Personalization as a Game: Equilibrium-Guided Generative Modeling for Physician Behavior in Pharmaceutical Engagement

We present $\mathbf{\text{EGPF}}$ (Equilibrium-Guided Personalization Framework), a mathematically rigorous architecture unifying Bayesian game theory, category theory, information theory, and generative AI for hyper-personalized physician engagement in the pharmaceutical domain. Our framework models the pharma–physician interaction as an incomplete-information Bayesian game where physician behavioral types are inferred via functorial mappings from observational categories, equilibrium strategies guide content generation through large language models (LLMs), and information-theoretic feedback loops ensure adaptive recalibration. We formalize behavior composition through category-theoretic functors, natural transformations, and monoidal structures, enabling modular, composable physician archetypes that respect structural invariants under domain shift. We introduce a novel $\mathit{\text{Rate-Distortion}}\mathit{\text{Equilibrium}}$ (RDE) criterion that bounds the personalization–privacy tradeoff, an $\mathit{\text{Evolutionary}}\mathit{\text{Game}}\mathit{\text{Dynamics}}$ layer for population-level behavior modeling, a $\mathit{\text{Mechanism}}\mathit{\text{Design}}$ module for incentive-compatible engagement, and a $\mathit{\text{Sheaf-Theoretic}}$ extension for multi-scale behavioral consistency. We prove convergence of our iterative belief-update mechanism at rate $O(\frac{K\log K}{t\cdot C_{\min }})$ and establish finite-sample regret bounds. Extensive experiments on synthetic pharma datasets and a real-world HCP engagement pilot demonstrate a 34% improvement in engagement prediction (AUC) and 28% lift in content relevance scores compared to state-of-the-art methods.