arxiv: 2605.04202 · v1 · submitted 2026-05-05 · 💻 cs.LG

Recognition: unknown

Sequential Strategic Classification with Multi-Stage Selective Classifiers

Ziyuan Huang , Lina Alkarmi , Mingyan Liu

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:24 UTC · model grok-4.3

classification 💻 cs.LG

keywords strategic classificationmulti-stage modelsselective classifiersmyopic policiesagent incentivessequential decisionsimprovement versus gaming

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The pith

In a multi-stage strategic classification model, sequences of selective classifiers can be tuned so that myopic agents gain more long-term utility by choosing genuine improvement over gaming.

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

This paper builds a model in which agents move through ordered stages of classification, each with its own selective classifier that can abstain. At every stage an agent may raise its true attributes through costly improvement or raise only the observable features through cheaper gaming. A positive decision moves the agent forward, a negative decision moves it backward, and abstention leaves it in place. The authors derive the exact best one-step choice the agent should make at any given stage. They then compare the two simple repeated policies of always refusing to improve versus always refusing to game, and identify conditions on the sequence of classifiers under which the no-gaming policy produces strictly higher long-run utility.

Core claim

We introduce a sequential stochastic multi-stage model of strategic classification using selective classifiers that abstain at low confidence. We fully characterize the agent's optimal instantaneous action under these classifiers and show that there exist design principles for the sequence of classifiers under which the myopic no-gaming policy yields higher long-term utility than the myopic no-improvement policy, thereby incentivizing genuine effort over time.

What carries the argument

The selective classifier at each stage, which promotes the agent on a positive prediction, demotes on a negative prediction, and holds position on abstention, together with the closed-form characterization of the agent's optimal myopic choice between separable improvement and gaming actions.

If this is right

Under suitably chosen sequences of selective classifiers, agents that follow the no-gaming myopic policy reach higher stages and accumulate greater cumulative reward than those that follow the no-improvement policy.
The instantaneous optimal action can be expressed in closed form once the current stage, the classifier parameters, and the two cost values are known.
Abstention creates a stationary point that allows the long-term comparison of the two policies without requiring the agent to plan across multiple future stages.
The same characterization lets a designer compare the progression speed of agents under each policy and adjust thresholds to widen the utility gap favoring no-gaming.

Where Pith is reading between the lines

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

In applied settings such as repeated hiring or credit decisions, inserting selective abstention at each stage could reduce the payoff to pure feature manipulation.
If agents later learn to anticipate the entire sequence, the incentive to improve may become even stronger than the myopic calculation predicts.
The closed-form action characterization supplies a concrete benchmark against which one can measure how much real-world agents deviate from myopic optimality.

Load-bearing premise

Agents optimize only their immediate next outcome and the costs of improvement and gaming actions remain separable across stages.

What would settle it

For a concrete sequence of classifier thresholds, calculate the long-term expected utilities under repeated no-gaming and under repeated no-improvement; the design claim holds if the former exceeds the latter for some sequences and not for others.

Figures

Figures reproduced from arXiv: 2605.04202 by Lina Alkarmi, Mingyan Liu, Ziyuan Huang.

**Figure 1.** Figure 1: The evolution of the agent’s level. The complete state is given by st := (it, xt). The process {st}t is a discrete-time Markov process with an uncountable state space and potentially time-homogeneous state transitions (if the action at time t is completely determined by st), as we will focus on for the rest of the paper. We will shortly show that when the agent adopts a myopic strategy, {st}t, conditioned … view at source ↗

**Figure 2.** Figure 2: Agent’s optimal actions (red arrows). Our result in this first step is a relatively minor generalization of a result in [17]. For this reason, we will focus on illustrating the intuition behind the result and have included the technical details in Appendix B. Fig. 2a illustrates a two-dimensional action space where the return on investment (ROI) differs: θ1 c1 > θ2 c2 . For an attribute x below the decis… view at source ↗

**Figure 3.** Figure 3: Illustration of G(i, ·) (left, with z being the x-axis) and a¯(i, ·) (right, with x being the x-axis). When G(i, ·) has multiple local maximizers, a¯(i, ·) can exhibit disconnected positive intervals on x, as shown in view at source ↗

