# A Passivity-Based Approach to Nash Equilibrium Seeking over Networks

arXiv: https://arxiv.org/abs/1705.02424 · HTML: https://mlsanity.com/paper/1705.02424/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 May 2017.

## Built on by (most cited first, 5 of 5)

- [The Confluence of Networks, Games and Learning](https://mlsanity.com/paper/2105.08158/) (2021)
- [Learning to Control Unknown Strongly Monotone Games](https://mlsanity.com/paper/2407.00575/) (2024)
- [Online Optimization in Games via Control Theory: Connecting Regret, Passivity and Poincar'e Recurrence](https://mlsanity.com/paper/2106.04748/) (2021)
- [Diffusion Stochastic Learning Over Adaptive Competing Networks](https://mlsanity.com/paper/2504.19635/) (2025)
- [Choose Your Battles: Distributed Learning Over Multiple Tug of War Games](https://mlsanity.com/paper/2509.20147/) (2025)

## Abstract (arXiv)

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.