Finite-Agent Stochastic Differential Games on Large Graphs: I. The Linear-Quadratic Case

TL;DR

This paper introduces a comprehensive framework for finite-agent linear-quadratic stochastic differential games on large graphs, establishing convergence of solution methods and characterizing Nash equilibria, with validation through numerical experiments.

Contribution

It extends existing models by incorporating heterogeneous, interpretable interactions and provides convergence results and equilibrium characterizations for general and vertex-transitive graphs.

Findings

Fictitious play converges for general graphs regardless of the number of players.

Semi-explicit Nash equilibrium characterization for vertex-transitive graphs.

Numerical experiments validate theoretical convergence and equilibrium properties.

Abstract

In this paper, we study finite-agent linear-quadratic games on graphs. Specifically, we propose a comprehensive framework that extends the existing literature by incorporating heterogeneous and interpretable player interactions. Compared to previous works, our model offers a more realistic depiction of strategic decision-making processes. For general graphs, we establish the convergence of fictitious play, a widely-used iterative solution method for determining the Nash equilibrium of our proposed game model. Notably, under appropriate conditions, this convergence holds true irrespective of the number of players involved. For vertex-transitive graphs, we develop a semi-explicit characterization of the Nash equilibrium. Through rigorous analysis, we demonstrate the well-posedness of this characterization under certain conditions. We present numerical experiments that validate our theoretical results and provide insights into the intricate relationship between various game dynamics and the underlying graph structure.