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 to take an action that pays if it is beneficial () and zero otherwise (). I take the action if its expected net benefit exceeds zero, where is my prior belief about . Thus, my decision rule is to take the action whenever exceeds the cost-benefit ratio .
Now suppose I can wait for a noisy signal with error rate I use my prior, the signal, and Bayes’ rule to form a posterior belief about . Then I take the action if its expected net benefit given exceeds zero. This happens with probability where the probability of receiving a positive signal depends on my prior and the error rate .
If or then the signal doesn’t affect whether I take the action, so I don’t need to wait. But if 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 where the discount factor 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 if and only if the expected net benefit under my prior exceeds . This happens precisely when my prior exceeds The following chart plots against when and . Increasing the discount factor or the cost-benefit ratio raises the option value of waiting, which raises the threshold prior above which I should take the action. Increasing the error rate makes the signal less informative, which lowers the option value of waiting and, hence, lowers . If then the signal is uninformative and so independently of .