Thinking and Deciding

238 NORMATIVE THEORY OF CHOICE UNDER UNCERTAINTY
Expected value
When a simple gamble involves money, the expected value of the gamble can easily
be computed mathematically by multiplying the probability of winning by the monetary
value of the payoff. For example, suppose I offer to draw a card from a shuffled
deck of playing cards and pay you $4 if it is a heart. The probability of drawing a
heart is .25, so the expected value of this offer is .25 · $4, or $1. If we played this
game many many times (shuffling the cards each time), you would, on the average,
win $1 per play. You can therefore “expect” to win $1 on any given play, and, on the
average, your expectation will be correct. To calculate the expected value of a more
complex gamble, with many possible outcomes, simply multiply the probability of
each outcome by the value of that outcome, and then sum across all the outcomes.
For example, if I offer you $4 for a heart, $2 for a diamond, and $1 for anything else,
the expected value is .25 · $4 + .25 · $2 + .50 · $1, or $2. This is the average amount
you would win over many plays of this game. Formally,
EV =
(10.1)
i
pi · vi
where EV stands for expected value; i stands for all of the different outcomes; pi
is the probability of the “ith” outcome; vi is the value of the ith outcome. pi · vi is
therefore the product of the probability and value of the ith outcome; and
i pi · vi
is the total of all of these products.
The use of expected value as a way of deciding about money gambles seems
reasonable. If you want to choose between two gambles, it would make sense to take
the one with the higher expected value, especially if the gamble you choose will be
played over and over, so that your average winning will come close to the expected
value itself. This rule has been known by gamblers for centuries. (Later, we shall see
why this might not always be such a good rule to follow.)
Expected utility
The same method can be used for computing expected utility rather than expected
monetary value. The philosopher and mathematician Blaise Pascal (1623–62) made
what many regard as the first decision analysis based on utility as part of an argument
for living the Christian life (Pascal, 1670/1941, sec. 233). His famous argument is
known as “Pascal’s wager” (Hacking, 1975). The question of whether God exists is
an ancient one in philosophy. Pascal asked whether, in view of the difficulty of proving
the existence of God by philosophical argument, it was worthwhile for people to
live a Christian life — as though they were believers — in the hope of attaining eternal
life (and of becoming a believer in the process of living that life). In answering
this question Pascal argued, in essence, that the Christian God either does exist or
does not. If God exists, and if you live the Christian life, you will be saved — which
has nearly infinite utility to you. If God exists, and if you do not live the Christian
life, you will be damned — an event whose negative utility is also large. If God does