Linear Strategic Classification with Endogenous Improvements

Classical models of strategic classification treat agents' responses to a deployed classifier as cosmetic: features change, but true labels do not. We study an improvement-aware variant in which strategic responses can change outcomes: agents choose post-deployment features strategically at a cost, and labels are then drawn from a fixed conditional outcome law. We formalize this problem for linear classifiers under a single-index qualification model and linear-decomposable costs. Under these assumptions, agents' best responses reduce to a closed-form shift of the decision boundary, and the resulting improvement-aware classification problem is learnable with linear sample complexity. As post-deployment labels are unavailable a priori, we begin by establishing a generalization bound with oracle access to the outcome law. One of our main contributions is a practical plug-in algorithm that replaces the oracle with an estimate, together with a bound on its improvement-aware error. We also show that the optimal classifier for standard strategic classification is a better proxy for the improvement-aware objective than the non-strategic Bayes-optimal classifier. Experiments on semi-synthetic and synthetic data support these findings and the plausibility of our assumptions.