Search in the Dark: The Normal Case

46 Pages Posted: 9 Jun 2022 Last revised: 21 Jul 2022

See all articles by Manel Baucells

Manel Baucells

University of Virginia - Darden School of Business

Sasa Zorc

University of Virginia - Darden School of Business

Date Written: June 1, 2022

Abstract

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.

Suggested Citation

Baucells, Manel and Zorc, Sasa, Search in the Dark: The Normal Case (June 1, 2022). Available at SSRN: https://ssrn.com/abstract=4125540 or http://dx.doi.org/10.2139/ssrn.4125540

Manel Baucells (Contact Author)

University of Virginia - Darden School of Business ( email )

P.O. Box 6550
Charlottesville, VA 22906-6550
United States

Sasa Zorc

University of Virginia - Darden School of Business ( email )

P.O. Box 6550
Charlottesville, VA 22906-6550
United States

HOME PAGE: http://https://www.darden.virginia.edu/faculty-research/directory/sasa-zorc

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