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Classification in Equilibrium: Structure of Optimal Decision Rules

Elizabeth Maggie Penn, John W. Patty

TL;DR

This paper analyzes how optimal classification rules should be designed when individuals adapt their behavior in response to the rule, using a Stackelberg framework with endogenous signaling. Under monotone likelihood ratio, the globally optimal rules are surprisingly simple: baseline optimal rules are single-threshold (positive or negative), while allowing cheating yields non-monotone two-cut rules; quota constraints and designer objectives can eliminate harmful equilibria. The authors provide a complete characterization across general objectives and capacity constraints, and they show that even with strategic responses the optimal mechanism remains low-dimensional and interpretable. The work bridges incentive-aware machine learning, performative prediction, and signaling/screening theories, with practical implications for policy design and fairness considerations in settings where capacity and manipulation costs matter.

Abstract

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.

Classification in Equilibrium: Structure of Optimal Decision Rules

TL;DR

This paper analyzes how optimal classification rules should be designed when individuals adapt their behavior in response to the rule, using a Stackelberg framework with endogenous signaling. Under monotone likelihood ratio, the globally optimal rules are surprisingly simple: baseline optimal rules are single-threshold (positive or negative), while allowing cheating yields non-monotone two-cut rules; quota constraints and designer objectives can eliminate harmful equilibria. The authors provide a complete characterization across general objectives and capacity constraints, and they show that even with strategic responses the optimal mechanism remains low-dimensional and interpretable. The work bridges incentive-aware machine learning, performative prediction, and signaling/screening theories, with practical implications for policy design and fairness considerations in settings where capacity and manipulation costs matter.

Abstract

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.

Paper Structure

This paper contains 24 sections, 5 theorems, 31 equations, 3 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Relation to Incentive-Aware Machine Learning and Performative Prediction.
  4. Relation to Signaling and Screening Models in the Social Sciences.
  5. Structure of the article.
  6. Baseline Model
  7. Timing of Decisions.
  8. Individual Incentives & Behavior
  9. Individual Payoffs.
  10. Individual $i$'s Incentives under $\delta$.
  11. The Designer's Payoff.
  12. Threshold & Two-Cut Classifiers
  13. Threshold Rules.
  14. Two-Cut Rules.
  15. Quotas and Threshold Structure
  16. ...and 9 more sections

Key Result

Theorem 3.1

Suppose $\delta$ can reward no more than $q\in(0, 1]$ of the individuals. The optimal classifier, $\delta_q^*$, is a threshold or negative threshold rule.

Figures (3)

  • Figure 1: Examples of Inner and Outer Two-Cut Rules
  • Figure 2: Comparing Threshold and Outer Two-Cut Rules
  • Figure 3: Comparing Threshold and Inner Two-Cut Rules

Theorems & Definitions (7)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.1
  • Theorem 4.1
  • Lemma A.1