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 single-shot "strategic classification" into a sequential mechanism composed of multi-level ternary classifiers (pass/abstain/fail = promotion/retention/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 reduced 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." A classic conclusion is highly negative: in a single-shot interaction, when the cost of gaming is strictly lower than the cost of real improvement (\(c^- < c^+\)), a rational agent will always choose to game the system unless external tools like subsidies or penalties are introduced.

Limitations of Prior Work: (1) Single-shot models treat individuals as memoryless optimizers, ignoring the intertemporal coupling where "yesterday's effort affects today's state"; (2) Existing sequential strategic classification research mostly focuses on "how to dynamically update classifier weights," lacking systematic characterization of threshold design, level progression, and "marginal gains from promotion"; (3) Classic work (Harris et al. 2021), while involving sequential regression, considers neither discrete classification feedback nor skill depreciation.

Key Challenge: To make agents voluntarily choose the more expensive honest improvement, there must exist some "future reward amplifier." Single-shot or weight-adjustment-only mechanisms lack an explicit representation of such an amplifier.

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

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

Core Idea: Construct a level progression mechanism using ternary classifiers (pass/abstain/fail), redefining the economic implication of "honest cost \(c^+\)" as \((1-\beta\gamma)c^+\), and using \(\delta\) to provide continuous upward pulling, 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, \(x_t \ge 0\) is the private attribute (unobservable), and \(z_t = x_t + a_t^+ + a_t^-\) is the observable feature. At each step, the agent simultaneously chooses an improvement amount \(a_t^+ \ge 0\) (unit cost \(c^+\)) and a gaming amount \(a_t^- \ge 0\) (unit cost \(c^- < c^+\)). Both contribute equally to the feature \(z_t\), but only \(a_t^+\) genuinely improves the attribute.

Overall Architecture

After a single-step action, the attribute immediately becomes \(x_{t_+} = x_t + a_t^+\). Before entering the next step, it undergoes two modifications: 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 (abstain); if \(\theta z_t \le \mu_l\), the agent is relegated (boundary levels are one-way). Without loss of generality, let \(\theta=1, \mu_1 \equiv 0\). 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 \(R_l = r(l-1)\) scales linearly with 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 \(\liminf_t x_{t_+} \ge M\); and (iii) the highest level is eventually reached.

Key Designs

1. Ternary Multi-level Mechanism + Three Intertemporal Effects:

   • Function: Decomposes the "single-shot incentive impossibility" into adjustable geometric conditions, allowing the mechanism designer to reshape the non-incentivizable region by controlling the number of levels \(L\), the threshold sequence \(\vec\mu\), and the reward rate \(r\).
   • Mechanism: Each level acts as a selective classifier capable of rejection (abstain keeps the agent at the current level), extending "pass/fail" into "promotion/retention/relegation." The three intertemporal effects provide economic interpretations: discount \(\beta\) denotes future-oriented agents; retention \(\gamma\) reflects skill depreciation; leg-up \(\delta\) is the resource spillover from higher levels. Proposition 2.1 proves that as long as \((1-\beta\gamma)c^+ < c^-\), a design exists that makes agents choose improvement—effectively shrinking the single-shot non-incentivizable region \(c^+ > c^-\) to \((1-\beta\gamma)c^+ > c^-\).
   • Design Motivation: In single-shot models, agents have no future returns, making gaming always dominant. Introducing the ternary abstraction (letting abstain lead to retention rather than just a non-decision) transforms thresholds into "short-term hurdles" rather than "ultimate outcomes," giving the designer leverage to map intertemporal effects into static constraints.

2. Full Characterization of Two-Level Agent Optimal Strategy:

   • Function: Condenses "how the agent will react" from a numerical problem into a readable phase diagram, providing atomic components for subsequent multi-level design.
   • Mechanism: Theorem 3.1 handles the low-threshold case—when \(\mu < \delta/(1-\gamma)\), there exists \(x^\circ \in [0, \mu]\) such that the agent purely games in the interval \([x^\circ, \mu]\) and mixes improvement and gaming in \([0, x^\circ)\). Theorem 3.2 handles the high-threshold case, identifying two constants \(\underline\mu, \overline\mu\) independent of \(\delta\), further dividing \(\mu \ge \delta/(1-\gamma)\) into three segments: (1) in \([\delta/(1-\gamma), \underline\mu+\delta/(1-\gamma))\), the agent purely improves when close enough, otherwise stays still; (2) in the middle segment, the agent chooses "improve if near, game if medium, stay still if far"; (3) in \([\overline\mu+\delta/(1-\gamma), \infty)\), improvement is no longer worthwhile, leading to either gaming or giving up. Notably, \(\underline\mu, \overline\mu\) increase with \(\beta, \gamma\), but \(\gamma \to 1\) pushes both to infinity, while \(\beta \to 1\) only pushes them to \(r/((1-\gamma)c^+)\)—retention is more effective than farsightedness in eliminating gaming.
   • Design Motivation: To use dynamic programming in multi-level design, one must first know if the agent will respond as intended at each level. The two-level case is the smallest solvable instance of a single classifier and serves as the subproblem for each step of the greedy algorithm (Theorem 5.1).

