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Stochastic Graphon Games with Memory

Eyal Neuman, Sturmius Tuschmann

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.

Stochastic Graphon Games with Memory

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.

Paper Structure

This paper contains 24 sections, 20 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Our main contributions.
  3. Structure of the paper.
  4. The Finite-Player Game
  5. Function spaces and integral operators
  6. Definition of the finite-player game
  7. Main results for the finite-player game
  8. The Infinite-Player Game
  9. Definition of the finite-player game on a weighted graph
  10. Definition of the infinite-player graphon game
  11. Main results for the graphon game
  12. Convergence Results
  13. Preliminaries
  14. Convergence results for given graph sequences
  15. Convergence results for sampled graph sequences
  16. ...and 9 more sections

Key Result

Proposition 2.7

Let Assumption assum:A-B-C hold and assume that the objective functionals $\alpha^{i,N} \mapsto J^{i,N}(\alpha^{i,N}; \alpha^{-i,N})$ are strictly concave in $\alpha^{i,N}\in\mathcal{A}$ for all $i\in\{1,\ldots, N\}$. Then, a set of strategies $\alpha^N=(\alpha^{1,N},\dots,\alpha^{N,N})\in\mathcal{A where $B$, $\bar{B}$, $\Lambda$ and $b$ are defined in eq:B-Bbar, eq:Lambda and eq:b, respectively.

Theorems & Definitions (65)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7
  • Remark 2.9
  • Definition 2.10
  • Definition 2.11
  • Theorem 2.12
  • ...and 55 more