A Hierarchical Bayesian Dynamic Game for Competitive Inventory and Pricing under Incomplete Information: Learning, Credible Risk, and Equilibrium

2026-03-06

AI Summary

This study addresses the dynamic inventory and pricing problem faced by firms operating under incomplete information, where both market demand and competitors’ private characteristics are unknown. The authors propose a hierarchical Bayesian dynamic game framework that integrates Bayesian learning, strategic belief updating, and operational optimization. A novel “credible risk criterion” is introduced as an equilibrium concept, incorporating posterior predictive dispersion as a regularization term to yield robust and conservative decisions. The equilibrium is computed via dynamic programming over belief states, Bayesian posterior updating, and high-dimensional inference techniques. Simulations demonstrate the critical role of Bayesian learning in performance enhancement and highlight the superior regularizing effect of the proposed criterion under uncertainty. Furthermore, the approach successfully identifies interpretable biological subgroups and latent states in protein expression data.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Sequential Decision Making
• Game Theory and Economic Paradigms: Imperfect Information

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We develop a hierarchical Bayesian dynamic game for competitive inventory and pricing under incomplete information. Two firms repeatedly choose order quantities and prices while facing two layers of uncertainty: unknown market demand and private rival characteristics. The framework combines Bayesian learning about demand and substitution with strategic belief updating about rival types. To make decisions robust to posterior uncertainty, we introduce a credible-risk criterion that rewards expected future profit while penalizing posterior predictive dispersion. This yields a conservative equilibrium concept in which firms learn, compete, and adapt simultaneously. The paper provides the model formulation, information structure, posterior updating mechanism, equilibrium definition, and a computational strategy based on belief-state dynamic programming. A simulation study shows that Bayesian learning is crucial for strong performance and that the credible-risk rule is especially effective as an operational regularizer under uncertainty. A real-data illustration on a high-dimensional protein-expression dataset demonstrates that the same uncertainty-aware Bayesian principle can produce biologically interpretable subgroup and latent-state findings. The proposed framework offers a unified bridge between Bayesian game theory and operations research, with practical relevance for competitive decision-making in uncertain and information-limited environments.

Problem

• incomplete information
• competitive inventory
• pricing
• Bayesian learning
• uncertainty

Innovation

• Hierarchical Bayesian dynamic game
• Credible-risk criterion
• Bayesian learning
• Incomplete information
• Belief-state dynamic programming