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 \(\delta\).
