Thinking and Deciding

260 DESCRIPTIVE THEORY OF CHOICE UNDER UNCERTAINTY
Table 11.1: The Allais paradox. Each option is a gamble described in terms of the
possible outcomes and the probability of each.
Situation X
Option 1 $1,000, 1.00
Option 2 $1,000, .89
$5,000, .10
Situation Y
$0, .01
Option 3 $1,000, .11
$0, .89
Option 4 $5,000, .10
$0, .90
when investigators have used small amounts of real money in place of hypothetical
outcomes.
The Allais paradox
Allais (1953) proposed the following hypothetical decision: Suppose you were offered
the choices (between different amounts of money) given in Table 11.1. You are
to make one choice in Situation X and one in Situation Y. Notice that the table also
gives the probability of each outcome.
Most people are inclined to choose Option 1 in Situation X and Option 4 in
Situation Y. In Situation X, they are not willing to give up the certainty of winning
$1,000 in Option 1 for the chance of winning $5,000 in Option 2: This extra possible
gain would expose them to the risk of winning nothing at all. (If you do not happen
to feel this way, try replacing the $5,000 with a lower figure, until you do. Then
use that figure in Option 4 as well.) In Situation Y, they reason that the difference
between the two probabilities of winning is small, so they are willing to try for the
larger amount.
Now suppose (as suggested by Savage, 1954) that the outcomes for these same
choices are to be determined by a lottery. Balls numbered from 1 to 100 are put into
an urn, which is shaken well before any ball is drawn. Then a ball will be taken out,
and the number on the ball will, together with your choice, determine the outcome,
as shown in Table 11.2. The dollar entries in the table represent the outcomes for
different balls that might be drawn. For example, in Option 2, you would get nothing
if ball 1 is drawn, $5,000 if ball 2, 3, 4, ...or 11 is drawn, and so forth. This situation
yields the same probabilities of each outcome, for each choice, as the original
gambles presented in Table 11.1. For example, in Option 1, the probability of $1,000
is .01 + .10 + .89, which is 1.00.