Thinking and Deciding

PROSPECT THEORY 265
asserts that one’s choices ought to depend on the situation itself, not on the way it is
described. In other words, when we can recognize two descriptions of a situation as
equivalent, we ought to make the same choices for both descriptions. Subjects seem
to violate this principle. 3 The invariance principle would seem to be a principle of
rational choice that is at least as fundamental as other principles we have assumed as
part of utility theory, such as transitivity and the sure-thing principle. Violations of
the invariance principle are also called framing effects, because the choice made is
dependent on how the situation is presented, or “framed.”
Note that π(p), unlike p itself, is not additive. In general, π(p)+π(1 − p) < 1.
Therefore, we cannot assume that the 1.00 probability of $1,000 in the Allais paradox
can be psychologically decomposed (as in Table 11.2) into .01 + .10 + .89. At least
part of the bias shown in the Allais paradox is caused by the certainty effect operating
on Option 1. 4
Is the certainty effect rational? Why should we not weigh certain (sure) outcomes
more than uncertain ones? One reason why not is that it leads us to more inconsistent
decisions, decisions that differ as a function of the way things are described to us (or
the way we describe things to ourselves). Second, our feeling of “certainty” about
an outcome is often, if not always, an illusion, or, to put it more precisely, another
sort of artifact of the way things are described (as we noted in Chapter 16). For
example, you may think of ($30) as a certain outcome: You get $30. Unless having
money is your only goal in life, though, the $30 is really just a means to other ends.
You might spend it on tickets to a football game, for example, and the game might
be close, and so exciting that you tell your grandchildren about it — or it might be
a terrible game, with the rain pouring down, and you, without an umbrella, having
to watch your team get slaughtered. You might use the money to buy a book that
enlightens you more than a year of college — or a book that turns out to be a lot of
trash. In short, most, if not all, “certain” outcomes can be analyzed further, and in
doing so one finds, on close examination, that the outcomes are themselves gambles.
The description of an outcome as certain is not certainty itself.
An important consequence of the certainty effect (McCord and de Neufville,
1985) is the conclusion that people do not conform to the assumptions underlying
the method of gambles when this method is used to measure utility. When people
say that they are “indifferent” between ($5) and ($20, .5), we cannot assume that their
utility for $5 is literally halfway between that of $0 and that of $20. They underweigh
3 The different patterns of choices in the Allais paradox, depending on whether the situation is presented
in a table or not, are another example of violation of the invariance principle.
4 Quiggin (1982), Segal (1984), and Yaari (1985) have shown that the major results ascribed to the
π function, including the certainty effect, can be accounted for by other transformations of p. These
transformations avoid the following problem: According to prospect theory taken literally, a person might
prefer ($5.01, .05; $5.02, .05) to ($5.03, .10), even though we might think of ($5.03, .10) as ($5.03, .05;
$5.03, .05), which is clearly better than the first option. This preference is possible if π(.05) is sufficiently
large compared to π(.10)/2. It is called a violation of stochastic dominance. The way to avoid violations
of stochastic dominance is to rank the outcomes in order of preference and calculate, for each outcome,
the probability Q of doing at least as well as that outcome. These Qs can be transformed freely, as long
as Q is 1 for the worse outcome. Reviews of other recent developments of this sort are found in Fishburn
(1986), Machina (1987), Sugden (1986), and Weber and Camerer (1987).