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Economics 142: Choice under Uncertainty (or Certainty) Winter 2008Vincent Crawford (with very large debts to Matthew Rabin and especially Botond Koszegi)
Background: Classical theory of choice under certainty
Rational choice (complete, transitive, and continuous preferences) over certain outcomes and
representation of preferences via maximization of an ordinal utility function of outcomes.
The individual makes choices “as if” to maximize the utility function; utility maximization is just
a compact, tractable way for us to describe the individual’s choices in various settings.
We can view the utility function as a compact way of storing intuition about behavior from
simple experiments or though-experiments and transporting it to new situations.
The preferences represented can be anything—self-interested or not, increasing in intuitive
directions (more income or consumption) or not—although there are strong conventions in
mainstream economics about what they are normally defined over—own income or consumption
rather than both own and others’, levels of final outcomes rather than changes.
Thus if you think the mainstream approach is narrow or wrong-headed, it may make as much or
more sense to complain about those conventions than about the idea of rationality per se.
,Background: Classical “expected utility” theory of choice under uncertainty
This is the standard way to describe people’s preferences over uncertain outcomes. The
Marschak reading on the reading list, linked on the course page, is a readable introduction.
The basic idea is that if an individual’s preferences satisfy certain axioms, discussed below, and
the uncertainty is over which of a given list of outcomes will happen, then a person’s preferences
over probability distributions over those outcomes can be described (much as for certain
outcomes, although there is an important difference) by assigning utility numbers (called “von
Neumann-Morgenstern utilities” in analyses of individual decisions or, equivalently, “payoffs” in
games), one to each possible outcome, and assuming that the person chooses as if to maximize
“expected utility”—the mathematical expectation of the utility of the realized outcome.
Example: Suppose that your initial lifetime wealth w is $2 million dollars, you are asked to
choose whether or not to accept a bet (investment opportunity, insurance contract, etc.) that will
add either x, y, or z (which could be negative) to your wealth, with respective probabilities p, q,
or 1 − p − q. Suppose further that you care only about your final lifetime wealth.
Then the claim is that, under the axioms mentioned about, the analyst can assign utilities to the
possible final outcomes w, w + x, w + y, and w + z, call them u(w), u(w + x), u(w + y), and u(w
+ z), such that the person will accept the bet if and only if (ignoring ties) u(w) < pu(w + x) +
qu(w + y) + (1 − p − q)u(w + z). (In other words, if the expected utility of “w for sure” is less
than that of a random addition to w of x, y, or z with probabilities p, q, or 1 − p − q.) The vN-M
utility function whose expectation the individual acts as if to maximize is a compact way to
describe the individual’s choices in various settings involving uncertainty.
,There are two important assumptions in this example:
● That the person’s preferences are “well-behaved” enough to be represented by a “preference
function” (so called to distinguish it from the utility function) over probability distributions that
(with outcomes fixed and distributions over the fixed outcomes described by lists of
probabilities) is linear in the probabilities (that is, it is an expected utility).
● That the person’s preferences respond only to final, level (as opposed to change) outcomes, in
this case of the person’s own lifetime wealth.
The first assumption is logically justified by a famous result known as the Von Neumann-
Morgenstern Theorem.
The theorem’s axioms are sometimes systematically violated in observed behavior, and the
axioms that the theorem uses to justify expected-utility maximization are not completely
uncontroversial. But these violations seem behaviorally and economically less important than
violations of the second assumption.
The second assumption is not at all logically necessary to use expected-utility maximization to
describe choice under uncertainty.
It is only a convention of mainstream economics, which could be replaced by an alternative
convention to yield an alternative expected-utility characterization of choice under uncertainty,
as we shall do below.
, First let’s record the logic of the first assumption. (This snapshot and others in this section are
from Machina, “Expected Utility Hypothesis,” linked on the course web page.)
Von Neumann-Morgenstern Theorem: Complete, transitive, and continuous preferences over
probability distributions of outcomes that satisfy the “independence axiom” can be represented
by the maximization of “expected utility” (just as complete, transitive, and continuous
preferences over certain outcomes can be represented by the maximization of standard utility).
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