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

Abstract: We present \textbf{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 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 \textit{Rate-Distortion Equilibrium} (RDE) criterion that bounds the personalization--privacy tradeoff, an \textit{Evolutionary Game Dynamics} layer for population-level behavior modeling, a \textit{Mechanism Design} module for incentive-compatible engagement, and a \textit{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.

• The paper introduces the Equilibrium-Guided Personalization Framework (EGPF), a rigorous multi-layer model integrating game theory, category theory, and generative AI for physician engagement.
• It models strategic physician behavior via Bayesian games and Stackelberg leadership, achieving a 34% improvement in AUC-ROC for engagement prediction.
• By employing category and sheaf theory for compositional transfer, EGPF enables 2.4× faster type inference and up to 17.8% gains in low-resource cross-therapeutic data efficiency.

Equilibrium-Guided Generative Personalization for Physician Engagement

Overview

The paper "Personalization as a Game: Equilibrium-Guided Generative Modeling for Physician Behavior in Pharmaceutical Engagement" (2604.06860) introduces the Equilibrium-Guided Personalization Framework (EGPF), a mathematically rigorous multi-layer framework for modeling and operationalizing hyper-personalized physician engagement strategies in the pharmaceutical domain. EGPF unifies Bayesian game theory, category theory, information theory, sheaf theory, and generative AI to address the complex, strategic, and compositional nature of physician-pharma interaction. The system models physicians as strategic agents with private behavioral types, leverages functorial composition for transfer across therapeutic areas, employs information-theoretic feedback to balance personalization and privacy, and integrates generative models (LLMs) for content synthesis conditioned on equilibrium strategies.

Theoretical Foundation: Game-Theoretic Modeling

EGPF formulates the physician-pharma interaction as a Bayesian game with incomplete information. Physicians are modeled as strategic agents whose private types are vectors over factors such as evidence sensitivity, peer influence, patient outcome orientation, formulary access sensitivity, risk aversion, inertia, cognitive bandwidth, and temporal discounting. Pharma is modeled as a Stackelberg leader, selecting engagement actions (content, messaging) while anticipating physician best responses.

The framework rigorously defines utility functions for both agents, capturing the multidimensional tradeoffs inherent in prescribing decisions and engagement. Physicians process evidence, peer signals, patient factors, system access, and inertia, while pharma optimizes engagement efficacy, cost, regulatory risk, and type learning (information gain). Bayesian Nash Equilibrium (BNE) and Stackelberg equilibrium are characterized, with formal proofs of existence, uniqueness (under strict concavity), and a formal demonstration of Stackelberg advantage in sequential settings. Bayesian belief updating utilizes a quantal response equilibrium (QRE) likelihood model to represent bounded rationality in physician choices.

Theoretical convergence is established: the iterative belief update mechanism converges to the true physician type at a rate bounded by $O\left(\frac{K \log K}{t \cdot C_{\min}}\right)$, where $K$ is the number of types and $C_{\min}$ is the minimal channel capacity among types. Finite-sample regret bounds of $O(\sqrt{KMT \log T})$ are given.

Compositional Structure: Category Theory and Sheaf Theory

To address physician behavior modularity across therapeutic areas, the framework employs category-theoretic abstractions. Behavioral observation data, archetype distributions, and engagement actions are formalized as categories. Functors map observation-derived features to posterior beliefs over archetypes, ensuring structural invariance and consistency under domain shifts. Natural transformations enable transfer (domain adaptation) between therapeutic areas or contexts, supporting few-shot generalization.

A monoidal structure allows composition of sub-behaviors and proper contextual weighting. Adjoint functors model encoding/decoding relationships akin to autoencoding, ensuring information-preserving mappings between data and type spaces.

Multi-scale consistency in physician behavior inference is enforced via a sheaf-theoretic framework. Beliefs and inferred types at micro (interaction), meso (weekly), and macro (quarterly) time scales must glue together, and inconsistencies are measured by sheaf cohomology. The approach enables detection of anomalous, scale-inconsistent behaviors—high-value targets for compliance or intervention.

Information-Theoretic Feedback and Privacy Controls

EGPF explicitly incorporates information-theoretic constructs: the physician engagement process is modeled as a noisy communication channel, with actions as inputs and responses as outputs, parameterized by physician type. Channel capacity for each type is computed (via Blahut-Arimoto), quantifying the maximum information that can be reliably transmitted by engagement strategies.

A key innovation is the Rate-Distortion Equilibrium (RDE): the framework formalizes and upper bounds the tradeoff between personalization (distortion minimization) and privacy (bounded information transfer about sensitive type attributes). Thus, the degree of achievable personalization is controlled by explicit privacy or regulatory constraints.

KL divergence and Rényi divergences are used for drift detection, characterizing when observed physician responses deviate substantially from model predictions, triggering automatic recalibration. Active learning strategies based on Fisher information and information gain facilitate optimal experiment selection for rapid type identification.

Generative AI Integration: Equilibrium-Conditioned Content

EGPF operationalizes its equilibrium strategies through LLMs fine-tuned with Reinforcement Learning from Human Feedback (RLHF). The generative personalization policy conditions on the current state, posterior mean of physician type, and equilibrium action as separate prompt fields. RLHF reward includes relevance, factual accuracy, regulatory compliance, bias minimization, and explicit alignment to the theoretically optimal action.

Exploration-exploitation tradeoff is handled dynamically according to belief entropy and information gain, guided by a formal decaying schedule. Regret is decomposed and bounded as a function of exploratory interactions and time to type identification.

Empirical Results

On synthetic (SynthRx) and real-world (HCPilot) datasets, EGPF achieves:

• 34% improvement in AUC-ROC for engagement prediction over static segmentation.
• 28% lift in human-rated content relevance compared to transformer and bandit baselines.
• 2.4× faster convergence to high-confidence type inference compared to deep sequential and linear bandit models.

Ablation studies demonstrate that the game-theoretic layer provides the largest performance contribution, followed by category theory (compositional transfer), and information-theoretic feedback (drift detection). Category-theoretic natural transformations enable up to 17.8% data efficiency gains for low-resource cross-therapeutic domain transfer.

Implications and Future Directions

The compositional, equilibrium-guided approach enables hyper-personalized, adaptive engagement strategies that respect strategic and informational constraints. Practically, this allows for:

• Real-time, data-driven engagement adaptation at individual and population levels.
• Transparent, auditable decision rationales (via equilibrium computation).
• Scalable cross-domain transfer and rapid deployment in new therapy areas.
• Formal privacy guarantees and multi-scale behavioral consistency.

The framework's formalism naturally generalizes to richer settings: continuous type and infinite games, non-stationary environments (formulary, guidelines dynamics), multi-agent and network-structured games (e.g., including payers or patient groups), and causal inference under partial observability.

Potential research directions include: mean-field extensions for infinite-type spaces, adversarial settings for regulatory compliance and adversarial network defense, and further fusion with causal inference methods for disentangled effect estimation.

Conclusion

The EGPF framework advances the precision and rigor of personalized physician engagement through a multi-disciplinary synthesis of mathematical frameworks and generative AI. It establishes theoretical and practical performance guarantees, enables compositional and transferable modeling, and provides explicit information-theoretic and ethical controls. This approach is readily extensible to personalized decision-making settings beyond pharma, involving strategic, high-dimensional interactions and privacy-constrained information transfer.