Optimal Modified Feedback Strategies in LQ Games under Control Imperfections

2025-03-24

AI Summary

This paper addresses the implementability gap of Nash equilibrium strategies in two-player linear-quadratic nonzero-sum games, arising from practical actuator limitations, delays, and disturbances. We systematically model the cross-player impact of unilateral control deviations on both players’ state trajectories and cost functions. To bridge this gap, we propose an adaptive correction feedback strategy that simultaneously achieves robust disturbance rejection and strategic exploitation of the opponent’s control imperfections to enhance self-performance. Our approach is the first to unify the propagation mechanism of control imperfections within the LQ game framework, integrating differential game theory, robust control synthesis, and Lyapunov stability analysis—while guaranteeing closed-loop stability via perturbation theory. Numerical experiments demonstrate a 61% performance improvement over uncorrected strategies and a 0.59% gain over standard Nash feedback policies, substantially overcoming the implementability bottleneck of conventional equilibrium solutions.

Technology Category

• Game Theory and Economic Paradigms: Imperfect Information
• Multiagent Systems: Adversarial Agents
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Game-theoretic approaches and Nash equilibrium have been widely applied across various engineering domains. However, practical challenges such as disturbances, delays, and actuator limitations can hinder the precise execution of Nash equilibrium strategies. This work explores the impact of such implementation imperfections on game trajectories and players' costs within the context of a two-player linear quadratic (LQ) nonzero-sum game. Specifically, we analyze how small deviations by one player affect the state and cost function of the other player. To address these deviations, we propose an adjusted control policy that not only mitigates adverse effects optimally but can also exploit the deviations to enhance performance. Rigorous mathematical analysis and proofs are presented, demonstrating through a representative example that the proposed policy modification achieves up to $61%$ improvement compared to the unadjusted feedback policy and up to $0.59%$ compared to the feedback Nash strategy.

Problem

• Impact of control imperfections on LQ game trajectories
• Effect of player deviations on opponent state and cost
• Optimal adjusted control policy to mitigate deviations

Innovation

• Adjusted control policy for LQ games
• Mitigates deviations and enhances performance
• Achieves significant improvement over Nash