Classification in Equilibrium: Structure of Optimal Decision Rules

This paper characterizes optimal classification when individuals adjust their behavior in response to the classification rule. We model the interaction between a designer and a population as a Stackelberg game: the designer selects a classification rule anticipating how individuals will comply, cheat, or abstain in order to obtain a favorable classification. Under standard monotone likelihood ratio assumptions, and for a general set of classification objectives, optimal rules belong to a small and interpretable family--single-threshold and two-cut rules--that encompass both conventional and counterintuitive designs. Our results depart sharply from prior findings that optimal classifiers reward higher signals. In equilibrium, global accuracy can be maximized by rewarding those with lower likelihood ratios or by concentrating rewards or penalties in a middle band to improve informational quality. We further characterize classification objectives that rule out socially harmful equilibria that disincentivize compliance for some populations.