Technical Summary: Sequential Search with Planning

1. Problem Formulation

The paper models a sequential search problem where a decision-maker (DM) explores a fixed sequence of $n$ options (or "boxes"), each containing an uncertain reward $X_i$ drawn from a known distribution $F_i$. The search is constrained by two distinct cost structures that differentiate this model from standard Pandora's Box problems:

1. Inspection Costs ($c_i$): The cost to observe the reward of a specific box.
2. Planning and Adjustment Costs:
   • Scope Cost ($\phi(j_0)$): Before the search begins, the DM must choose an initial "scope" $j_0 \in \{1, \dots, n\}$, representing the number of stages made accessible. This incurs an ex-ante cost $\phi(j_0)$, which is strictly increasing and convex.
   • Adjustment Cost ($\psi$): During the search, the DM may revise the scope. Expanding the scope from $j$ to $j' > j$ at stage $i$ incurs an additional marginal cost $\psi(i, j, j')$. Contraction is costless.

The DM is risk-neutral and aims to maximize the expected net payoff, defined as the maximum of the initial endowment and the highest observed reward, minus the sum of all inspection, initial scope, and adjustment costs. The sequence of boxes is exogenously given; the DM cannot skip stages.

2. Methodology

The authors employ Dynamic Programming (DP) to solve the sequential decision problem. The state space is defined by three variables:

• $i$: The current stage (box index).
• $z$: The current "guarantee" (the maximum of the initial endowment and all observed rewards so far).
• $j$: The current "paid scope" (the furthest stage for which access costs have already been incurred).

The value function $V(i, z, j)$ represents the optimal expected continuation value after observing stage $i$ with guarantee $z$ and paid scope $j$. The Bellman equation compares the option to stop (yielding payoff $z$) against the option to continue, which involves selecting a new scope $j' \geq i+1$, paying the adjustment cost $\psi(i, j, j')$ and the next inspection cost $c_{i+1}$, and receiving the expected future value.

A key methodological innovation is the introduction of scope-dependent reservation values. Unlike standard search models where a reservation value is a fixed property of a specific box, here the threshold for continuing depends on the state variable $j$.

3. Key Contributions and Results

A. Scope-Dependent Reservation Values

The paper establishes the existence and uniqueness of a family of reservation values, denoted $r_i(j)$, for each stage $i$ and each feasible paid scope $j$.

• Definition: $r_i(j)$ is the unique value such that the DM is indifferent between stopping and continuing optimally given the current paid scope $j$.
• Optimal Strategy: The optimal policy is a threshold rule:
  • Stop if the current guarantee $z \geq r_i(j)$.
  • Continue if $z < r_i(j)$. If continuing, the DM chooses a new scope $j'$ to maximize the net expected continuation value.
• Implication: The optimal stopping rule is no longer "indexable" in the traditional sense (a single value per stage). Two DMs at the same stage with the same observed history but different paid scopes will make different stopping decisions.

B. Comparative Statics

The authors derive several monotonicity results regarding the reservation values and optimal scope choices:

1. Scope Effect: The reservation value $r_i(j)$ is weakly increasing in the paid scope $j$. A DM with a broader prepaid scope is willing to continue searching at higher guarantees because the marginal cost of accessing future stages is lower.
2. Cost Effect: Reservation values are weakly decreasing in inspection costs ($c$) and adjustment costs ($\psi$).
3. Submodularity and Scope Choice: Under the assumption that adjustment costs are submodular (i.e., the marginal cost of expansion decreases as the current scope increases), the optimal next scope is weakly increasing in the current scope. This implies that a firm with a broader current scope will choose a (weakly) broader future scope.

C. Economic Forces

The paper identifies and characterizes three interacting economic forces that shape the optimal strategy:

1. Guarantee Effect: A higher current best offer ($z$) reduces the expected improvement from the next stage, inducing earlier stopping. This parallels the "fall-back value" effect in existing literature.
2. Paid-Scope Effect (Novel): A larger prepaid scope lowers the marginal cost of future access, raising the continuation value and supporting continuation at higher guarantees. This effect has no counterpart in models where future access is free (e.g., Weitzman, 1979; Boodaghians et al., 2020).
3. Remaining-Horizon Effect: As the search progresses, the option value of continuing shrinks because fewer stages remain.

The interaction of these forces generates two distinct regimes:

• Under-planning: The firm chooses a narrow initial scope and expands adaptively as outcomes are realized (common when rewards are normal and costs are convex).
• Over-planning: The firm commits to a broad scope ex-ante but stops early (common under specific conditions with fat-tailed rewards or specific cost structures).

D. Costless Planning Benchmark

The paper demonstrates that when planning and adjustment costs vanish ($\phi = 0, \psi = 0$), the model reduces to the standard ordered-search benchmark (Boodaghians et al., 2020). In this limit, the scope-adaptive formulation coincides with the dynamic programming value function, and the reservation values become independent of planning history.

4. Significance and Claims

The paper claims to fill a gap in the literature on sequential search and R&D planning by explicitly modeling costly ex-ante planning and adaptive scope revision.

• Theoretical Contribution: It generalizes the Pandora's Box problem to settings where access to future options is not free but requires sunk investments. It proves that the "reservation value" is not a static property of an option but a dynamic state variable dependent on past planning commitments.
• Economic Insight: The authors argue that the "paid-scope effect" fundamentally alters search behavior. In standard models, a firm's decision to continue depends only on the current reward and the distribution of future rewards. In this model, the decision also depends on how much the firm has already invested in making those future stages accessible.
• Application to R&D and Exploration: The framework is motivated by real-world scenarios such as multi-stage R&D (e.g., AI development, clinical trials) and natural resource exploration, where firms must commit to infrastructure or regulatory approvals (scope) before outcomes are known, and where plans are often revised adaptively based on interim results.
• Modesty: The paper does not claim to solve all strategic interactions in industrial organization but provides a tractable single-agent framework that captures the tension between initial commitment and adaptive flexibility. It acknowledges that extensions to correlated rewards or competitive environments are left for future work.

The paper concludes that the interplay between the guarantee effect, the paid-scope effect, and the remaining-horizon effect generates realistic adaptive behaviors (both under- and over-planning) that cannot be captured by models assuming costless planning or fixed horizons.