Criteria for Normative Applications

Criteria for Normative Applications
Savage’s omelet example effectively shows how representing nature’s underlying uncertainty by a set of “states,” then representing one’s alterative courses of action as “acts” that map these states into their respective consequences, can serve to organize a decision problem
and make it easier to see exactly how one’s beliefs (the state likelihoods)
and risk preferences should enter into the problem. For proper normative application, this first step—namely, the specification of the states—must satisfy three properties:
1) The alternative states must be mutually exclusive—that is, no two distinct states can simultaneously occur. Thus, it would not have been correct to list “egg is rotten” and “five-egg omelet” as two distinct states, since is it possible that these could simultaneously occur.
States of the World and the State of Decision Theory 31
2) The family of states must be exhaustive—that is, whatever happens,
at least one of the states can be said to have occurred. Although it is at the same logical level as the previous criterion (mutual exclusivity), the exhaustiveness criterion is much more difficult—and some would argue, actually impossible—to guarantee
in practice. For example, if you cracked the sixth egg into the bowl and found that it was actually hollow, then neither of the two states in the Savage table could be said to have occurred, since neither of the first-row consequences would be realized (you would not have a six-egg omelet, nor would you have destroyed the other five eggs). When the decision maker has reason
to “expect the unexpected,” the exhaustivity requirement cannot necessarily be achieved, and the best one can do is specify
a final, catch-all state, with a label like “none of the above,” and a very ill-defined consequence.
3) The states must represent nature’s exogenous uncertainty, so their likelihoods cannot be affected by the individual’s choice of act. This issue can be illustrated by a simple example involving the decision whether or not to install a lightning rod on one’s house. Naturally, the relevant occurrences are the two mutually exclusive
results {“house burns down,” “house doesn’t burn down”}. But since installing a lightning rod will clearly alter the respective
likelihoods of these occurrences, can we really specify states of nature that are independent of the decision maker’s action? The answer is illustrated in the following table, which makes it clear that “house burns down” and “house doesn’t burn down” are not the states at all, but rather, part of the consequences, and clarifies that the effect of installing a lightning rod—as with any subjective act—is the outcome of an interaction between the act and an exogenous state of nature.
State
Big lightning
Small lightning
No lightning
Act
strike
strike
strike
Lightning
House burns down, House doesn’t
House doesn’t burn,
rod
paid for rod
burn, paid for rod
paid for rod
No lightning
House burns down, House burns down,
House doesn’t burn,
rod
didn’t pay for rod
didn’t pay for rod
didn’t pay for rod