Sequential Search with Planning

2026-06-09

AI Summary

This study addresses sequential decision-making in product development or resource exploration under an ordered-stage framework, where upfront planning costs must be balanced against uncertain future rewards. To this end, the authors propose an ordered Pandora’s box model with planning costs: the decision maker first selects an initial scope and incurs a cost, then may expand this scope over time, employing a threshold-based search policy that depends on a newly introduced state variable—the already-paid-for scope. Using dynamic programming and optimal stopping theory, they formulate a state-dependent reservation value model, establish its existence and uniqueness, and characterize the optimal policy as a family of thresholds indexed by the already-paid-for scope. The analysis reveals intricate interactions among the commitment effect, the already-paid-for scope effect, and the remaining-stage effect, with illustrative examples under normal and heavy-tailed reward distributions demonstrating distinct planning and search behaviors across contexts.

Abstract

Sequential development of a new product or technology, or natural resource exploration, often progresses through ordered stages with uncertain rewards and requires costly (ex ante) planning to make future stages accessible. We model this process as an ordered Pandora's box problem where a decision-maker first chooses an initial scope, paying a cost that rises with the number of stages made accessible, and may later expand the scope at a marginal adjustment cost. Since the paid planning costs are sunk, the continuation values depend on the state variable ``paid scope''. We prove existence and uniqueness of scope-dependent reservation values, characterize the optimal search strategy as a threshold rule indexed by paid scope, and derive comparative statics. Interactions among three economic forces shape the optimal behavior -- a guarantee effect (a higher current best offer reduces the expected improvement from the next stage and induces earlier stopping), a paid-scope effect (a larger prepaid scope lowers the marginal cost of future access, raises the continuation value, and supports continuation at higher guarantees), and a remaining-horizon effect (fewer stages remaining shrink the option value of continuing). Two examples illustrate how these forces generate distinct planning and search patterns under normal and fat-tailed rewards.

Problem

• sequential search
• planning cost
• Pandora's box
• reservation value
• uncertain rewards

Innovation

• Sequential Search
• Pandora's Box
• Planning Cost
• Scope-Dependent Reservation Value
• Optimal Stopping