**Figure 4.** Figure 4: Relationship between wi and wi . Proposition 3.2 says that (as a consequence of adopting Assumption 3.1) for non-terminal levels (i < I), the agent exerts effort (either improvement or gaming) primarily for promotion. For the terminal level I, however, demotion probability may exceed promotion probability, reflecting the agent’s shifted focus to status maintenance instead of advancement. Reduction to cou… view at source ↗

**Figure 5.** Figure 5: A countable state space with xt plotted along the horizontal axis (increasing to the right) and it along the vertical axis (increasing downward). Each rectangle corresponds to the interval [µi , µi ], though µi and µi themselves are in general not part of the state space. 4 Incentivizing Improvement in the Long Term In this section we will analyze and compare two types of greedy agents (or policies): one t… view at source ↗

**Figure 6.** Figure 6: Long-term properties across different numbers of levels ( view at source ↗

**Figure 7.** Figure 7: Long-term properties under Mix(ρ) versus ρ (probability of no gaming). As a further generalization, we consider the optimal pure strategy policy where the agent can choose either a +, a −, or both, to maximize the following discounted reward over a large finite horizon: E " η T X−1 t=0 view at source ↗

**Figure 8.** Figure 8: Average level across different level counts with fixed M = 3 under No Gaming (NG) and No Improvement (NI) strategies ±2 STD view at source ↗

**Figure 9.** Figure 9: Average utility across different level counts with fixed M = 3 under No Gaming (NG) and No Improvement (NI) strategies ±2 STD view at source ↗

**Figure 10.** Figure 10: Average qualification across different level counts with fixed M = 3 under No Gaming (NG) and No Improvement (NI) strategies ±2 STD. 27 view at source ↗

**Figure 11.** Figure 11: Truly optimal policies across different numbers of levels. view at source ↗

**Figure 12.** Figure 12: Stationary distributions over levels under view at source ↗

**Figure 13.** Figure 13: Stationary distributions over levels under view at source ↗

**Figure 14.** Figure 14: Stationary distributions over levels under view at source ↗

**Figure 15.** Figure 15: Agent’s average level and utility versus view at source ↗

**Figure 16.** Figure 16: Average level versus α for NG and NI agents under different abstention functions ±2 STD. 4 5 6 7 8 9 10 11 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Average Utility Entropy Absval Parab (a) NG agents 4 5 6 7 8 9 10 11 8.02 8.04 8.06 8.08 8.10 8.12 8.14 Average Utility Entropy Absval Parab (b) NI agents view at source ↗

**Figure 17.** Figure 17: Average utility versus α under different abstention functions ±2 STD. 4 5 6 7 8 9 10 11 0.48 0.50 0.52 0.54 0.56 Average Qualification Entropy Absval Parab (a) NG agents 4 5 6 7 8 9 10 11 0.04 0.02 0.00 0.02 0.04 Average Qualification Entropy Absval Parab (b) NI agents view at source ↗

**Figure 18.** Figure 18: Average qualification versus α under different abstention functions ±2 STD. 31 view at source ↗

**Figure 19.** Figure 19: Average level versus β˜ for NG and NI agents under different abstention functions ±2 STD. 0.1 0.2 0.3 0.4 0.5 0.6 7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4 Average Utility Entropy Absval Parab (a) NG agents 0.1 0.2 0.3 0.4 0.5 0.6 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Average Utility Entropy Absval Parab (b) NI agents view at source ↗

**Figure 20.** Figure 20: Average utility versus β˜ for NG and NI agents under different abstention functions ±2 STD. 0.1 0.2 0.3 0.4 0.5 0.6 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Average Qualification Entropy Absval Parab (a) NG agents 0.1 0.2 0.3 0.4 0.5 0.6 0.04 0.02 0.00 0.02 0.04 Average Qualification Entropy Absval Parab (b) NI agents view at source ↗

