Incentivizing Compliance with Algorithmic Instruments

The evolution of causal inference has seen the rise of randomized controlled trials (RCTs) as the gold standard. However, real-world applications often face non-compliance, where individuals do not follow assigned treatments, leading to biased estimates. Instrumental variables (IV) methods, notably Angrist and Imbens, address static biases but struggle in multi-round, dynamic environments. Recent advances like Mansour et al.'s incentivizing exploration (IE) techniques use payments to encourage exploration but face ethical and cost issues. This paper innovates by integrating IV with dynamic recommendation mechanisms, enabling causal estimation amid evolving compliance behaviors, thus extending the applicability of IV in complex, adaptive settings.

The core challenge is how to design a mechanism that incentivizes individuals to follow treatment recommendations over multiple rounds, ensuring accurate causal effect estimation despite initial non-compliance. Static IV methods cannot directly handle the dynamic nature of compliance, which varies based on beliefs and observed outcomes. Furthermore, existing exploration strategies relying on payments are costly and ethically contentious. The problem becomes more complex when individuals' private types influence their baseline rewards and responses, creating selection bias. The key question is how to create a scalable, incentive-compatible system that gradually aligns individual behavior with the true treatment effects.

This paper's innovations include: 1) a history-dependent recommendation policy that maps past interactions into valid IVs, ensuring independence from private types; 2) a theoretical framework providing high-probability error bounds for treatment effect estimates; 3) a mechanism that transitions from complete non-compliance to high compliance over multiple rounds without costly payments. Unlike prior static IV or payment-based exploration methods, this approach dynamically incentivizes compliance by leveraging information asymmetry and Bayesian persuasion principles, enabling consistent causal inference in complex, evolving environments.

• �� Design a randomized recommendation policy π that maps the history Ht−1 to a treatment suggestion zt, ensuring IV relevance and independence.
• �� Use the observed interactions (zt, xt, yt) to perform two-stage least squares (2SLS) regression, estimating θ with confidence bounds.
• �� In the fully non-compliant phase, initially let individuals choose freely, then gradually introduce recommendations that incentivize compliance based on prior estimates.
• �� In the partially compliant phase, iteratively refine the recommendation policy using accumulated data, employing active arm elimination techniques to identify the optimal treatment.
• �� Theoretical analysis guarantees that the IV estimator’s error shrinks at a rate of O(1/√t), with high probability bounds ensuring reliable estimates.
• �� Simulation experiments validate the mechanism’s ability to achieve high compliance and accurate treatment effect estimation within 10^4 rounds, significantly reducing cumulative regret.

Simulations used synthetic datasets with T=10,000 rounds, modeling various compliance scenarios. Baselines included standard IV and non-incentivized recommendations. Metrics evaluated were compliance rate, treatment effect estimation error, and cumulative regret. Hyperparameters like exploration probability ρ and sample sizes were tuned for optimal performance. Results showed the proposed mechanism achieved over 85% compliance, with treatment effect confidence intervals narrowing to one-third of initial width. The error bounds matched theoretical predictions, and the mechanism maintained robustness across different bias levels and noise conditions, demonstrating scalability and effectiveness in complex environments.

The mechanism consistently outperformed baseline IV methods, achieving compliance rates above 85% within 10^4 rounds. The treatment effect estimates converged at a rate consistent with O(1/√t), with confidence intervals shrinking proportionally. The cumulative regret was bounded by O(√T), indicating near-optimal long-term performance. Simulations confirmed the theoretical error bounds, with the treatment effect estimation error decreasing as sample size increased. The results demonstrated the mechanism’s robustness to different prior beliefs and noise levels, validating its practical applicability.

Applicable in public policy evaluations, clinical trials, and personalized recommendation systems, especially where participant compliance is uncertain or costly to enforce. The mechanism can be used to design adaptive interventions that gradually improve compliance, enabling accurate long-term causal estimates. Its ability to operate without costly payments makes it suitable for large-scale, real-world deployments in healthcare, social programs, and digital platforms, where ethical considerations and resource constraints are critical.

The model assumes independent private types with known distributions, which may not hold in real settings. Parameter tuning, such as exploration probability, is sensitive and may affect performance if mis-specified. Extending the framework to high-dimensional treatments increases computational complexity. The approach relies on accurate prior knowledge, and deviations can impair the mechanism’s effectiveness. Future work should address these limitations by developing adaptive parameter tuning and modeling dependencies among individuals.