# Gradient-Tracking over Directed Graphs for solving Leaderless Multi-Cluster Games

arXiv: https://arxiv.org/abs/2102.09406 · HTML: https://mlsanity.com/paper/2102.09406/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Feb 2021.

## Built on by (most cited first, 1 of 1)

- [Competing Adaptive Networks](https://mlsanity.com/paper/2103.15664/) (2021)

## Abstract (arXiv)

We are concerned with finding Nash Equilibria in agent-based multi-cluster games, where agents are separated into distinct clusters. While the agents inside each cluster collaborate to achieve a common goal, the clusters are considered to be virtual players that compete against each other in a non-cooperative game with respect to a coupled cost function. In such scenarios, the inner-cluster problem and the game between the clusters need to be solved simultaneously. Therefore, the resulting inter-cluster Nash Equilibrium should also be a minimizer of the social welfare problem inside the clusters. In this work, this setup is cast as a distributed optimization problem with sparse state information. Hence, critical information, such as the agent's cost functions, remain private. We present a distributed algorithm that converges with a linear rate to the optimal solution. Furthermore, we apply our algorithm to an extended cournot game to verify our theoretical results.