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

TL;DR

This work studies sequential, multi-level strategic classification where a principal designs a ladder of increasingly difficult, thresholded classifiers to incentivize honest improvement over gaming. The agent's dynamics incorporate inter-temporal incentives via discounting $\\beta$, retention $\\gamma$, and a leg-up effect $\\delta$, which together shape long-run strategies and feasible incentive schemes. The paper fully characterizes the two-level (L=2) agent strategy, analyzes the principal's feasibility conditions, and derives a constructive, threshold-based design that, under mild conditions, can drive arbitrarily high, honest attainment through threshold sequencing (e.g., $\\mu_l = \\\delta l/(1-\\gamma)$ with $L = ceil((1-\\gamma)M/\\delta)$). Numerical results, including a greedy threshold algorithm and CMA-ES optimization on real data, illustrate the mechanism's practical potential and parameter sensitivities, such as the dominance of retention in expanding incentivizable levels and the nuanced role of leg-up. Overall, the work informs robust, long-horizon classifier design by accounting for strategic, self-interested agent behavior and inter-temporal incentives with real-world relevance to admissions, hiring, and credit scoring contexts.

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. While existing studies on sequential strategic classification primarily focus on optimizing dynamic classifier weights, we depart from these weight-centric approaches by analyzing the design of classifier thresholds and difficulty progression within a multi-level promotion-relegation framework. Our model captures the critical inter-temporal incentives driven by an agent's farsightedness, skill retention, and a leg-up effect where qualification and attainment can be self-reinforcing. We characterize the agent's optimal long-term strategy and demonstrate that a principal can design a sequence of thresholds to effectively incentivize honest effort. Crucially, we prove that under mild conditions, this mechanism enables agents to reach arbitrarily high levels solely through genuine improvement efforts.

Figures (12)

• Figure 1: Dynamics of the Agent's Decision Process.
• Figure 2: The evolution of the agent's level.
• Figure 3: General impossibility region and the agent's optimal long-term strategy in the two-level case. (a) Reduction of the non-incentivizable region under the multilevel mechanism compared to the single-shot problem. (b) The agent's optimal (Markov) strategy for given threshold $\mu$ and attribute $x$. (c) The agent's steady state distribution for given initial state under the optimal strategy.
• Figure 4: Agent state trajectories and strategy types. The dashed red lines mark the largest level ($L$) and threshold ($\mu_L$). (a) Successful incentive alignment leads to monotonic honest improvement. (b) Total failure, where high costs and low rewards prevent genuine improvement efforts despite advancement in levels. (c) Partial failure where the agent achieves the target, yet mainly through gaming.
• Figure 5: Pictorial interpretation of $W$ that satisfies the fixed-point equation.

Theorems & Definitions (25)

• Proposition 2.1
• Theorem 3.1
• Theorem 3.2
• Theorem 4.1
• Theorem 4.2
• Theorem 5.1
• Proposition 6.1
• Claim 3.1
• proof
• Claim 3.2