Longcheng Liu

Game theory has emerged as a fundamental framework for modeling and analyzing strategic interactions and decision-making among multiple agents, and has witnessed rapidly growing impact in cyber–physical systems over the past decade. Its integration with dynamic systems has driven major theoretical and technological advances in a wide range of applications, including smart grids, autonomous driving, robotic swarms, and networked control systems. In particular, distributed games in dynamic systems and their equilibrium learning mechanisms have attracted increasing attention due to their scalability, lightweight information exchange, and real-time implementability. This article provides a comprehensive survey of distributed games in dynamic systems, where agents interact only with local neighbors while collectively achieving global equilibrium and stability. First, the foundational theories of distributed dynamic games under three representative classes of systems: linear dynamic systems, nonlinear dynamic systems, and uncertain dynamic systems, are presented. Then, state-of-the-art distributed equilibrium learning and control methods are reviewed, including gradient-based dynamics, payoff-based learning, best-response dynamics, and learning-based approaches. To demonstrate the practical relevance and impact of distributed games in dynamic systems, representative application domains are discussed in detail. Finally, several promising future research directions are outlined, highlighting open challenges at the intersection of distributed games, learning, and dynamic systems.

This article presents a novel model-free distributionally robust framework for a challenging equilibrium-seeking problem (ESP) under fully unknown coupled dynamics. We consider a scenario in the ESP where the state transitions of players are governed by an unknown coupled dynamic system, and each player aims to minimize its own cost function. By predicting the stochastic distribution of player states through Gaussian process regression, we propose a novel distributionally robust approximation (DRA) that transforms the complex ESP with unknown coupled dynamic system into a solvable distributionally robust optimization problem. The gradient of the DRA's objective function is quantified, ensuring solvability. The effectiveness of the proposed DRA framework is evaluated through a nonlinear system, demonstrating comparable performance to model-based methods without requiring any dynamic model.

This article investigates a multileader Stackelberg game where leaders lack critical information about the follower’s objective function and face random disturbances with unknown distributions. Unlike conventional approaches requiring complete follower information, we consider leaders who manipulate physical plant states while observing the follower’s strategy through private tracking responders. To address distributional uncertainty in the follower’s best response, we reformulate the game as a distributionally robust equilibrium-seeking problem and develop a fully distributed FL algorithm. The proposed data-driven approach operates without prior knowledge of system models or disturbance distributions, enabling leaders to estimate states through neighbor communication and local gradient updates. We characterize equilibrium existence in nonconvex settings. The relationship between communication and gradient errors and the energy function of the dynamic system is established. The upper bound of the regret based on the proposed algorithm is rigorously analyzed. A case study demonstrates the framework’s effectiveness in achieving distributionally robust solutions against uncertain stochastic perturbations.