Optimal Sequential Screening Mechanisms

• Optimal sequential screening is a decision-making protocol that uses threshold-based rules to select candidates under uncertainty.
• It integrates dynamic programming, Bayesian inference, and mechanism design to ensure incentive compatibility and robust performance.
• Its applications span economics, machine learning, and high-dimensional variable selection, optimizing outcomes and resource allocation.

An optimal sequential screening mechanism is a principled protocol for decision-making in settings where options, agents, or variables are revealed and evaluated sequentially and must be screened or selected under uncertainty and constraints. Such mechanisms appear in economics (contract theory, auctions), online selection (secretary problems, multi-armed bandits), statistical inference, and machine learning pipelines, providing rigorous strategies for maximizing objectives such as utility, profit, accuracy, or speed subject to Bayesian, strategic, or stochastic considerations. The optimality of these mechanisms refers to maximizing an explicit criterion (e.g., expected utility, welfare, or information gain) while satisfying incentive compatibility, individual rationality, or error control.

1. Mathematical Foundations and Core Models

Sequential screening mechanisms are fundamentally optimal stopping or selection procedures under uncertainty and information asymmetry. A central instantiation is the classical secretary problem and its generalizations, where a decision-maker (DM) sequentially observes a finite set of randomly ordered options and must select or screen the best one on the spot, with irreversibility and no recall.

In the framework of "Sequential selections with minimization of failure" (Szajowski, 2022), the DM faces $N$ applicants arriving in random order and, at each time $k$, observes if the $k$th applicant is the best among those seen so far (a "record"). The DM receives payoffs depending on whether the selected applicant is the overall best ($+\alpha$), a suboptimal choice ($-\beta$), or if no choice is made ($-\gamma$). The DM maximizes

$$U(\tau) = \alpha\,\mathbb{P}(Z_\tau=1, \tau \le N) - \beta\,\mathbb{P}(Z_\tau \ne 1, \tau \le N) - \gamma\,\mathbb{P}(\tau=N,\,\text{no choice})$$

where $\tau$ is a stopping time. Reduction via dynamic programming shows that optimal choices can be restricted to record times, enabling a recursion for value functions $V(k)$ and a characterization of the unique optimal threshold $k^*$ such that the DM adopts the rule: "skip the first $k^* - 1$ records, then select the first subsequent record." Asymptotically, the optimal threshold scales as $t^* \sim \exp(-(\alpha+\gamma)/(\alpha+\beta))$ for large $N$.

In contract theory and information economics, the sequential screening problem generalizes via multi-stage mechanisms, endogenous information acquisition, and multi-dimensional screening (see (Ozbek, 8 Dec 2025, Gao, 29 Dec 2025)). Here, agents possess or acquire information in stages and mechanisms are designed to maximize welfare or principal revenue while ensuring incentive compatibility (IC) and individual rationality (IR).

2. Economic and Game-Theoretic Mechanisms

Optimal sequential screening is a central paradigm in mechanism design with endogenous information acquisition costs. In "Optimal Auction Design under Costly Learning" (Ozbek, 8 Dec 2025), each agent (with private value $v_i$ and information cost $c_i$) first announces their cost parameter and pays a screening transfer $t(c_i)$, determines the precision $s_i$ of information to acquire at cost $C(s_i,c_i)$, and then participates in an efficient second-stage Vickrey-Clarke-Groves (VCG) auction. The screening transfer $t(c_i)$ is uniquely pinned down by the welfare-maximizing equilibrium and the envelope theorem to extract agents’ information rents while implementing Bayesian IC and IR.

The approach generalizes to multidimensional screening (multiple goods) with coupled post-contract valuations. Under the "dynamic decoupling" property, if the dependence structure over goods conditional on pre-contract learning does not vary, the optimal mechanism is to run parallel optimal single-good sequential screening schemes for each dimension, frontloading all information rents into a scalar up-front membership fee (see (Gao, 29 Dec 2025)). This result leverages the separability of the virtual value (or marginal revenue) representations across coordinates: $$\phi^j(\gamma, \theta^j) = \theta^j + \frac{F^j_\gamma(\theta^j | \gamma)}{f^j(\theta^j | \gamma)} \frac{1-G(\gamma)}{g(\gamma)}$$ with ex-post allocation using cutoff pricing at the unique critical $\hat\theta^j(\gamma)$ where $\phi^j(\gamma, \hat\theta^j(\gamma))=0$.

In information design, "Optimal Disclosure of Information to a Privately Informed Receiver" (Candogan et al., 2021) shows that screening mechanisms can be optimized by laminar partitional signaling, effectively partitioning the state space into nested convex sets, with IC and MPC constraints. Screening strictly dominates non-screening, achieving up to a $1/|\Theta|$ factor improvement in designer payoff.

