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 (ω=1) and zero otherwise (ω=0). I take the action if its expected net benefit E[ωbc]=pbc exceeds zero, where p=Pr(ω=1) is my prior belief about ω. 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{0,1} with error rate Pr(sωω)=ε[0,0.5]. I use my prior, the signal, and Bayes’ rule to form a posterior belief qsPr(ω=1s)={εp(1ε)(1p)+εpif s=0(1ε)pε(1p)+(1ε)pif s=1 about ω. Then I take the action if its expected net benefit E[ωbcs]=qsbc given s exceeds zero. This happens with probability Pr(qsbc0)={1if c/bq0Pr(s=1)if q0<c/bq10if q1<c/b, where the probability Pr(s=1)=ε(1p)+(1ε)p of receiving a positive signal depends on my prior p and the error rate ε.

If c/bq0 or q1<c/b then the signal doesn’t affect whether I take the action, so I don’t need to wait. But if q0<c/bq1 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 WδE[E[max{0,qsbc}s]]={δ(pbc)if c/bq0δ(q1bc)Pr(s=1)if q0<c/bq10if q1<c/b, where the discount factor δ[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 (pbc) under my prior exceeds W. This happens precisely when my prior exceeds p(1δε)cbδ((1ε)b(12ε)c). The following chart plots p against δ when c/b{0.1,0.3,0.5} and ε{0,0.25,0.5}. Increasing the discount factor δ 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 ε makes the signal less informative, which lowers the option value of waiting and, hence, lowers p. If ε=0.5 then the signal is uninformative and so p=c/b independently of δ.