Thinking and Deciding

BIAS IN DECISIONS UNDER UNCERTAINTY 261
Table 11.2: The Allais paradox as a lottery
Ball numbers
1 2–11 12–100
Situation X
Option 1 $1,000 $1,000 $1,000
Option 2
Situation Y
$0 $5,000 $1,000
Option 3 $1,000 $1,000 $0
Option 4 $0 $5,000 $0
Table 11.3: Expected utilities of options in the Allais paradox
Situation X
Option 1 .01 u($1, 000) + .10 u($1, 000) + .89 u($1, 000)
Option 2
Situation Y
.01 u($0) + .10 u($5, 000) + .89 u($1, 000)
Option 3 .01 u($1, 000) + .10 u($1, 000) + .89 u($0)
Option 4 .01 u($0) + .10 u($5, 000) + .89 u($0)
It is apparent from Table 11.2 that Options 1 and 2 are identical to Options 3 and
4, except for the outcome for balls 12 through 100. Moreover, the outcome in the
rightmost column does not depend on the choice made in each situation. In Situation
X, whether you choose Option 1 or Option 2, you would get $1,000 if the ball drawn
is numbered between 12 and 100. Situation Y is the same, except that the common
outcome for the two choices is $0.
By the sure-thing principle, described in Chapter 10, we should ignore these
common outcomes in making such a decision. We should choose Options 1 and 3, or
Options 2 and 4. When choices are presented in this tabular form, it becomes easier
for us to see what is at issue. Many subjects do in fact change their choices to make
them consistent with the expected-utility axioms when they are shown in the table
(Keller, 1985). Other subjects stick to their choices of Options 1 and 4, even after
being shown the table (Slovic and Tversky, 1974). As the economist Paul Samuelson
put it (1950, pp. 169–170), they “satisfy their preferences and let the axioms satisfy
themselves.”
Because the choice of Options 1 and 4 violates the sure-thing principle, we cannot
account for these choices in terms of expected-utility theory. The expected utility
of each choice (using data from Table 11.1) is calculated in Table 11.3. If we prefer
Option 1 to Option 2 and if we did follow expected-utility theory, then the expected