Expected value of including uncertainty

In decision theory and quantitative policy analysis, the expected value of including information (EVIU) is the expected difference in the value of a decision based on a probabilistic analysis versus a decision based on an analysis that ignores uncertainty.

Formalization

Let

${\begin{array}{ll}d\in D&{\mbox{the decision being made, chosen from space }}D\\x\in X&{\mbox{the uncertain quantity, with true value in space }}X\\U(d,x)&{\mbox{the utility function}}\\p(x)&{\mbox{your prior subjective probability distribution (density function) on }}x\end{array}}$

When not including uncertainty, you find the optimal decision using only $E[x]$, the expected value of the uncertain quantity. Hence, the decision ignoring uncertainty is given by:

Failed to parse (unknown function "\argmax"): {\displaystyle d_{iu} = {\argmax_{d}} ~ U(d,E[x]) }

The optimal decision taking uncertainty into account is

Failed to parse (unknown function "\argmax"): {\displaystyle d^* = {\argmax_{d}} \int U(d,x) f(x) dx }

The EVIU is thus given as

Example

(in progress)

Relation to Expected Value of Perfect Information (EVPI)

Both EVIU and EVPI compare the expected value of the Bayes' decision with another decision made without uncertainty. For EVIU this other decision is made when the uncertainty is ignored, although it is there, while for EVPI this other decision is made after the uncertainty is removed by obtaining perfect information about x.

The EVPI is the expected cost of being uncertain about x, while the EVIU is the additional expected cost of pretending you are not uncertain.

The EVIU, like the EVPI, gives expected value in terms of the units of the utility function.

See also

• Expected value of perfect information (EVPI)
• Expected value of sample information