Prediction-Intervention Games and Invariant Sets

Overview

The paper formalizes prediction as a leader–follower game: a predictor is chosen from observational data, then another agent intervenes on covariates to optimize a separate objective. It investigates invariant predictors that remain reliable after this strategic response.

What the paper contributes

• Introduces prediction–intervention games as a causal Stackelberg-game framework.
• Proves that stable-blanket predictors dominate causal-parent predictors for two common classes of follower objectives.
• Connects post-intervention performance to worst-case risk and gives conditions for worst-case optimality.
• Provides practical strategies for known and unknown causal graphs, tested on simulated and real data.

Abstract

We consider the following two-player game: using observational data, the leader chooses a prediction function for a response variable Y from given covariates. The follower then reacts with an intervention on some covariates in the underlying structural causal model to maximize their own objective. The leader knows the intervention targets, but may have limited knowledge of the follower's objective. We call this setup a prediction-intervention game, a special case of a Stackelberg game. Finding an optimal strategy for the leader is generally difficult. To avoid severe performance loss, the leader may base their prediction on the causal parents of Y, or more generally on an invariant subset of covariates. We prove, for two common classes of follower objectives, that predictors based on the stable blanket, a specific invariant subset, are always better or as good as those based on the causal parents. We further upper bound the leader's post-intervention risk by a worst-case risk over allowed interventions and strengthen existing distribution generalization results to analyze this bound: we give sufficient conditions under which stable-blanket predictors are worst-case optimal, and show by examples that these conditions cannot in general be dropped. Finally, we discuss practical strategies for settings with known and unknown graph, and test them on simulated and real-world data.

Citation

@misc{kuehne2026predictionintervention,
  title  = {Prediction-Intervention Games and Invariant Sets},
  author = {Kühne, Linus and Schur, Felix and Peters, Jonas},
  year   = {2026},
  eprint = {2605.16828},
  archivePrefix = {arXiv},
  primaryClass = {stat.ML}
}