This post is about waiting for information before taking an action. It uses a simple model to explain when and why waiting is valuable. It formalizes some ideas discussed in my posts on climate change and pandemic policy.

Suppose it costs `\(c>0\)`

to take an action that pays `\(b>c\)`

if it is beneficial (`\(\omega=1\)`

) and zero otherwise (`\(\omega=0\)`

).
I take the action if its expected net benefit
`$$\newcommand{\E}{\mathrm{E}} \E[\omega b-c]=pb-c$$`

exceeds zero, where `\(p=\Pr(\omega=1)\)`

is my prior belief about `\(\omega\)`

.
Thus, my decision rule is to take the action whenever `\(p\)`

exceeds the cost-benefit ratio `\(c/b\)`

.

Now suppose I can wait for a noisy signal `\(s\in\{0,1\}\)`

with error rate
`$$\renewcommand{\epsilon}{\varepsilon} \Pr(s\not=\omega\mid \omega)=\epsilon\in[0,0.5].$$`

I use my prior, the signal, and Bayes’ rule to form a posterior belief
`$$\begin{align} q_s &\equiv \Pr(\omega=1\mid s) \\ &= \begin{cases} \frac{\epsilon p}{(1-\epsilon)(1-p)+\epsilon p} & \text{if}\ s=0 \\ \frac{(1-\epsilon)p}{\epsilon(1-p)+(1-\epsilon)p} & \text{if}\ s=1 \end{cases} \end{align}$$`

about `\(\omega\)`

.
Then I take the action if its expected net benefit
`$$\begin{align} \E[\omega b-c\mid s] &= q_sb-c \end{align}$$`

given `\(s\)`

exceeds zero.
This happens with probability
`$$\Pr(q_sb-c\ge0)=\begin{cases} 1 & \text{if}\ c/b\le q_0 \\ \Pr(s=1) & \text{if}\ q_0<c/b\le q_1 \\ 0 & \text{if}\ q_1<c/b, \end{cases}$$`

where the probability
`$$\Pr(s=1)=\epsilon(1-p)+(1-\epsilon)p$$`

of receiving a positive signal depends on my prior `\(p\)`

and the error rate `\(\epsilon\)`

.

If `\(c/b\le q_0\)`

or `\(q_1<c/b\)`

then the signal doesn’t affect whether I take the action, so I don’t need to wait.
But if `\(q_0<c/b\le q_1\)`

then waiting gives me a real option not to take the action if I learn it isn’t beneficial.
So the expected benefit of waiting equals
`$$\begin{align} W &\equiv \delta\,\E\left[\E[\max\{0,q_sb-c\}\mid s]\right] \\ &= \begin{cases} \delta(pb-c) & \text{if}\ c/b\le q_0 \\ \delta(q_1b-c)\Pr(s=1) & \text{if}\ q_0<c/b\le q_1 \\ 0 & \text{if}\ q_1<c/b, \end{cases} \end{align}$$`

where the discount factor `\(\delta\in[0,1]\)`

captures
(i) my patience and
(ii) my confidence that the action will still be available if I wait.

I should take the action *before* receiving `\(s\)`

if and only if the expected net benefit `\((pb-c)\)`

under my prior exceeds `\(W\)`

.
This happens precisely when my prior exceeds
`$$p^*\equiv\frac{(1-\delta\epsilon)c}{b-\delta((1-\epsilon)b-(1-2\epsilon)c)}.$$`

The following chart plots `\(p^*\)`

against `\(\delta\)`

when `\(c/b\in\{0.1,0.3,0.5\}\)`

and `\(\epsilon\in\{0,0.25,0.5\}\)`

.
Increasing the discount factor `\(\delta\)`

or the cost-benefit ratio `\(c/b\)`

raises the option value of waiting, which raises the threshold prior `\(p^*\)`

above which I should take the action.
Increasing the error rate `\(\epsilon\)`

makes the signal less informative, which *lowers* the option value of waiting and, hence, lowers `\(p^*\)`

.
If `\(\epsilon=0.5\)`

then the signal is uninformative and so `\(p^*=c/b\)`

independently of `\(\delta\)`

.