Doubly Robust Estimation of Causal Effects in Strategic Equilibrium Systems

TL;DR

This work addresses causal effect estimation in environments where agents act strategically, causing endogenous treatment assignments. It introduces the Strategic Doubly Robust ($SDR$) estimator, which embeds strategic equilibrium modeling into the doubly robust paradigm by using a strategic propensity score $e(X,S)$ and a strategic outcome model $\mu(t,X,S)$, and proves consistency and asymptotic normality under strategic unconfoundedness. Theoretical results establish SDR’s double robustness, asymptotic theory, and convergence of equilibrium estimation, while experiments on synthetic data show bias reductions from 7.6% to 29.3% and favorable scalability as the agent population grows and covariate dimensionality varies. The framework provides a principled, robust approach for reliable causal inference in strategic settings, enabling better policy evaluation when agents adapt their behavior to interventions.

Abstract

We introduce the Strategic Doubly Robust (SDR) estimator, a novel framework that integrates strategic equilibrium modeling with doubly robust estimation for causal inference in strategic environments. SDR addresses endogenous treatment assignment arising from strategic agent behavior, maintaining double robustness while incorporating strategic considerations. Theoretical analysis confirms SDR's consistency and asymptotic normality under strategic unconfoundedness. Empirical evaluations demonstrate SDR's superior performance over baseline methods, achieving 7.6\%-29.3\% bias reduction across varying strategic strengths and maintaining robust scalability with agent populations. The framework provides a principled approach for reliable causal inference when agents respond strategically to interventions.

Figures (4)

• Figure 1: Relationship between number of agents ($N$) and estimation bias across different methods. The results show that when $N > 100$, estimation bias stabilizes across all methods, indicating that larger agent populations provide more reliable causal effect estimates. SDR demonstrates superior scalability, with its performance advantages becoming increasingly prominent in larger strategic environments where traditional methods struggle.
• Figure 2: SDR performance across varying numbers of covariates. SDR maintains robust performance regardless of covariate dimensionality, with advantages in both sparse and high-dimensional settings due to its strategic equilibrium modeling and doubly robust properties.
• Figure 3: Performance comparison of SDR and baseline methods across four model specification scenarios: correct models, misspecified outcome models, misspecified propensity models, and both models misspecified. SDR demonstrates robust performance across all scenarios, maintaining competitive bias levels comparable to AIPW while significantly outperforming methods lacking double robustness properties, particularly in challenging misspecification settings.
• Figure 4: Comparison of bias between SDR and DR across varying strategic strength ($\alpha$) and number of agents ($N$). SDR consistently outperforms DR, with the performance gap widening as strategic interactions become more pronounced

Theorems & Definitions (7)

• Definition 1: Strategic Causal Model
• Definition 2: Strategic Propensity Score
• Definition 3: Strategic Outcome Model
• Definition 4: Strategic Doubly Robust Estimator
• Theorem 1: Strategic Double Robustness
• Theorem 2: Consistency
• Theorem 3: Asymptotic Normality