The option value of waiting

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 $$E[\omega b-c]=pb-c$$ 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 $$Pr(s\ne \omega \mid \omega )=\epsilon \in [0,0.5].$$ I use my prior, the signal, and Bayes’ rule to form a posterior belief $$\begin{array}{cc}{q}_s& \equiv Pr(\omega =1\mid s)\\ & =\{\begin{array}{cc}\frac{\epsilon p}{(1-\epsilon )(1-p)+\epsilon p}& \text{if}s=0\\ \frac{(1-\epsilon )p}{\epsilon (1-p)+(1-\epsilon )p}& \text{if}s=1\end{array}\end{array}$$ about $\omega$. Then I take the action if its expected net benefit $$\begin{array}{cc}E[\omega b-c\mid s]& =q_{s}b-c\end{array}$$ given $s$ exceeds zero. This happens with probability $$Pr(q_{s}b-c\ge 0)=\{\begin{array}{cc}1& \text{if}c/b\le q_0\\ Pr(s=1)& \text{if}{q}_0<c/b\le q_1\\ 0& \text{if}{q}_1<c/b,\end{array}$$ 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 $$\begin{array}{cc}W& \equiv \delta E\left[E\right[max\{0,q_{s}b-c\}\mid s\left]\right]\\ & =\{\begin{array}{cc}\delta (pb-c)& \text{if}c/b\le q_0\\ \delta (q_{1}b-c)Pr(s=1)& \text{if}{q}_0<c/b\le q_1\\ 0& \text{if}{q}_1<c/b,\end{array}\end{array}$$ 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 $$p^{\ast }\equiv \frac{(1-\delta \epsilon )c}{b-\delta \left(\right(1-\epsilon )b-(1-2\epsilon \left)c\right)}.$$ The following chart plots $p^{\ast }$ 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 $p^{\ast }$ 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 $p^{\ast }$. If $\epsilon =0.5$ then the signal is uninformative and so $p^{\ast }=c/b$ independently of $\delta$.