**Figure 21.** Figure 21: Average qualification versus β˜ for NG and NI agents under different abstention functions ±2 STD view at source ↗

read the original abstract

Strategic classification studies the problem where self-interested individuals or agents manipulate their response to obtain favorable decision outcomes made by classifiers, typically turning to dishonest actions when they are less costly than genuine efforts. Prior works have demonstrated a fundamental inability to get out of this conundrum by only focusing on the design of a classifier. We note that prior work also heavily focuses on either one-shot settings or repeated interaction with the same classifier. Real-world decision making is often multi-stage, involving a sequence of potentially different classifiers as an agent progresses. This paper introduces a sequential, stochastic, multi-stage model of strategic classification, by capturing how agents adapt their behavior, through improvement actions (enhancing both observable features and true attributes) and gaming actions (enhancing only observable features), over multiple levels of classification with increasing difficulty as well as reward. For each level, we adopt a selective classifier that can abstain from making a prediction at low confidence. Consequently, a positive (resp. negative) outcome leads to promotion (resp. demotion) of the agent to the next higher (resp. lower) level, while abstention keeps the agent at the same level. We fully characterize the agent's optimal instantaneous action under selective classifiers and compare the long-term properties and utility of the agent repeatedly following an optimal myopic policy of either no-improvement (never choose the improvement action) or no-gaming (never choose the gaming action). We further examine design principles over the sequence of classifiers that yield higher long-term utility for the latter policy, thereby effectively incentivizing genuine effort in the long run.

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.

Desk Editor's Note private letter to a colleague

This extends strategic classification to multi-stage settings with abstention and compares myopic no-improvement versus no-gaming policies, but the closed forms require myopic agents and separable costs.

read the letter

The paper's core move is to model agents progressing through a sequence of classifiers with increasing difficulty, where each can abstain and agents either improve their true attributes or just game the features. Promotion or demotion follows the outcome, creating a Markov chain across levels. They characterize the agent's best instantaneous choice between improvement and gaming under the selective classifier, then compare long-term utilities when the agent sticks to one of two myopic rules: never improve, or never game. They also sketch how to set the sequence of classifiers so the no-gaming rule yields higher long-run utility, which could push agents toward real effort instead of manipulation. That sequential formulation with abstention and the explicit policy comparison is the new piece relative to the one-shot literature. The selective classifier adds a practical touch by letting low-confidence cases hold the agent in place rather than forcing a binary call. The main limitation is that the clean characterization and the design principles both rest on agents optimizing myopically at each stage and on improvement and gaming having fully separable costs. If agents anticipate future stages or if costs overlap, the instantaneous best response changes and the long-term comparisons no longer hold in closed form. The abstract states the results as full characterizations, so the derivations will need to be checked for hidden assumptions. This is useful reading for people building progressive decision systems in education or hiring, where agents face repeated evaluations. It deserves a serious referee because it addresses a realistic gap, even though the assumptions will require explicit discussion and sensitivity checks in revision.

Referee Report

2 major / 2 minor

Summary. The paper introduces a multi-stage sequential model of strategic classification in which agents interact with a sequence of selective classifiers (that may abstain) of increasing difficulty. At each stage agents may take an improvement action (affecting both observable features and the true label) or a gaming action (affecting only features), with promotion or demotion determined by the classifier outcome. The central claims are that the agent's optimal instantaneous (myopic) action can be fully characterized in closed form, that the long-term utilities of the repeated no-improvement and no-gaming myopic policies can be compared, and that design principles on the sequence of classifiers exist that yield higher long-term utility for the no-gaming policy, thereby incentivizing genuine effort.

Significance. If the closed-form characterization and long-term comparisons hold under the stated assumptions, the work provides a concrete mechanism-design lens on how selective classifiers can be sequenced to favor improvement over gaming in repeated interactions. The use of abstention to induce a Markovian promotion/demotion process and the explicit comparison of myopic policy utilities are technically distinctive contributions to the strategic-classification literature.

major comments (2)

[Abstract; characterization section] Abstract and the section deriving the optimal instantaneous action: the claimed full characterization of the agent's best response under selective classifiers is obtained only under the joint assumptions of myopic per-stage optimization and separable cost structures for improvement versus gaming actions. When agents solve a finite-horizon dynamic program that anticipates future stages, or when improvement effort reduces the marginal cost of subsequent gaming, the instantaneous best response depends on continuation values and the closed-form expressions no longer hold; this directly undermines the subsequent long-term utility comparisons and the design principles for incentivizing no-gaming.
[Long-term properties section] Section on long-term properties and design principles: the stationary-utility comparison between the repeated no-improvement and no-gaming myopic policies, as well as the claimed classifier-sequence rules that favor genuine effort, are derived under the Markovian process induced by the myopic best responses. The paper should supply either a robustness argument or an explicit counter-example showing when forward-looking behavior or non-separable costs invalidate these stationary comparisons.

