Prediction-Intervention Games and Invariant Sets
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, an invariant subset, are at least 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.
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BibTeX
@misc{kuhne2026predictionintervention,
title = {{Prediction-Intervention Games and Invariant Sets}},
author = {Linus Kühne and Felix Schur and Jonas Peters},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.16828},
eprint = {2605.16828},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.16828},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Prediction-Intervention Games and Invariant Sets
AU - Linus Kühne
AU - Felix Schur
AU - Jonas Peters
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2605.16828
UR - https://arxiv.org/abs/2605.16828
AB - 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, an invariant subset, are at least 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.
ER -