Thinking and Deciding

244 NORMATIVE THEORY OF CHOICE UNDER UNCERTAINTY
greater extent, in the long run, than any other rule. Although this argument turns out
to be weak, it may help to explain the immediate appeal of expected-utility theory.
To see this, consider the analogy between expected value (formula 1), and expected
utility (formula 2). In both cases, we are trying to maximize something. In
one case, it is wealth; in the other, it is utility. In both cases, we will do better in the
long run by making all of our decisions in agreement with the formula than by any
other method at our disposal. (Remember, we cannot see into the future.)
If, for example, I have a choice between $4.00 if a heart is drawn and $1.00 if any
red card (heart or diamond) is drawn, the expected value of the first option ($1.00) is
higher than that of the second ($0.50). In the long run — if I am offered this choice
over and over — I am bound to do better by taking the first option every time than by
any other policy. I will win 25% of the times when I choose the first option, so I will
average $1.00 each time I choose it. I will win 50% of the times when I choose the
second option, but my average winning will be only $0.50. Any way of playing that
tells me to choose the second option on some plays of the game will lead to a lower
total payoff on those plays. In a sufficiently large number of plays, I will do best to
choose the first option every time.
The same reasoning can be applied if I am faced with many different kinds of
decisions. Even if the amount to be won and the probability of winning change from
decision to decision, my total wealth will be highest at the end if I choose the larger
expected value every time.
Similarly, if utility measures the extent of my goal achievement, if I can add up
utilities from different decisions just as I can add up my monetary winnings from
gambling, and if I maximize expected utility for every decision, then, over the long
run, I will achieve my goals more fully than I could by following any other policy.
The same reasoning applies when we consider decisions made by many people.
If a great many people make decisions in a way that maximizes the expected utility
of each decision, then all these people together will achieve their goals more fully
than they will with any other policy. This extension of the argument to many people
becomes relevant when we reflect on the fact that sometimes the “long run” is not so
long for an individual. Decisions like Pascal’s Wager are not repeated many times
in a lifetime. Even for once-in-a-lifetime decisions, the expected-utility model maximizes
everyone’s achievement of his goals, to the extent to which everyone follows
the model. Moreover, if we give advice to many people (as I am implicitly doing as
I write this), the best advice is to maximize expected utility, because that will lead to
the best outcomes for the group of advice recipients as a whole.
Can we add utilities from the outcomes of different decisions, just as we add
together monetary winnings? Three points need to be made. First, this idea requires
that we accept a loss in utility at one time — or for one person — for the sake of a
greater gain at another time — or for another person. If I am offered the game I have
described, in which I have a .25 chance of winning $4.00, I ought to be willing to
pay some money to play it. If I have to pay $0.50 each time I play, I will still come
out ahead, even though I will lose the $0.50 on three out of every four plays.