Multi-Level Strategic Classification: Incentivizing Improvement Through Promotion and Relegation Dynamics¶

Conference: ICML 2026
arXiv: 2602.11439
Code: None
Area: Strategic Classification / Mechanism Design / Algorithmic Fairness
Keywords: Strategic Classification, Multi-level Mechanisms, Promotion-Relegation, Markov Decision Processes, Incentive Compatibility

TL;DR¶

This paper extends traditional one-shot "strategic classification" into a sequential mechanism composed of multi-level ternary classifiers (pass/abstain/fail = promotion/stay/relegation). It proves that by leveraging three intertemporal effects — the discount factor \(\beta\), skill retention rate \(\gamma\), and "leg-up gain" \(\delta\) — the non-incentivizable region can be shrunk from \(c^+>c^-\) to \((1-\beta\gamma)c^+>c^-\). Furthermore, it provides a steady-state threshold sequence \(\mu_l = \delta(l-1)/(1-\gamma)\), demonstrating that under mild conditions, honest effort can be incentivized to push attributes to arbitrarily high levels.

Background & Motivation¶

Background: The main body of strategic classification research involves decision-makers deploying classifiers while self-interested individuals choose between "honest improvement" or "low-cost gaming." The classical conclusion is highly negative: in a single-shot interaction, when the cost of gaming is strictly lower than the cost of real improvement (\(c^- < c^+\)), rational agents will always choose gaming unless additional tools like external subsidies or penalties are introduced.

Limitations of Prior Work: (1) Single-shot models treat agents as memoryless optimizers, ignoring the intertemporal coupling in real-world scenarios where "yesterday's effort affects today's state." (2) Most existing sequential strategic classification studies focus on "how to dynamically update classifier weights" but lack a systematic characterization of threshold design, level progression, and the "marginal gain from promotion." (3) Classic works (e.g., Harris et al. 2021) involve sequential regression but do not consider discrete feedback from classification nor introduce skill depreciation.

Key Challenge: To make agents voluntarily choose the more expensive honest improvement, there must exist a "future reward magnifier." Single-shot mechanisms or those relying solely on weight adjustments lack an explicit expression of such a magnifier.

Goal: (1) Formalize a multi-level sequential mechanism with promotion-relegation dynamics. (2) Fully characterize the agent's optimal long-term strategy under two levels (a single classifier). (3) Provide feasibility conditions and optimal solutions for multi-level threshold designs, proving that honest improvement can push attributes to an arbitrary level \(M\).

Key Insight: The authors observe that positive feedback like "exam \(\to\) promotion \(\to\) more resources \(\to\) easier to pass higher levels" naturally exists in reality. They explicitly characterize this as a "leg-up" factor \(\delta\). By overlaying retention \(\gamma\) and farsightedness \(\beta\), these three factors collectively lower the "effective long-term unit cost" of improvement.

Core Idea: Construct a level-advancement mechanism using ternary classifiers (pass/abstain/fail), rewriting the economic meaning of "honest cost \(c^+\)" as \((1-\beta\gamma)c^+\), and using \(\delta\) to provide a continuous upward pull, thereby incentivizing improvement without external subsidies.

Method¶

All dynamics occur on a continuous-state, continuous-action MDP \(\{(l_t, x_t)\}_{t\ge0}\): \(l_t\) is the discrete level, \(x_t\ge0\) is the private attribute (unobservable), and \(z_t = x_t + a_t^+ + a_t^-\) is the observable feature. In each step, the agent simultaneously chooses the improvement amount \(a_t^+\ge0\) (unit cost \(c^+\)) and gaming amount \(a_t^-\ge0\) (unit cost \(c^-<c^+\)). Both contribute equally to the feature \(z_t\), but only \(a_t^+\) truly improves the attribute.

