Frontiers in Operations: Bounding Counterfactual Outcomes of Health Insurance Delay-and-Deny Practices

Problem definition: Health insurance delay-and-deny practices, such as requiring prior authorization, frequently restrict access to timely and necessary medical care. These policies can have severe consequences, including delayed diagnosis and treatment, leading to poor patient outcomes or even death. Counterfactual analysis offers a way to assess the impact of these practices by bounding the likelihood that a patient would have survived under alternative policies given the severe consequence under the original policy (probability of necessity). However, quantifying counterfactual probabilities is challenging in dynamic systems, such as those arising in healthcare, where disease progression is often latent and evolves over time. Methodology and results: We develop a principled framework to bound counterfactual probabilities, specifically the probability of necessity (PN), in generalized hidden Markov models (GHMMs). Our approach leverages the structure of GHMMs to construct a feasible space of structural causal models and employs polynomial optimization to compute tight bounds on PN. We integrate domain-specific knowledge to further refine these bounds. A data-driven case study on breast cancer screening and treatment demonstrates the power of our framework. We show that incorporating domain-specific knowledge can reduce the width of PN bounds significantly, thus providing actionable insights. Our computational techniques are scalable, enabling high-quality solutions for time horizons as large as 100 periods within a few hours. Managerial implications: Our framework offers a rigorous tool for evaluating the impact of healthcare policies, such as requiring prior authorization or incorrectly denying a medical screening, on patient outcomes. The ability to estimate counterfactual probabilities has applications in policy and legal analysis and healthcare management. For instance, the bounds on PN can inform court cases by quantifying the impact of an (incorrect) delay-and-deny insurance policy on a particular patient who experienced a detrimental outcome. Beyond healthcare, our methodology is applicable to dynamic latent-state systems in other domains. History: This paper has been accepted in the Manufacturing & Service Operations Management Frontiers in Operations Initiative. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2025.0030 .