The Worst-Case Weighted Multi-Objective Game with an Application to Supply Chain Competitions

Area of Science:

• Game Theory
• Operations Research
• Decision Sciences

Background:

• Traditional game models often struggle with players having multiple, competing objectives.
• Existing weighted approaches may lack robustness in complex, uncertain environments.
• The need for equilibrium concepts that account for objective uncertainty is critical.

Purpose of the Study:

• To propose a novel worst-case weighted approach for multi-objective n-person non-zero sum games.
• To introduce and define the 'robust-weighted Nash equilibrium' concept.
• To demonstrate the applicability and robustness of the proposed model in supply chain risk management.

Main Methods:

• Development of the 'worst-case weighted multi-objective game' model.
• Mathematical proof of the existence of robust-weighted Nash equilibria.
• Formulation of a robust-weighted Nash equilibrium as a mathematical program with equilibrium constraints (MPEC) for polytope weight sets.

Main Results:

• Guaranteed existence of robust-weighted Nash equilibria, even with unbounded weight sets.
• Demonstration that robust-weighted Nash equilibria can be found by solving MPECs for specific game structures.
• Validation of the model's robustness and efficiency in a supply chain risk management application.

Conclusions:

• The worst-case weighted approach provides a robust equilibrium concept for complex games with competing objectives.
• Robust-weighted Nash equilibrium offers a more stable and practical solution compared to existing methods.
• The model is effective for real-world applications, particularly in managing supply chain risks under demand uncertainty.