Overall Architecture¶

After a single-step action, the attribute immediately becomes \(x_{t_+} = x_t + a_t^+\). It then undergoes two corrections before the next step: depreciation \(\gamma\in(0,1)\) scales the attribute to \(\gamma x_{t_+}\), and a level-dependent leg-up gain \(\delta(l_{t+1}-1)\) is added. Combined: \(x_{t+1}=\gamma x_{t_+}+\delta(l_{t+1}-1)\). The classifier is a ternary function indexed by level: if \(\theta z_t \ge \mu_{l+1}\), the agent is promoted; if \(\mu_l \le \theta z_t < \mu_{l+1}\), the agent stays; if \(\theta z_t \le \mu_l\), the agent is relegated (boundary levels are one-way). Without loss of generality, let \(\theta=1, \mu_1\equiv0\). The agent's goal is to maximize the infinite-horizon discounted total reward \(\sum_t \beta^t (R_{l_{t+1}} - \vec c^\top \vec a_t)\), where \(R_l = r(l-1)\) is linearly related to the level. The Principal's task is to design the shortest threshold sequence \(\vec\mu\) such that (i) the agent never games; (ii) the long-term attribute \(\liminf_t x_{t_+}\ge M\); (iii) the agent eventually reaches the highest level.

Key Designs¶

1. Ternary Multi-Level Mechanism: Translating "One-shot Non-incentivizable" into Adjustable Geometric Constraints

In single-shot models, agents have no future returns; if gaming is cheaper (\(c^-<c^+\)), it is always dominant. This is the most classic and pessimistic impossibility result in strategic classification. The breakthrough here is making each level a selective classifier that allows for abstention: passing \(\theta z_t\ge\mu_{l+1}\) leads to promotion, \(\mu_l\le\theta z_t<\mu_{l+1}\) allows staying, and \(\theta z_t\le\mu_l\) results in relegation. The original binary "pass/fail" is extended to a ternary "promotion/stay/relegation"—abstention is no longer a statistical choice but an economic decision to "stay at the current level." This abstraction degrades thresholds from "ultimate outcomes" to "short-term hurdles," giving the mechanism designer leverage to map intertemporal effects into static constraints.

This leverage is provided by three intertemporal effects with clear economic interpretations: the discount factor \(\beta\) represents the agent's concern for the future, the retention rate \(\gamma\) reflects skill depreciation, and the leg-up \(\delta\) represents resource overflow from higher levels. Proposition 2.1 compresses these into a clean effective cost expression—as long as \((1-\beta\gamma)c^+<c^-\), a design exists that makes the agent voluntarily improve. This strictly shrinks the non-incentivizable region from \(c^+>c^-\) to \((1-\beta\gamma)c^+>c^-\). In other words, future rewards depress the "effective long-term unit cost" of honest improvement from \(c^+\) to \((1-\beta\gamma)c^+\).

2. Full Phase Diagram of Optimal Two-Level Strategies: Providing Solvable Atomic Components

To use dynamic programming on a multi-level mechanism, one must first know "how the agent responds at a certain level." The two-level (single classifier) case is the smallest solvable instance and serves as the sub-problem for each step of the subsequent greedy algorithm (Theorem 5.1). This paper solves it completely into a phase diagram. Theorem 3.1 handles low-threshold scenarios: when \(\mu<\delta/(1-\gamma)\), a critical point \(x^\circ\in[0,\mu]\) exists; the agent purely games in the \([x^\circ,\mu]\) interval and uses a mix of improvement and gaming in \([0,x^\circ)\). Theorem 3.2 handles high-threshold scenarios, providing two constants \(\underline\mu,\overline\mu\) independent of \(\delta\), further dividing \(\mu\ge\delta/(1-\gamma)\) into three segments: pure improvement near the threshold, otherwise inaction; middle segment follows "improve when near, game when intermediate, stay when far"; when \(\mu\ge\overline\mu+\delta/(1-\gamma)\), improvement is no longer worth it, leaving only gaming or quitting.

The phase diagram hides a key asymmetry: both \(\underline\mu\) and \(\overline\mu\) increase monotonically with \(\beta\) and \(\gamma\), but \(\gamma\to1\) pushes both to infinity, completely eliminating the pure gaming zone, while \(\beta\to1\) only pushes them to a finite upper bound \(r/((1-\gamma)c^+)\). This means skill retention is more effective than farsightedness in eliminating gaming.