3. Steady-State Threshold Sequence \(\mu_l = \delta(l-1)/(1-\gamma)\) and Feasibility Boundaries:

   • Function: Provides a simple closed-form threshold sequence, proving that as long as reward \(r\) and gaming cost \(c^-\) exceed explicit thresholds, the agent can be incentivized to honestly improve attributes to any \(M\) within \(L = \lceil(1-\gamma)M/\delta\rceil\) levels.
   • Mechanism: Theorem 4.2 proves that when \(\delta > 0\) and \(r < \frac{1-\beta}{1-\gamma}c^+\delta\), the problem is infeasible for any \(M\). Conversely, when \(r \ge \frac{1-\beta}{1-\gamma}c^+\delta\) and \(c^- \ge \max\{(1+\beta\gamma/2)(1-\beta\gamma)c^+, \beta\gamma(1-\beta^2\gamma^2)c^+\}\), it is feasible with the sequence \(\mu_l = \delta(l-1)/(1-\gamma)\); this sequence is optimal when \(r\) is at the boundary. Its economic meaning: setting each level's threshold exactly at the attribute steady-state where the agent would naturally converge—where depreciation \(-\gamma\mu_l\) and leg-up \(+\delta(l-1)\) exactly cancel out—ensures attributes do not drop after promotion. The Principal leverages the natural equilibrium rather than fighting it. For the case without leg-up (\(\delta=0\)), Theorem 4.1 gives an absolute infeasibility boundary \(M \ge r/((1-\beta)(1-\gamma)^2 c^+)\), highlighting the quadratic effect of \(\gamma\).
   • Design Motivation: Naive intuition might suggest "small, fast steps" to encourage climbing, but too high density allows agents to snowball via leg-up alone (promotion begets promotion), while too low density lets depreciation consume the attribute. The steady-state threshold is the unique fixed point where the agent is neither pulled down nor incentivized to stagnate, avoiding tedious parameter tuning.

Loss & Training

No learning loss. The Agent solves the MDP using ValueIterate (value iteration + attribute space discretization + linear interpolation), with a proved convergence rate of \(O(\log(1/\varepsilon)/|\log\beta|)\) and a value function error bound of \(c^+\Delta x/(2(1-\beta))\). The Principal uses CMA-ES for black-box optimization under relaxed objectives, paired with a greedy threshold search algorithm (Algorithm 1), guaranteed by Theorem 5.1 to return a feasible sequence 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. Fixed parameters: \(\beta=\gamma=0.8, \delta=0.01, \alpha=0.95, \xi=0.01, \lambda=5\). 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-cheat	(0.8, 0.7)	6	1.80	10.76	630.4
II High-cost for both	(1.5, 1.2)	7	2.51	11.92	629.9
III Easy-to-learn, Easy-to-cheat	(0.8, 0.4)	2	4.48	11.98	628.8
IV Hard-to-learn, Easy-to-cheat	(1.5, 0.4)	8	0.63	7.98	107.9

Ablation Study

Configuration	Key Phenomenon	Description
Full Mechanism (Case I)	Pure improvement throughout, monotonic attribute rise	Incentive alignment holds, reaching ideal trajectory
Lacking Rewards (Case IV)	\(r^*\) squeezed to 0.63, gaming dominates	Mechanism degrades when Assumption 2.2 fails
Discount \(\beta \to 0\)	Non-incentivizable region returns to \(c^+ > c^-\)	Loss of intertemporal effects
Retention \(\gamma \to 1\)	\(\underline\mu, \overline\mu \to \infty\), gaming zone disappears	Maximum freedom for Principal when skills don't decay
\(\delta = 0\)	Constrained by Theorem 4.1 bound \(r/((1-\beta)(1-\gamma)^2c^+)\)	Hard limit exists without leg-up

