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 $$\delta$$. 