Thinking and Deciding

PROSPECT THEORY 263
Probability: The pi function
Pi and the certainty effect. In essence, prospect theory begins with the premise
that we do not treat the probabilities as they are stated. Instead, we distort them,
according to a particular mathematical function that Kahneman and Tversky named
the “pi function,” using the Greek letter π instead of the usual p for probability.
Instead of multiplying our utilities by p, the researchers proposed, people multiply
by π(p). 1 The function is graphed in Figure 11.1. For example, in a gamble some
people prefer ($30) to ($45, .80), but they prefer ($45, .20) to ($30, .25). Here
($30) means $30 for sure; ($45, .80) means $45 with probability .80 and $0 with
probability .20, and so forth. If you prefer ($30) to ($45, .80), this would imply —
using the expected-utility formula — that
u($30) >.80 u($45)
If you prefer ($45, .20) to ($30, .25), this implies that
.25 u($30)