Key Findings

• The non-incentivizable region \(c^+ > c^-\) in single-shot problems is reduced to \((1-\beta\gamma)c^+ > c^-\) in the multi-level mechanism. This tight geometric reduction is empirically validated by the phase transition in FICO experiments (incentive capability suddenly vanishes when gaming cost crosses \((1-\beta\gamma)c^+\)).
• \(\gamma\) (skill retention) is more effective than \(\beta\) (discount factor)—the former drives quadratic expansion of \(\underline\mu, \overline\mu\) to eliminate gaming zones, while the latter only provides linear approximation to a finite limit.
• The threshold sequence \(\mu_l = \delta(l-1)/(1-\gamma)\) is approximately optimal for large \(M\), and empirically loses little efficiency even when leg-up is weak (small \(\delta\)), suggesting that pinning thresholds to natural equilibria is a robust choice.
• Case III reveals a counter-intuitive phenomenon: when gaming costs are much lower than improvement costs, the optimal design collapses levels to \(L=2\) and sets an extremely high threshold, using a single massive hurdle rather than gradual steps to deter continuous gaming.

Highlights & Insights

• Packaging "ternary classification + multi-level" as a mechanism design primitive is a clever abstraction: "abstain" naturally maps to "retention," translating statistical rejection into economic meaning for the agent.
• The expression \((1-\beta\gamma)c^+\) for effective cost in Proposition 2.1 is clean enough to serve directly as policy guidance—by calculating the system's discount and depreciation rates, one can immediately judge if incentives are feasible.
• Formalizing the intuition of "pinning thresholds at natural steady-states" into the closed-form \(\mu_l = \delta(l-1)/(1-\gamma)\) avoids re-running convex optimization for every \(M\), making it highly deployable.
• The economic interpretation is pervasive: every theorem provides intuitive explanations, allowing readers from ML backgrounds to easily map the theory to real-world scenarios (degrees, credit ratings, professional certifications).

Limitations & Future Work

• The model assumes attributes and features are scalars. While the authors mention generalization to multi-dimensional cases, no such analysis is provided. In reality, "qualification" for credit or education is almost certainly multi-dimensional; cross-dimensional coupling may make leg-up and retention hard to estimate separately.
• The ternary classifier assumes uniform model weights \(\theta\), implicitly assuming \(\theta\) is estimated from non-strategic data. If strategic feedback pollutes the training distribution (the feedback loop problem by Hardt et al.), estimation bias in \(\theta\) could invalidate theoretical guarantees.
• Experiments are limited to synthetic and FICO data, lacking longitudinal validation on real sequential tasks (e.g., repeated exams, credit renewals). Furthermore, there is almost no discussion of agent heterogeneity or group fairness, leaving a fairness blind spot.
• Future Directions: Incorporating weights \(\theta\) and thresholds \(\vec\mu\) into a joint sequential design and explicitly modeling the distribution of agent types; re-characterizing steady-states under realistic "reset" events (e.g., changing jobs or platforms).

Related Work & Insights

• 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 three intertemporal effects into an MDP and provides analytical feasibility boundaries.
• vs Hardt et al. 2015 / Milli et al. 2019: The negative conclusion \(c^+ > c^-\) for single-shot strategic classification is strictly weakened to \((1-\beta\gamma)c^+ > c^-\), marking a rare result where mechanism design alone breaks the incentive wall.
• vs Jin et al. 2022: They rely on external subsidies to break non-incentivizability, requiring additional budget; this paper proves that multi-level mechanisms + natural leg-up achieve the same effect without external monetary transfers.
• vs Kleinberg & Raghavan 2018: They focus on topological characterizations of effort incentives; this paper provides computable threshold sequences and numerical experiments, offering higher operational utility for real-world policy.

Rating

• Novelty: ⭐⭐⭐⭐⭐ Integrating "multi-level, ternary, leg-up + retention + farsightedness" into a single sequential framework with closed-form optimal solutions is rare in strategic classification.
• Experimental Thoroughness: ⭐⭐⭐ FICO and synthetic experiments sufficiently validate the theory, but lack real longitudinal data and multi-dimensional attribute extensions.
• Writing Quality: ⭐⭐⭐⭐ Theorems and economic interpretations are well-coordinated, with highly intuitive phase diagrams (Fig 3). The only drawback is that some key proofs are in the appendix, requiring frequent jumping.
• Value: ⭐⭐⭐⭐ Provides a directly applicable analytical framework and design principles for algorithmic mechanisms in multi-level decision systems like education, credit, and certification.

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