3. Steady-State Threshold Sequence \(\mu_l=\delta(l-1)/(1-\gamma)\): Pinning Thresholds to Natural Equilibria

The remaining problem is how to arrange the threshold sequence to push attributes to a target \(M\). Naive intuition suggests "small steps and fast runs" with dense thresholds to encourage climbing, but too much density allows agents to snowball just via leg-up (promotion begets promotion), while too much sparsity allows depreciation to consume attributes. The simplest closed-form sequence provided in this paper sets the threshold exactly at the attribute steady-state where the agent would naturally converge if they stayed at that level:

\[\mu_l = \frac{\delta(l-1)}{1-\gamma}\]

The beauty of this value is that the depreciation term \(-\gamma\mu_l\) and the leg-up term \(+\delta(l-1)\) cancel out exactly—the Principal is leveraging natural equilibrium rather than fighting it. Theorem 4.2 gives the feasibility boundary: when \(\delta>0\) and \(r<\frac{1-\beta}{1-\gamma}c^+\delta\), it is infeasible for any \(M\); otherwise, when \(r\ge\frac{1-\beta}{1-\gamma}c^+\delta\) and

\[c^-\ge\max\Big\{(1+\tfrac{\beta\gamma}{2})(1-\beta\gamma)c^+,\ \beta\gamma(1-\beta^2\gamma^2)c^+\Big\}\]

it is feasible. The required number of levels is only \(L=\lceil(1-\gamma)M/\delta\rceil\), and when \(r\) takes the boundary value, this sequence is optimal. As a contrast, without leg-up (\(\delta=0\)), Theorem 4.1 gives a hard infeasibility upper bound \(M\ge r/((1-\beta)(1-\gamma)^2c^+)\), where the square of \((1-\gamma)\) further confirms the dominant role of retention.

Loss & Training¶

There is no learning loss. The Agent's side uses ValueIterate (value iteration + attribute space discretization + linear interpolation) to solve the MDP, proving a convergence rate of \(O(\log(1/\varepsilon)/|\log\beta|)\) and a value function error bound of \(c^+\Delta x/(2(1-\beta))\). The Principal's side uses CMA-ES for black-box optimization under relaxed targets, complemented by a greedy threshold search algorithm (Algorithm 1, which Theorem 5.1 guarantees to be feasible when \(M\le \mu_L\)).

Key Experimental Results¶

Main Results¶

FICO credit score data (normalized to \([0,10]\)) simulates a multi-level credit product system. Fixing \(\beta=\gamma=0.8, \delta=0.01, \alpha=0.95, \xi=0.01, \lambda=5\), searching for the Principal's optimal design for \(L\in[2,8]\):

Case	\((c^+, c^-)\)	\(L^*\)	\(r^*\)	\(\mu_L^*\)	\(U^*\)
I Easy to learn, hard to game	(0.8, 0.7)	6	1.80	10.76	630.4
II High costs for both	(1.5, 1.2)	7	2.51	11.92	629.9
III Easy to learn, easy to game	(0.8, 0.4)	2	4.48	11.98	628.8
IV Hard to learn, easy to game	(1.5, 0.4)	8	0.63	7.98	107.9

Ablation Study¶

Config	Key Phenomenon	Explanation
Full Mechanism (Case I)	Pure improvement throughout, monotonic attribute rise	Incentive alignment achieved, ideal trajectory reached
Lack of Reward (Case IV)	\(r^*\) suppressed to 0.63, gaming dominates	Mechanism degrades when Assumption 2.2 is violated
Discount \(\beta\to0\)	Non-incentivizable region reverts to \(c^+>c^-\)	Loss of intertemporal effects
Retention \(\gamma\to1\)	\(\underline\mu,\overline\mu\to\infty\), pure gaming zone disappears	Maximum freedom for Principal when skills don't depreciate
\(\delta=0\)	Constrained by Theorem 4.1 bound \(r/((1-\beta)(1-\gamma)^2c^+)\)	Hard upper limit exists without leg-up

