A Passivity-Based Approach to Nash Equilibrium Seeking over Networks

Abstract: In this paper we consider the problem of distributed Nash equilibrium (NE) seeking over networks, a setting in which players have limited local information. We start from a continuous-time gradient-play dynamics that converges to an NE under strict monotonicity of the pseudo-gradient and assumes perfect information, i.e., instantaneous all-to-all player communication. We consider how to modify this gradient-play dynamics in the case of partial, or networked information between players. We propose an augmented gradient-play dynamics with correction in which players communicate locally only with their neighbours to compute an estimate of the other players' actions. We derive the new dynamics based on the reformulation as a multi-agent coordination problem over an undirected graph. We exploit incremental passivity properties and show that a synchronizing, distributed Laplacian feedback can be designed using relative estimates of the neighbours. Under a strict monotonicity property of the pseudo-gradient, we show that the augmented gradient-play dynamics converges to consensus on the NE of the game. We further discuss two cases that highlight the tradeoff between properties of the game and the communication graph.

• The paper presents a passivity-based method that reformulates Nash equilibrium seeking as a multi-agent coordination problem.
• It introduces a continuous-time gradient-play model with Laplacian feedback that ensures convergence under strict monotonicity conditions.
• Numerical results demonstrate reduced communication overhead and improved performance in dynamic, networked systems.

Passivity-Based Approaches for Nash Equilibrium Seeking over Networked Games

The paper by Gadjov and Pavel addresses the problem of distributed Nash Equilibrium (NE) seeking in networked multi-agent systems with a focus on scenarios where communication among agents is limited to local interactions. The authors propose an innovative use of a passivity-based framework to reformulate the NE seeking problem as a multi-agent coordination problem, leveraging the concept of incremental passivity. This approach is particularly valuable in dynamic environments, such as those encountered in wireless communication networks, optical networks, and distributed optimization, where agents cannot reliably access global information.

Overview of Proposed Methodology

The core contribution of the paper is the development of a continuous-time, gradient-play dynamic model augmented with a correction mechanism. This model allows agents to estimate other players' actions based solely on local communication with their direct neighbors, which contrasts with traditional methods that assume perfect knowledge of all agents' actions. The augmentation involves the introduction of auxiliary state variables, which are estimated by each player through local interactions, thus reducing the necessity for global information exchange. The new methodology relies on a Laplacian feedback derived from relative estimates, ensuring that these new dynamics align with the strict monotonicity conditions of the pseudo-gradient.

The authors' strategy is split across two principal fronts:

1. Single-Timescale Consensus and Optimization Approach: By embracing the incremental passivity properties of the pseudo-gradient, the proposed dynamics converge to the NE leveraging the natural tendency for synchrony induced by the Laplacian feedback over any connected graph. The authors demonstrate that under strict monotonicity conditions, the equilibrium points of these augmented dynamics are achieved when agents align their strategies to the NE.
2. Two-Timescale Decomposition: This method highlights that by accelerating the auxiliary state dynamics relative to the action dynamics—achieved through a singular perturbation technique—agents can converge to NE even with weaker connectivity requirements. This approach underscores the flexibility of the proposed model in balancing between game properties and network structure for achieving equilibrium.

Strong Numerical Insights

The analytical claims are supported by robust numerical results that validate the effectiveness of the proposed dynamics across various network topologies, showcasing reductions in both computational complexity and communication overhead when juxtaposed with all-to-all communication requirements of existing models. The strategies perform well even under communication constraints, indicating practical applicability in real-time, networked systems.

Implications and Future Directions

The paper sheds light on several critical implications:

• Theoretical Intersections: The connection between passivity theory and game-theoretic NE problems suggests a fertile ground for further development in both theoretical and practical directions. Particularly, this study opens pathways to explore passivity-based control in broader classes of dynamic, non-cooperative games.
• Algorithmic Development: By solidifying the theoretical underpinnings through rigorous proof of convergence, the proposed model can inspire the design of more refined algorithms in constrained environments, offering potential efficiency improvements for applications requiring real-time decision-making.
• Expanding Scope: Although the paper primarily addresses quadratic games and linear communication dynamics, future work may explore extensions to non-linear games and more complex network dynamics. Moreover, understanding the impact of approximate communication and estimation errors on convergence and system behavior could yield more resilient algorithms in unpredictable network conditions.

In summary, this paper provides a well-grounded contribution to the field of networked dynamic games, using passivity-based control to tackle the restrictive information assumptions of conventional NE seeking methods. The proposed dynamics not only theoretically enrich the existing literature but also offer tangible benefits for designing distributed systems in modern, communication-efficient applications.