3. Algorithmic and Statistical Sequential Screening

Sequential correct screening is prominent in high-dimensional inference, where the goal is to iteratively eliminate variables not among the top-m based on adaptively updated confidence bounds. In "Sequential Correct Screening and Post-Screening Inference" (Toyoda et al., 20 Aug 2025), the Sequential Correct Screening (SCS) algorithm maintains a candidate set $\widehat S_T$ at each time $T$, uses anytime-valid confidence sequences, and screens out variables whose upper bound falls below the m-th greatest lower bound:

• At each step, $\widehat S_T$ shrinks monotonically and always contains the true top-m variables with probability at least $1-\alpha$.
• The construction is anytime-valid, permitting reporting $\widehat S_T$ at any stopping time with prescribed false coverage control (FCR).
• The methodology provides post-screening inference with FCR guarantees using e-BY style and Bonferroni-PSI confidence intervals.

The algorithm is computationally efficient ($O(k)$ per step with $k$ variables), supports adaptive or asynchronous sampling, and outperforms LUCB-style batch elimination procedures in empirical simulations.

For large-scale sparse learning problems, sequential screening can be embedded within optimization routines (see (Wang et al., 2016)). Here, "feedback-controlled sequential screening" utilizes problem-dependent, adaptive cutoffs between Lasso subproblems, using geometric dual bounds and feedback to efficiently reduce problem size while guaranteeing provable safe feature elimination and strong empirical speedups.

4. Robustness and Strategic Behavior

The design of robust sequential screening mechanisms anticipates strategic manipulation and adversarial behavior. In "Sequential Strategic Screening" (Cohen et al., 2023), individuals may optimally "zig-zag" through a sequence of classifiers to strategically pass each stage without meeting the most stringent conjunctive criterion. The optimal defense is the "conservative defense," which $τ$-shifts each classifier's acceptance region inward by the maximum allowed manipulation, thereby ensuring zero false positives and maximizing true positive retention: $\tilde h_i(x) = 1 \{ w_i^T x \geq b_i + \tau \}$ Both theory and simulation confirm that this method yields tight separation against adversarial manipulation in both sequential and simultaneous screening regimes, providing a blueprint for designing robust multi-stage filters in adversarial environments.

5. Warm-Starting, Information, and Partial Observability

Sequential screening mechanisms are also optimized under partial information or mixed information regimes. In "Optimal Multiple Stopping Rule for Warm-Starting Sequential Selection" (Fekom et al., 2020), the decision-maker must fill or upgrade postings through sequential selection, possibly with partially filled slots at the outset ("warm start"). The optimal rule is derived via dynamic programming, leveraging Bellman recursions: $$V_j^{X,Y} = \mathbb E[\max\{ V_{j+1}^{X,Y}, S_j + \max\{ V_{j+1}^{X-1,Y}, V_{j+1}^{X,Y-1} \} \}]$$ With this, threshold policies are derived that specify when to hire/replace. Extensions provide plug-in strategies for partial knowledge of score distributions (adaptive estimation), and pure rank-based versions for total lack of numerical scoring information, yielding near-optimal regret and learning performance guarantees.

6. Practical Implementations and Applications

Optimal sequential screening mechanisms have concrete algorithmic instantiations, as summarized below:

Setting	Mechanism Type	Key Properties
Secretary-type selection	Harmonic threshold rule (Szajowski, 2022)	Explicit threshold, risk tuning
Costly learning auctions	Two-stage VCG with welfare screening (Ozbek, 8 Dec 2025)	Ex-post IC, revenue max
Multidimensional contract screening	Frontload membership + option pricing (Gao, 29 Dec 2025)	Decoupling, virtual value cuts
Variable selection/statistical inference	SCS (anytime-valid, FCR) (Toyoda et al., 20 Aug 2025)	Exact error control, fast update
Adversarial pipelines	Conservative defense (Cohen et al., 2023)	Tight, robust, zero FP
Warm-start hiring/online assignment	Dynamic programming thresholds (Fekom et al., 2020)	Handles partial/no info
Large sparse learning	Feedback-controlled screening (Wang et al., 2016)	Adaptive, optimal computation

These mechanisms are critical in real-world hiring, online portfolio allocation, resource scheduling, mechanism design for marketplaces, sequential testing protocols, and high-dimensional model selection.

7. Limitations and Research Frontiers

Despite their breadth, current optimal sequential screening mechanisms generally rely on simplifying assumptions such as independence or invariant dependence, known pool size, and full commitment to the screening protocol. Open problems include:

• Extending to environments with unknown or variable pool sizes, random horizon, or non-stationary and delayed feedback.
• Incorporating multi-dimensional or multidimensional-dependent information structures with non-invariant copula.
• Integrating dynamic agent beliefs and learning, and multidimensional subjective cost and utility parameters.
• Developing computationally tractable, minimax-optimal rules under arbitrary adaptive adversaries or partial monitoring.

Recent work suggests promising connections between optimal sequential screening mechanisms, dynamic contract design, robust machine learning pipelines, and information-theoretic inference, setting a rich research agenda for future advances.