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

Debashis Chatterjee

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.

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

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.
Paper Structure (57 sections, 7 theorems, 63 equations, 25 figures, 9 tables, 1 algorithm)

This paper contains 57 sections, 7 theorems, 63 equations, 25 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Research Objective & Novelty
  3. Novelty of the Proposed Paper
  4. Model Formulation
  5. Players, Horizon, and Actions
  6. Private Types
  7. Latent Demand State
  8. Demand System
  9. Information Structure and Belief States
  10. Histories
  11. Posterior Beliefs
  12. Hierarchical Bayesian Specification
  13. Prior Structure
  14. Likelihood
  15. Posterior Updating with Data Augmentation
  16. ...and 42 more sections

Key Result

Proposition 10.1

Suppose the action space $\mathcal{A}_i$ is compact, the one-period credible-risk return is continuous in own action, and the continuation value is bounded and continuous. Then, for any fixed measurable rival strategy $\sigma_j$, player $i$'s interim optimization problem admits at least one maximize

Figures (25)

  • Figure 1: Mean cumulative discounted market profit over time. The proposed method corresponds to Proposed_Bayesian_CredibleRisk.
  • Figure 2: Distribution of final discounted market profit across replications.
  • Figure 3: Dynamic operational behavior of the three competing policies.
  • Figure 4: Learning diagnostics over time.
  • Figure 5: Objective surface of the proposed method over the price--quantity action grid.
  • ...and 20 more figures

Theorems & Definitions (22)

  • Remark 2.1
  • Definition 3.1: Belief-state Markov representation
  • Remark 4.1
  • Remark 6.1
  • Definition 7.1: Credible-risk Markov Perfect Bayesian Nash Equilibrium
  • Proposition 10.1: Existence of interim best response
  • proof
  • Theorem 10.2: Existence of CR-MPBNE under strengthened regularity conditions
  • proof
  • Remark 15.1: Key findings
  • ...and 12 more