Prediction-Intervention Games and Invariant Sets

TL;DR

This paper introduces a game-theoretic framework for decision-making under causal interventions, demonstrating that invariant predictors based on stable sets outperform traditional causal predictors in certain scenarios.

Contribution

It establishes the superiority of stable-blanket predictors over causal parent-based predictors in prediction-intervention games and provides bounds and strategies for practical implementation.

Findings

Stable-blanket predictors are always better or equal to causal parent predictors.

The paper provides bounds on post-intervention risk and conditions for worst-case optimality.

Practical strategies are tested on simulated and real-world 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.