minor comments (2)

[Model section] Notation for the selective classifier's abstention threshold and the promotion/demotion transition probabilities should be introduced with a single consolidated table or diagram to improve readability.
[Related work] The paper would benefit from a brief discussion of how the selective-classifier abstention rule relates to existing work on selective classification in non-strategic settings.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below, clarifying the scope of our myopic analysis while agreeing to strengthen the presentation of assumptions.

read point-by-point responses

Referee: [Abstract; characterization section] Abstract and the section deriving the optimal instantaneous action: the claimed full characterization of the agent's best response under selective classifiers is obtained only under the joint assumptions of myopic per-stage optimization and separable cost structures for improvement versus gaming actions. When agents solve a finite-horizon dynamic program that anticipates future stages, or when improvement effort reduces the marginal cost of subsequent gaming, the instantaneous best response depends on continuation values and the closed-form expressions no longer hold; this directly undermines the subsequent long-term utility comparisons and the design principles for incentivizing no-gaming.

Authors: We agree that the closed-form characterization applies specifically under myopic per-stage optimization and separable costs, as the paper consistently frames the agent's behavior in terms of instantaneous myopic best responses (see abstract: 'optimal instantaneous action' and 'repeatedly following an optimal myopic policy'). The long-term utility comparisons are between the stationary utilities obtained by repeating these myopic policies, which induce the Markovian promotion/demotion process. We acknowledge that forward-looking agents solving a dynamic program or agents with non-separable costs would generally have best responses that depend on continuation values, rendering the closed-form expressions invalid. To address this, we will revise the abstract and characterization section to more explicitly foreground these assumptions and add a dedicated paragraph in the long-term properties section discussing the limitations and positioning non-myopic extensions as future work. This does not alter the validity of the results within the myopic setting but improves clarity. revision: partial
Referee: [Long-term properties section] Section on long-term properties and design principles: the stationary-utility comparison between the repeated no-improvement and no-gaming myopic policies, as well as the claimed classifier-sequence rules that favor genuine effort, are derived under the Markovian process induced by the myopic best responses. The paper should supply either a robustness argument or an explicit counter-example showing when forward-looking behavior or non-separable costs invalidate these stationary comparisons.

Authors: The stationary comparisons and design principles are derived under the myopic best-response Markov chain. We will add a robustness paragraph explaining that the myopic assumption is maintained for tractability, enabling closed-form characterization and explicit design rules on classifier sequences; relaxing it to finite-horizon dynamic programming would require solving a more complex MDP whose stationary utilities lack the same closed-form structure. While we do not construct an explicit counter-example (as that would necessitate specifying particular continuation values and non-separable cost functions outside the current model), the added discussion will delineate the precise conditions under which the results hold and note that the incentive-design principles are intended for the myopic separable-cost regime. revision: partial

Circularity Check

0 steps flagged

No circularity: optimal-action characterization derived from explicit cost/utility primitives and myopic optimization

full rationale

The paper defines improvement and gaming costs as separable primitives, specifies per-stage utilities, and adopts a selective classifier with explicit abstention rule. It then solves the agent's per-stage best-response optimization under these inputs to obtain the closed-form instantaneous action characterization. The subsequent long-term utility comparisons for repeated no-improvement vs. no-gaming policies follow directly from iterating that best response in the induced Markov chain. No step equates a derived quantity to a fitted parameter, renames an input as a prediction, or relies on a self-citation whose content is itself unverified; the results are therefore self-contained consequences of the stated model rather than tautological restatements of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard game-theoretic assumptions about rational myopic agents and separable action costs; no free parameters or invented entities are mentioned in the abstract.

axioms (2)

domain assumption Agents choose actions myopically at each stage based on instantaneous utility.
Long-term policy comparisons rely on repeated application of the instantaneous optimal action.
domain assumption Improvement and gaming actions have distinct cost structures that permit closed-form characterization of optimal responses to selective classifiers.
The full characterization of instantaneous actions depends on this separability.

pith-pipeline@v0.9.0 · 5581 in / 1373 out tokens · 31494 ms · 2026-05-08T17:24:51.809158+00:00 · methodology

discussion (0)

Reference graph

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