Stochastic Graphon Games with Memory

TL;DR

This paper develops explicit solutions for stochastic graphon games with memory, analyzing how finite-player games converge to continuum models and applying the results to systemic risk and network games.

Contribution

It introduces a method to explicitly solve non-Markovian stochastic graphon games and proves convergence of finite-player equilibria to the continuum limit.

Findings

Explicit Nash equilibrium solutions for stochastic graphon games.

Finite-player game equilibria converge to graphon game equilibria as number of agents increases.

Convergence rates depend on graph sequence properties and sampling methods.

Abstract

We study finite-player dynamic stochastic games with heterogeneous interactions and non-Markovian linear-quadratic objective functionals. We derive the Nash equilibrium explicitly by converting the first-order conditions into a coupled system of stochastic Fredholm equations, which we solve in terms of operator resolvents. When the agents' interactions are modeled by a weighted graph, we formulate the corresponding non-Markovian continuum-agent game, where interactions are modeled by a graphon. We also derive the Nash equilibrium of the graphon game explicitly by first reducing the first-order conditions to an infinite-dimensional coupled system of stochastic Fredholm equations, then decoupling it using the spectral decomposition of the graphon operator, and finally solving it in terms of operator resolvents. Moreover, we show that the Nash equilibria of finite-player games on graphs converge to those of the graphon game as the number of agents increases. This holds both when a given graph sequence converges to the graphon in the cut norm and when the graph sequence is sampled from the graphon. We also bound the convergence rate, which depends on the cut norm in the former case and on the sampling method in the latter. Finally, we apply our results to various stochastic games with heterogeneous interactions, including systemic risk models with delays and stochastic network games.