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arxiv: 2606.10438 · v1 · pith:RMOLQVHJnew · submitted 2026-06-09 · 💰 econ.TH

Sequential Search with Planning

Ruhi Sonal , Saptarshi Mukherjee , Abhinaba Lahiri , Aniruddha Ghosh This is my paper

Pith reviewed 2026-06-27 11:14 UTC · model grok-4.3

classification 💰 econ.TH
keywords sequential searchPandora's box problemplanning costsreservation valuesthreshold strategycomparative staticseconomic theorystaged development
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The pith

Scope-dependent reservation values determine optimal stopping in sequential search with upfront planning costs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models staged development or exploration as an ordered Pandora's box problem in which a decision maker first selects a planning scope at a cost that increases with the number of accessible stages and can later expand that scope at an adjustment cost. Because the planning costs are sunk, the value of continuing depends on the current paid scope. The authors establish existence and uniqueness of reservation values that vary with this paid scope and show that the optimal policy is a threshold rule: stop when the current best offer exceeds the reservation value associated with the current scope. Three interacting forces govern behavior: a guarantee effect that favors stopping when the current offer is high, a paid-scope effect that favors continuing when more stages are already prepaid, and a remaining-horizon effect that reduces the value of continuing as fewer stages remain. Two illustrative examples demonstrate how these forces produce different planning and search patterns under normal and fat-tailed reward distributions.

Core claim

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 paid-scope effect, and a remaining-horizon effect. Two examples illustrate how these forces generate distinct planning and search patterns under normal and fat-tailed rewards.

What carries the argument

Scope-dependent reservation values that act as thresholds for stopping search, with the threshold level depending on the current paid planning scope.

If this is right

  • A higher current best offer reduces the expected gain from the next stage and induces earlier stopping via the guarantee effect.
  • A larger prepaid scope lowers the marginal cost of accessing future stages, raises continuation value, and supports search at higher current guarantees via the paid-scope effect.
  • Fewer stages remaining shrink the option value of continuing via the remaining-horizon effect.
  • The optimal initial scope choice balances the upfront cost against the option value created by the paid-scope effect on later thresholds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could be extended to endogenous choice of expansion timing rather than fixed adjustment costs.
  • Empirical tests could examine whether firms in resource exploration or R&D commit to larger initial scopes when reward distributions are more fat-tailed.
  • The model suggests that observed differences in search persistence across industries may partly reflect differences in typical planning-cost structures rather than differences in beliefs alone.

Load-bearing premise

Planning costs are sunk once paid, so continuation values are determined by the state variable paid scope and the cost structure rises with the number of stages made accessible.

What would settle it

In a controlled experiment where subjects choose initial scope and then observe sequential draws, check whether their stopping thresholds rise with the level of prepaid scope exactly as the comparative statics predict.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript models sequential search as an ordered Pandora's box problem in which a decision maker pays an increasing cost to choose an initial scope of accessible stages and can later expand scope at a marginal cost. Planning costs are sunk, so continuation values depend on the state variable 'paid scope.' The paper proves existence and uniqueness of scope-dependent reservation values, shows that the optimal policy is a threshold rule indexed by paid scope, and derives comparative statics driven by three forces: a guarantee effect, a paid-scope effect, and a remaining-horizon effect. Two examples with normal and fat-tailed rewards illustrate the resulting planning and search patterns.

Significance. If the results hold, the paper cleanly extends the Pandora's box framework to environments with ex-ante planning costs that remain sunk and affect future access. The decomposition into three monotone forces and the threshold characterization indexed by paid scope provide a transparent way to analyze optimal behavior in finite-horizon sequential development problems. The finite ordered stages permit direct backward induction under standard assumptions, yielding unique value functions and reservation values without additional restrictions.

minor comments (2)
  1. [Model Setup] The model statement would benefit from an explicit declaration of the finite number of stages N and the precise functional form of the planning-cost schedule in the setup section.
  2. [Examples] In the examples, the specific parameter values chosen for the normal and fat-tailed cases should be reported so that the distinct patterns can be reproduced.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript. The referee's summary and significance evaluation accurately reflect the paper's contributions, and we appreciate the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results—existence and uniqueness of scope-dependent reservation values, threshold characterization of the optimal strategy, and comparative statics—are obtained via standard backward induction on a finite-horizon dynamic program with sunk planning costs and monotone continuation values. The derivation begins from the model primitives (ordered stages, increasing access costs, reward distributions) and applies the usual single-crossing and indifference arguments without invoking fitted parameters, self-referential equations, or load-bearing self-citations. No step reduces by construction to its own inputs; the three economic forces (guarantee, paid-scope, remaining-horizon) are shown to preserve monotonicity directly from the value-function recursion. This is the expected non-circular outcome for a self-contained theoretical characterization.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard search-theoretic assumptions plus the specific cost and sunk-cost structure; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Planning costs rise with the number of stages made accessible.
    Explicitly stated in the abstract as the cost structure for choosing initial scope.
  • domain assumption Paid planning costs are sunk and continuation values depend on the paid-scope state variable.
    Core modeling choice stated in the abstract that enables scope-dependent reservation values.

pith-pipeline@v0.9.1-grok · 5735 in / 1297 out tokens · 28051 ms · 2026-06-27T11:14:33.482404+00:00 · methodology

discussion (0)

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