# Prediction-Intervention Games and Invariant Sets

arXiv: https://arxiv.org/abs/2605.16828 · HTML: https://mlsanity.com/paper/2605.16828/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 56 references that are in the map.
- First posted on arXiv 16 May 2026.

## Builds on (most cited first, 30 of 32)

- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [Performative Prediction](https://mlsanity.com/paper/2002.06673/) (2020)
- [The Risks of Invariant Risk Minimization](https://mlsanity.com/paper/2010.05761/) (2020)
- [Strategic Classification](https://mlsanity.com/paper/1506.06980/) (2015)
- [Invariant Causal Prediction for Nonlinear Models](https://mlsanity.com/paper/1706.08576/) (2017)
- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Domain Adaptation by Using Causal Inference to Predict Invariant Conditional Distributions](https://mlsanity.com/paper/1707.06422/) (2017)
- [Invariance, Causality and Robustness](https://mlsanity.com/paper/1812.08233/) (2018)
- [Anchor regression: heterogeneous data meets causality](https://mlsanity.com/paper/1801.06229/) (2018)
- [Does Invariant Risk Minimization Capture Invariance?](https://mlsanity.com/paper/2101.01134/) (2021)
- [Strategic Classification is Causal Modeling in Disguise](https://mlsanity.com/paper/1910.10362/) (2019)
- [How Do Classifiers Induce Agents To Invest Effort Strategically?](https://mlsanity.com/paper/1807.05307/) (2018)
- [A causal framework for distribution generalization](https://mlsanity.com/paper/2006.07433/) (2020)
- [Causal Strategic Linear Regression](https://mlsanity.com/paper/2002.10066/) (2020)
- [Performative Prediction: Past and Future](https://mlsanity.com/paper/2310.16608/) (2023)
- [An Introduction to Bi-level Optimization: Foundations and Applications in Signal Processing and Machine Learning](https://mlsanity.com/paper/2308.00788/) (2023)
- [Anticipating Performativity by Predicting from Predictions](https://mlsanity.com/paper/2208.07331/) (2022)
- [Stabilizing Variable Selection and Regression](https://mlsanity.com/paper/1911.01850/) (2019)
- [Causal Strategic Classification: A Tale of Two Shifts](https://mlsanity.com/paper/2302.06280/) (2023)
- [Exploiting Independent Instruments: Identification and Distribution Generalization](https://mlsanity.com/paper/2202.01864/) (2022)
- [A Unifying Causal Framework for Analyzing Dataset Shift-stable Learning Algorithms](https://mlsanity.com/paper/1905.11374/) (2019)
- [Invariant Policy Learning: A Causal Perspective](https://mlsanity.com/paper/2106.00808/) (2021)
- [Linear Classifiers that Encourage Constructive Adaptation](https://mlsanity.com/paper/2011.00355/) (2020)
- [Fair lending needs explainable models for responsible recommendation](https://mlsanity.com/paper/1809.04684/) (2018)
- [Incentivizing High-Quality Content in Online Recommender Systems](https://mlsanity.com/paper/2306.07479/) (2023)
- [Causality-oriented robustness: exploiting general noise interventions](https://mlsanity.com/paper/2307.10299/) (2023)
- [Geometric and computational hardness of bilevel programming](https://mlsanity.com/paper/2407.12372/) (2024)
- [Model-based causal feature selection for general response types](https://mlsanity.com/paper/2309.12833/) (2023)
- [Adversaries With Incentives: A Strategic Alternative to Adversarial Robustness](https://mlsanity.com/paper/2406.11458/) (2024)
- [Maximum Risk Minimization with Random Forests](https://mlsanity.com/paper/2512.10445/) (2025)

## Abstract (arXiv)

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.