Thinking and Deciding

250 NORMATIVE THEORY OF CHOICE UNDER UNCERTAINTY
200 and 310 in state A just offsets the difference between 100 and 0 in state B,
in terms of achieving your goals. This takes into account both your utility for the
money and your personal probability for states A and B. Likewise, indifference
between W and X (lower left) implies that the difference between 400 and 540 in
state A just offsets the difference between 100 and 0 in state B. If, as assumed,
you are basing these judgments on goal achievement, then we can conclude that the
difference between 200 and 310 matters just as much as the difference between 400
and 540 in terms of your goals, because they both offset the same thing. We are, in
essence, using the 100–0 difference in B as a measuring stick to mark off units in
state A.
Now suppose that you are indifferent between U ′ and V ′ (upper right). This
means that the difference between 205 and 100 in state B just offsets the difference
between 200 and 310 in state A. So it must also offset the difference between 400
and 540 in state A (lower right). You must be indifferent here too. If you are not
indifferent, then we cannot assign utilities to outcomes consistently. If you are, it
turns out, then we can assign probabilities to states and utilities to outcomes in a way
that is consistent with expected-utility theory.
Note that we are also assuming that the idea of differences in goal achievement is
meaningful. But it must be meaningful if we are to make such choices at all in terms
of goal achievement. For example, if states A and B are equally likely, then any
choice between U and V must depend on which difference is larger, the difference
between the outcomes in A (which favor option V ) or the difference between the
outcomes in B (which favor U). It makes sense to say that the difference between
200 to 310 has as much of an effect on the achievement of your goals as the difference
between 0 and 100.
The utility of money
Let us apply expected-utility theory to gambles that involve money. In analyzing
money gambles, we may think that we ought to choose on the basis of expected
(monetary) value, for we may assume that the utility of money is the same as its
monetary value. This, as we shall see, is not true, since it neglects an important
factor; but let us assume, for the moment, that the expected utility of a gamble is
simply its expected value.
If I offer you a chance to win $4 if a coin comes up heads on two out of two
tosses, the expected value of this gamble is $1 (since there is a .25 probability that
both tosses will be heads, and .25 · $4 is $1). Therefore, you ought to have no
preference between $1 and this gamble. Taking this argument one step farther, you
even ought to be willing to pay me any amount less than $1 for a chance to play the
gamble. If you have a ticket allowing you to play the gamble, you ought to be willing
to sell it for any amount over $1.
Most people would not pay anything close to $1 to play this game, and many
would sell it for less than $1, given the chance. It does not seem to be the case that