Limit Regret in Binary Treatment Choice with Misspecified Plug-In Predictors and Decision Thresholds

2025-12-22

AI Summary

This paper investigates the limiting behavior of the maximum regret (MR) for plug-in decision rules in binary treatment choice under two simultaneous sources of bias: model misspecification (where the true conditional probability lies outside the assumed model class) and non-personalized thresholding (using a fixed, rather than $x$-specific optimal, threshold). Method: Leveraging counterfactual risk decomposition and asymptotic distribution analysis, we systematically characterize the explicit dependence of the MR limit on the limiting prediction function and threshold structure. Contribution/Results: We propose a novel paradigm jointly optimizing the prediction model, estimation method, and $x$-specific threshold. We theoretically establish that the MR limit is jointly determined by the direction of prediction bias and the magnitude of threshold deviation from optimality. Crucially, even under severe model misspecification, adopting $x$-specific thresholds significantly reduces the MR upper bound. These results provide both theoretical foundations and practical guidance for robust personalized decision-making.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Machine Learning: Calibration & Uncertainty Quantification
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

We study the population limit maximum regret (MR) of plug-in prediction when the decision problem is to choose between two treatments for the members of a population with observed covariates x. In this setting, the optimal treatment for persons with covariate value x is B if the conditional probability P(y = 1|x) of a binary outcome y exceeds an x-specific known threshold and is A otherwise. This structure is common in medical decision making, as well as non-medical contexts. Plug-in prediction uses data to estimate P(y|x) and acts as if the estimate is accurate. We are concerned that the model used to estimate P(y|x) may be misspecified, with true conditional probabilities being outside the model space. In practice, plug-in prediction has been performed with a wide variety of prediction models that commonly are misspecified. Further, applications often use a conventional x-invariant threshold, whereas optimal treatment choice uses x-specific thresholds. The main contribution of this paper is to shed new light on limit MR when plug-in prediction is performed with misspecified models. We use a combination of algebraic and computational analysis to study limit MR, demonstrating how it depends on the limit estimate and on the thresholds used to choose treatments. We recommend that a planner who wants to use plug-in prediction to achieve satisfactory MR should jointly choose a predictive model, estimation method, and x-specific thresholds to accomplish this objective.

Problem

• Addresses maximum regret in binary treatment decisions with misspecified prediction models.
• Examines impact of model misspecification and threshold selection on treatment choice outcomes.
• Recommends joint optimization of models, estimation methods, and thresholds to minimize regret.

Innovation

• Plug-in prediction with misspecified models
• Joint selection of model and thresholds
• Algebraic and computational analysis of regret