which all acts are constant acts, and the set of consequences consists of probability measures or lotteries on a set of outcomes are referred to as “decisions under risk;” if the set of acts includes nonconstant acts they are decisions under uncertainty. Given the primitives of the theory, a complete and transitive binary relation on the set of consequences is a natural ingredient necessary to guide the decision maker's choices among acts. In the theory of choice under certainty, there is a one-to-one correspondence between acts and consequences.