Thinking and Deciding

264 DESCRIPTIVE THEORY OF CHOICE UNDER UNCERTAINTY
Figure 11.1: π, the weight applied to the utility of each outcome, as a function of p,
the probability of the outcome, according to prospect theory (based on Tversky and
Kahneman, 1992).
pay for, say, a fourth ticket if they already had three (which would only increase an
intermediate possibility of winning).
An interesting consequence of the π function is shown in the following problem
(based on Tversky and Kahneman, 1981, p. 455):
Consider the following two-stage game. In the first stage, there is a
.75 probability of ending the game without winning anything, and a .25
chance to move into the second stage. If you reach the second stage, you
have a choice between ($30) and ($45, .80). However, you must make
this choice before either stage of the game is played.
Most subjects think about this problem in the same way as they thought about the
choice of ($30) versus ($45, .80). They therefore choose ($30). These are the same
subjects who chose ($45, .20) over ($30, .25), however. Think about the two-stage
gambles for a moment, though. If you calculate the overall probability of $30, assuming
that that option is chosen, it is .25, the probability of getting to the second
stage. Likewise, the probability of $45, if that option is chosen, is (.25) (.80) (the
probability of getting to the second stage multiplied by the probability of winning if
you get there), or .20. Therefore, the probabilities of the outcomes in the two-stage
gambles are identical to the probabilities of the outcomes in the gambles ($30, .25)
and ($45, .20), yet the common pattern of choices is reversed.
Subjects who show this kind of reversal (as many do) are violating what Kahneman
and Tversky (1984) call the principle of invariance. The invariance principle