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Strategic Decision-Making Under Uncertainty through Bi-Level Game Theory and Distributionally Robust Optimization

Jiachen Shen, Jian Shi, Lei Fan, Chenye Wu, Dan Wang, Choong Seon Hong, Zhu Han

TL;DR

This work tackles hierarchical decision-making under deep uncertainty by integrating bi-level game theory with distributionally robust optimization (DRO). By modeling leader–follower interactions and guarding against worst-case probability distributions within a Wasserstein ambiguity set, the authors transform the bi-level problem into a single-level DRO via KKT reformulations and dualization. They develop a proximal-dual, cutting-plane algorithm to solve the resulting non-smooth convex problem and prove convergence properties, supported by numerical experiments on networked transport scenarios showing up to 22% cost reductions while maintaining high service levels. The framework demonstrates strong potential for robust decision-making in complex networks such as transportation and communications, where uncertainty and hierarchy interact intricately.

Abstract

In strategic scenarios where decision-makers operate at different hierarchical levels, traditional optimization methods are often inadequate for handling uncertainties from incomplete information or unpredictable external factors. To fill this gap, we introduce a mathematical framework that integrates bi-level game theory with distributionally robust optimization (DRO), particularly suited for complex network systems. Our approach leverages the hierarchical structure of bi-level games to model leader-follower interactions while incorporating distributional robustness to guard against worst-case probability distributions. To ensure computational tractability, the Karush-Kuhn-Tucker (KKT) conditions are used to transform the bi-level challenge into a more manageable single-level model, and the infinite-dimensional DRO problem is reformulated into a finite equivalent. We propose a generalized algorithm to solve this integrated model. Simulation results validate our framework's efficacy, demonstrating that under high uncertainty, the proposed model achieves up to a 22\% cost reduction compared to traditional stochastic methods while maintaining a service level of over 90\%. This highlights its potential to significantly improve decision quality and robustness in networked systems such as transportation and communication networks.

Strategic Decision-Making Under Uncertainty through Bi-Level Game Theory and Distributionally Robust Optimization

TL;DR

This work tackles hierarchical decision-making under deep uncertainty by integrating bi-level game theory with distributionally robust optimization (DRO). By modeling leader–follower interactions and guarding against worst-case probability distributions within a Wasserstein ambiguity set, the authors transform the bi-level problem into a single-level DRO via KKT reformulations and dualization. They develop a proximal-dual, cutting-plane algorithm to solve the resulting non-smooth convex problem and prove convergence properties, supported by numerical experiments on networked transport scenarios showing up to 22% cost reductions while maintaining high service levels. The framework demonstrates strong potential for robust decision-making in complex networks such as transportation and communications, where uncertainty and hierarchy interact intricately.

Abstract

In strategic scenarios where decision-makers operate at different hierarchical levels, traditional optimization methods are often inadequate for handling uncertainties from incomplete information or unpredictable external factors. To fill this gap, we introduce a mathematical framework that integrates bi-level game theory with distributionally robust optimization (DRO), particularly suited for complex network systems. Our approach leverages the hierarchical structure of bi-level games to model leader-follower interactions while incorporating distributional robustness to guard against worst-case probability distributions. To ensure computational tractability, the Karush-Kuhn-Tucker (KKT) conditions are used to transform the bi-level challenge into a more manageable single-level model, and the infinite-dimensional DRO problem is reformulated into a finite equivalent. We propose a generalized algorithm to solve this integrated model. Simulation results validate our framework's efficacy, demonstrating that under high uncertainty, the proposed model achieves up to a 22\% cost reduction compared to traditional stochastic methods while maintaining a service level of over 90\%. This highlights its potential to significantly improve decision quality and robustness in networked systems such as transportation and communication networks.

Paper Structure

This paper contains 33 sections, 8 theorems, 22 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Foundational Concepts in Hierarchical and Robust Optimization
  4. Advances in Integrating Bi-Level Optimization with Distributionally Robust Optimization
  5. Applications in Complex Network Systems
  6. Model Formulation
  7. Upper-Level Problem (Leader)
  8. Lower-Level Problem (Follower)
  9. Integrative Formulation
  10. Dual Reformulation
  11. Simulation Scenario Description
  12. Upper-Level Problem
  13. Lower-Level Problem
  14. Integration of the Bi-Level Problem with DRO and Concrete Formulation
  15. Algorithmic Framework
  16. ...and 18 more sections

Key Result

Theorem 1

If the convexity Assumption 1 holds, then for any $\varepsilon \geq 0$ the worst-case expectation (8) equals the optimal value of the finite convex program

Figures (5)

  • Figure 1: Applications of DRO in General Networks.
  • Figure 2: Total Logistics Costs and Service Level Under Different Uncertainty Levels
  • Figure 3: Distribution of Total Costs Under Different Forecast Error Scenarios
  • Figure 4: Inventory Variance and Unmet Demand as a Function of Forecast Error
  • Figure 5: Computational Time vs. Problem Size

Theorems & Definitions (15)

  • Theorem 1: Convex reduction
  • Lemma 1: Feasible Iterates
  • proof
  • Lemma 2: Bounded Subgradients
  • proof
  • Lemma 3: Objective Descent
  • proof
  • Lemma 4: Subsequential Convergence
  • proof
  • Theorem 2: Convergence of the Whole Sequence
  • ...and 5 more