Thinking and Deciding

WHY EXPECTED-UTILITY THEORY IS NORMATIVE 249
Many writers regard the last paragraph as the most important point. They think
of utility theory as a set of conditions for constructing a utility function. A utility
function is an assignment of numbers to outcomes; each number is the utility of its
respective outcome. If the conditions are met for a set of choices, then the numbers
can be assigned, and utility is this assignment. By this view, utility is something
we infer from choices. (The choices may be real or hypothetical.) I have taken a
somewhat different view. My view has been that utility is, in a sense, already there.
It is the amount of good that the outcome does, according to all the relevant goals.
By this view, utility theory is a way of inferring choices from utilities (and from
probabilities). My view leaves us with the problem of how to discover the utilities.
But we have that problem anyway, because, in fact, people do not follow the theory.
So we cannot use people’s choices to discover their utilities. The next chapter will
explain why people do not follow expected-utility theory, and Chapter 13 will explain
how we might measure utility anyway.
Note that following expected-utility theory means following probability theory
as well. The probabilities we assign to the states must be additive and must add
up to one (because the states are assumed to be mutually exclusive and exhaustive).
The arguments for expected-utility theory therefore provide additional support for
probability as a normative model of belief, as I discussed in Chapter 5.
An alternative principle: Tradeoff consistency
We can replace the sure-thing principle with another principle, which implies the
expected-utility formula with no other principles except weak ordering, tradeoff consistency
(Köbberling and Wakker, 2003; Wakker, 1989). A slightly simplified version
of the idea concerns two states of the world, A and B and two options at a
time, such as the following four choices, in which the numbers represent amounts of
money.
State
Option A B
U 200 100
V 310 0
State
Option A B
W 400 100
X 540 0
State
Option A B
U ′ 200 205
V ′ 310 100
State
Option A B
W ′ 400 205
X ′ 540 100
Suppose you are indifferent between U and V , and you are indifferent between
W and X. And suppose you make your decisions in terms of goal achievement.
Indifference between U and V (upper left table) implies that the difference between