Key Findings¶

While the non-incentivizable region in single-shot problems is \(c^+>c^-\), multi-level mechanisms shrink it to \((1-\beta\gamma)c^+>c^-\). The tight geometric shrinkage given by the theorem is empirically validated by phase transitions in FICO experiments.
\(\gamma\) (skill retention) is more effective than \(\beta\) (discount factor)—the former drives a quadratic expansion of \(\underline\mu, \overline\mu\) to eliminate gaming, while the latter only linearly approaches a finite upper bound.
The threshold sequence \(\mu_l = \delta(l-1)/(1-\gamma)\) is approximately optimal for large \(M\), and empirically loses almost no efficiency even when leg-up is weak (\(\delta\) is small), indicating that pinning thresholds to natural equilibria is a robust choice.
Case III reveals a counter-intuitive phenomenon: when gaming costs are far below improvement costs, the optimal design is to compress levels to \(L=2\) and set extremely high thresholds, using a large one-time hurdle rather than a gradual ladder to prevent persistent gaming.

Highlights & Insights¶

Using "ternary classification + multi-level" as a mechanism design primitive is a clever encapsulation: abstention naturally maps to "staying at the level," translating the statistical motivation of abstaining into economic meaning for the agent.
The effective cost expression \((1-\beta\gamma)c^+\) in Proposition 2.1 is clean enough to serve directly as policy guidance—by calculating a system's discount and depreciation rates, one can immediately judge if incentive alignment is feasible.
Writing the intuition of "pinning thresholds to natural steady-states" as a closed-form \(\mu_l = \delta(l-1)/(1-\gamma)\) avoids re-running convex optimization for every \(M\), making it engineering-ready.
Economic interpretation permeates the text: every theorem provides intuitive explanations, allowing readers with ML backgrounds to map results to real-world scenarios (degrees, credit ratings, certifications).

Limitations & Future Work¶

The model assumes attributes and features are scalars. While the authors state this can be generalized to multi-dimensional cases, no such analysis is provided. In reality, qualifications for credit or education are almost certainly high-dimensional vectors.
The ternary classifier assumes uniform model weights \(\theta\), implicitly assuming \(\theta\) is estimated from non-strategic data; when strategic feedback pollutes the training distribution, the bias in \(\theta\) could void the theoretical guarantees.
Experiments are restricted to synthetic and FICO data, lacking longitudinal validation on real sequential tasks. There is also no discussion on agent heterogeneity or group fairness.
Future directions: Incorporate weights \(\theta\) and thresholds \(\vec\mu\) into a joint sequential design, and explicitly model agent type distributions. Re-characterize steady-states under realistic "reset" events (e.g., job hopping).

vs. Harris et al. 2021: They focus on sequential regression + effort accumulation without discrete classification feedback or modeling depreciation and leg-up; this paper explicitly incorporates these intertemporal effects into an MDP.
vs. Hardt et al. 2015 / Milli et al. 2019: The negative conclusion \(c^+>c^-\) in single-shot strategic classification is strictly weakened to \((1-\beta\gamma)c^+>c^-\) here, marking a rare result where mechanism design itself breaks the incentive wall.
vs. Jin et al. 2022: They rely on external subsidies to break non-incentivizability, requiring extra budget; this paper proves that a multi-level mechanism with natural leg-up can achieve similar effects without monetary transfers.
vs. Kleinberg & Raghavan 2019: They focus on the topological characterization of effort incentives, while this paper provides computable threshold sequences and numerical experiments.

Rating¶

Novelty: ⭐⭐⭐⭐⭐ Integrating "multi-level, ternary, leg-up + retention + farsightedness" into a single sequential framework with closed-form optimal solutions is highly distinct.
Experimental Thoroughness: ⭐⭐⭐ FICO and synthetic experiments validate the theory well, but lack real longitudinal data and multi-dimensional attribute extensions.
Writing Quality: ⭐⭐⭐⭐ Theorems and economic explanations coordinate well, and the phase diagrams are intuitive. Some key proofs are in the appendix, requiring jumping.
Value: ⭐⭐⭐⭐ Provides a directly applicable analytical framework and design principles for multi-level decision systems like education, credit, and certification.