Sequential Search with Acquisition Uncertainty

66 Pages Posted: 28 Jan 2022

See all articles by David B. Brown

David B. Brown

Duke University - Decision Sciences

Cagin Uru

Duke University - Decision Sciences

Date Written: January 21, 2022

Abstract

In many applications involving sequential search, uncertainty in the ability to acquire alternatives complicates the search problem and erodes value. We study a variation of the classical Pandora’s problem in which a decision-maker (DM) sequentially explores alternatives from a given set and learns their values while trying to acquire the best alternative. The variations in the model we study are (i) alternatives randomly become unavailable during exploration and (ii) the DM’s ability to acquire a remaining alternative is uncertain and depends on a chosen offer price. Such acquisition uncertainties arise in many applications but greatly complicate the resulting stochastic dynamic program, as the number of possible states grows exponentially in the number of alternatives and the state and action spaces are in general continuous. We develop and study a class of greedy threshold policies that make offers only on the most recently explored alternative, provided its value exceeds a given threshold. The threshold in these policies can be calculated rapidly using bisection and, in some cases, analytically. We show that these threshold policies are asymptotically optimal as the number of alternatives grows large with a convergence rate that we characterize and that in general, these policies obtain at least 1-1/e ≈ 63.2% of the optimal value. We also show that threshold policies based on discrete approximations of the value distribution - which may arise when estimating the value distribution from data or when exploration provides incomplete information on values - are close in performance to those based on the exact value distributions. We illustrate these methods on examples from housing search using models calibrated on data from the online brokerage Redfin. Finally, we discuss how these results extend to some generalizations of the problem.

Keywords: sequential search, approximation algorithms, Lagrangian relaxations.

Suggested Citation

Brown, David B. and Uru, Cagin, Sequential Search with Acquisition Uncertainty (January 21, 2022). Available at SSRN: https://ssrn.com/abstract=4014841 or http://dx.doi.org/10.2139/ssrn.4014841

David B. Brown (Contact Author)

Duke University - Decision Sciences ( email )

Durham, NC 27708-0120
United States

Cagin Uru

Duke University - Decision Sciences ( email )

Durham, NC 27708-0120
United States

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