Thinking and Deciding

246 NORMATIVE THEORY OF CHOICE UNDER UNCERTAINTY
than Y for our achieving our goals. The tricky part of the theory is that X or Y can
consist of two or outcomes that happen in different states of the world. In some
states, the outcome of Y might be better than X. But we want to compare X and Y in
terms of their overall betterness.
The weak ordering principle simply asserts that we can do this, that it make sense,
even when X is a mixture of different things. It is an idealization. Sometimes we
feel we cannot do this in real life, but that doesn’t matter. This is the framework that
we impose. The principle of weak ordering has two parts. First, our choices must
be connected: For any two choices X and Y, we must either prefer X to Y, Y to X,
or we must be indifferent between them. In terms of betterness, either one is better
than the other or they are the same. We are not allowed to say that they cannot be
compared, although in real life we may sometimes feel that way. The idea that two
things can always be compared in betterness is part of the idealization imposed by
the normative framework, just like the idea of addition that we discussed in Chapter
2 in connection with raindrops. At the beginning of this chapter, I argued that this
was a reasonable idealization.
Second, our choices must be transitive, a mathematical term that means, roughly,
capable of being placed in order. More precisely, if we prefer X to Y and Y to Z, then
we must prefer X to Z. I cannot simultaneously prefer apples to bananas, bananas to
carrots, and carrots to apples. In other words, in terms of achieving our goals, we
cannot have X better than Y, Y better than Z, and Z better than X. Weak ordering
clearly is required if we are to assign utilities to our choices. The rule we adopt is
that we should choose the option with the highest expected utility, so we must assign
a number to every option representing its expected utility. (Numbers are connected
and transitive.)
In sum, connectedness and transitivity are consequences of the idea that expected
utility measures the extent to which an option achieves our goals. Any two options
either achieve our goals to the same extent, or else one option achieves our goals
better than the other; and if X achieves our goals better than Y, and Y achieves them
better than Z, then it must be true that X achieves them better than Z. 4
An apparent counterexample may help to understand utility theory more deeply
(Petit, 1991). Consider the following three choices offered (on different days) to a
well-mannered person:
1. Here is a (large) apple and an orange. Take your pick; I will have the other.
2. Here is an orange and a (small) apple. Take your pick; I will have the other.
4Another way to understand the value of transitivity is to think about what happens if one has fixed
intransitive choices over an extended period. Suppose X, Y, and Z are three objects, and you prefer owning
X to owning Y, Y to Z, and Z to X. Each preference is strong enough so that you would pay a little money,
at least 1 cent, to indulge it. If you start with Z (that is, you own Z), I could sell you Y for 1 cent plus
Z. (That is, you pay me 1 cent, then I give you Y, and you give me Z.) Then I could sell you X for 1
cent plus Y; but then, because you prefer Z to X, I could sell you Z for 1 cent plus X. If your preferences
stay the same, we could do this forever, and you will have become a money pump. Following the rule of
transitivity for stable preferences avoids being a money pump. This money-pump argument, of course,
applies only to cases that involve money or something like it.