CHAPTER 7: DEALING WITH UNCERTAINTY: EXPECTED VALUE, SENSITIVITY
ANALYSIS, AND THE VALUE OF INFORMATION
Purpose:
Develop the concepts of expected value, sensitivity analysis, and the value of
information.
EXPECTED VALUE ANALYSIS
Expected value analysis consists of modeling uncertainty as contingencies with specific
probabilities of occurrence.
It begins with the specification of a set of contingencies that are
exhaustive and mutually exclusive.
In practice, this means the contingencies capture the full
range of likely variation in net benefits and accurately represent possible outcomes between the
extremes. Once the analyst identifies representative contingencies, the next step is to assign
probabilities to each of them.
The probabilities must be non-negative and sum to one.
The
probabilities can be based on historically observed frequencies, subjective assessments, or
experts (based on information, theory, or both).
Calculating the Expected Value of Net Benefits
Calculate the net benefits of each contingency and then multiply by that contingency's
probability of occurrence.
Then sum all of the weighted benefits.
E(NB) = Σ P
i
(B
i
- C
i
)
Games Against Nature (Normal Form) have the following elements: states of nature,
probabilities of occurrence, actions available to the decision maker facing nature, and payoffs to
the decision maker under each combination of state of nature and action.
In CBA it is common practice to treat expected values as if they were certain (specific) amounts,
even though the actual results rarely equal the expected value.
This is not conceptually correct
when measuring the WTP in situations where individuals face uncertainty.
In practice, however,
treating them as commensurate is reasonable when either the pooling of risk over the collection
of policies, or the pooling of risk over the collection of persons affected by a policy, will make
the actual realized values of costs and benefits close to their expected values.
Unpooled risk
may require an adjustment to expected net benefits called an option-value, which is addressed in
Chapter 8.
Decision Trees and Expected Net Benefits
Basic expected value analysis takes the weighted average over all contingencies.
This can be
extended to situations where costs and benefits accrue over several years, as long as the risks in
each year are independent of the actions in the previous year.
This cannot be done when either
the net benefits or probability of a contingency depends on contingencies that have previously
occurred.
Decision analysis is used in these situations.
Decision analysis can be thought of as
an extended-form game against nature.
It has two stages.
First, one specifies the logical
structure of the decision problem in terms of sequences of decisions and realizations of
contingencies using a diagram (called a decision tree) that links an initial decision to final
Boardman, Greenberg, Vining, Weimer / Cost-Benefit Analysis, 3
rd
Edition
Instructor's Manual 7-1