Search in the Dark: The Normal Case
46 Pages Posted: 9 Jun 2022 Last revised: 21 Jul 2022
Date Written: June 1, 2022
The classic sequential search problem rewards the decision-maker with the highest sampled value, minus the sampling cost. If the sampling distribution is unknown, then a Bayesian decision-maker faces a complex balance between learning and optionality. We solve the stopping problem of sampling from a Normal distribution with unknown mean and unknown variance using a conjugate prior, a riddle that has remained open for half a century. We find that reservation prices---prevalent in search theory---are no longer optimal. Structurally, the optimal stopping region may be empty or comprise one or two bounded intervals. We also introduce the so-called internal cost function, which provides a computationally practical way to identify the optimal stopping rule for any given prior, sampling history, and remaining samples, and that can also be applied to the case of known variance.
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