such decisions, one has to consider two factors, the desirability of the potential
outcomes and their probability of occurrence. Indeed, decision theory is con-
cerned with the question of how these factors are, or should be, combined.
Consider a choice between a risky prospect that offers a 50 percent chance
to win $200 (and a 50 percent chance to win nothing) and the alternative
of receiving $100 for sure. Most people prefer the sure gain over the gamble,
although the two prospects have the same expected value. The expected value
of a gamble is a weighted average where each possible outcome is weighted
by its probability of occurrence. The expected value of the gamble above is
.50$200þ.50 0 ¼$100. A preference for a sure outcome over a risky pros-
pect that has higher or equal expected value is calledrisk averse; a preference
for a risky prospect over a sure outcome that has higher or equal expected
value is calledrisk seeking.
As illustrated above, people tend to be risk averse when choosing between
prospects with positive outcomes. This tendency toward risk aversion can be
explained by appealing to the notion of diminishing sensitivity. Just as the im-
pact of a candle is greater when it is brought into a dark room than into a room
that is well lit, so the impact of an additional $100 is greater when it is added to
a gain of $100 than when it is added to a gain of $800. This principle was first
formalized by Daniel Bernoulli and Gabriel Cramer, who proposed early in the
eighteenth century that subjective value, or utility, is a concave function of
money, as illustrated in figure 26.1. (A function is concave if a line joining any
two points on the curve lies entirely below the curve.) Notice that according to
such a function the utility difference,u($200)u($100), is greater than the util-
ity difference,u($900)u($800), though the dollar differences are the same.
Bernoulli and Cramer proposed that a person has a concave utility function
that captures his or her subjective value for money, and that preferences should
be described using expected utility instead of expected value. According to
expected utility, the worth of a gamble offering a 50 percent chance to win $200
(and a 50 percent chance to win nothing) is .50u($200), whereuis the per-
son’s utility function. (Assume thatuð 0 Þ¼0.) As can be seen from figure 26.2, it
follows from such a function that the subjective value attached to a gain of $100
Figure 26.1
For gains, subjective value, or utility, is a concave function of money. A gain (or loss) of $100, for
example, has a different subjective value depending on whether you have $100 or $800 to begin
with.
602 Eldar Shafir and Amos Tversky