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

TL;DR

This paper presents a novel framework combining bi-level game theory with distributionally robust optimization to improve strategic decision-making under uncertainty in hierarchical network systems, ensuring robustness and computational efficiency.

Contribution

It introduces an integrated model that handles hierarchical interactions and distributional uncertainty, transforming complex bi-level DRO problems into tractable single-level formulations with a new solution algorithm.

Findings

Achieves up to 22% cost reduction under high uncertainty.

Maintains over 90% service level in simulations.

Demonstrates effectiveness in transportation and communication networks.

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.