Sufficient and Necessary Open-Loop Stackelberg Strategy for Two-Player Game With Time Delay

Area of Science:

• Control Theory
• Game Theory
• Optimization

Background:

• Leader-follower games with time delays present significant challenges due to noncausality in strategy design.
• Existing research on delay-free leader-follower games is extensive, but the introduction of time delays complicates solutions.

Purpose of the Study:

• To develop a robust methodology for solving difference and differential leader-follower games with time delays.
• To establish conditions for the existence and uniqueness of Stackelberg strategies in delayed game systems.

Main Methods:

• Introduction of novel co-states to represent future control information and new states for past effects.
• Development of sufficient and necessary solvability conditions for the difference game.
• Derivation of explicit open-loop Stackelberg strategies using decoupled and symmetric Riccati equations.

Main Results:

• A unique open-loop Stackelberg strategy is guaranteed to exist for the difference game under the derived conditions.
• Explicit solutions for the open-loop strategies are provided in terms of Riccati equations.
• A unique strategy is also presented for continuous-time systems.

Conclusions:

• The proposed techniques effectively overcome the noncausality issue in delayed leader-follower games.
• The study provides a complete framework for analyzing and solving both discrete and continuous time delayed Stackelberg games.
• The findings contribute to advancing the theory and application of game theory in systems with time delays.