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Stochastic Games on Large Sparse Graphs

Eyal Neuman, Sturmius Tuschmann

TL;DR

A framework for stochastic games on large sparse graphs, covering continuous-time and discrete-time dynamic games as well as static games, is introduced and existence and uniqueness of Nash equilibria are proved and convergence of Nash equilibria along locally weakly convergent graph sequences are proved.

Abstract

We introduce a framework for stochastic games on large sparse graphs, covering continuous-time and discrete-time dynamic games as well as static games. Players are indexed by the vertices of simple, locally finite graphs, allowing both finite and countably infinite populations, with asymptotics described through local weak convergence of marked graphs. The framework allows path-dependent utility functionals that may be heterogeneous across players. Under a contraction condition, we prove existence and uniqueness of Nash equilibria and establish exponential decay of correlations with graph distance. We further show that global equilibria can be approximated by truncated local games, and can even be reconstructed exactly on subgraphs given information on their boundary. Finally, we prove convergence of Nash equilibria along locally weakly convergent graph sequences, including sequences sampled from hyperfinite unimodular random graphs.

Stochastic Games on Large Sparse Graphs

TL;DR

A framework for stochastic games on large sparse graphs, covering continuous-time and discrete-time dynamic games as well as static games, is introduced and existence and uniqueness of Nash equilibria are proved and convergence of Nash equilibria along locally weakly convergent graph sequences are proved.

Abstract

We introduce a framework for stochastic games on large sparse graphs, covering continuous-time and discrete-time dynamic games as well as static games. Players are indexed by the vertices of simple, locally finite graphs, allowing both finite and countably infinite populations, with asymptotics described through local weak convergence of marked graphs. The framework allows path-dependent utility functionals that may be heterogeneous across players. Under a contraction condition, we prove existence and uniqueness of Nash equilibria and establish exponential decay of correlations with graph distance. We further show that global equilibria can be approximated by truncated local games, and can even be reconstructed exactly on subgraphs given information on their boundary. Finally, we prove convergence of Nash equilibria along locally weakly convergent graph sequences, including sequences sampled from hyperfinite unimodular random graphs.
Paper Structure (22 sections, 17 theorems, 152 equations, 2 figures)

This paper contains 22 sections, 17 theorems, 152 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our main contributions.
  3. Structure of the paper.
  4. Preliminaries
  5. Graphs
  6. Local Convergence of Rooted Graphs
  7. Local Weak Convergence
  8. Limits of Sparse Graph Sequences
  9. Hyperfiniteness and Sampling
  10. Model Setup
  11. Formulation of the Game
  12. Examples
  13. Main Results
  14. Existence and Uniqueness
  15. Correlation Decay
  16. ...and 7 more sections

Key Result

Lemma 2.10

Let $\{G_n\}_{n\in{\mathbb N}}\subset\mathcal{G}$ be a sequence of finite random graphs, and let $G\in\mathcal{G}_\ast$ be a random rooted graph. If $\{G_n\}_{n\in\mathbb N}$ converges in distribution in the local weak sense to $G$, then $G$ is unimodular.

Figures (2)

  • Figure 1: Left: the circle graph with 30 vertices $G=C_{30}$ with two induced subgraphs $H_1\subset G$ (red) and $H_2\subset G$ (green). It holds that $d_G(H_1,H_2)=9$ and thus $k=5$ in \ref{['eq:graph-distance']}. Right: an excerpt of the 4-regular tree $G'=T_4$ with two induced subgraphs $H'_1\subset G'$ (red) and $H'_2\subset G'$ (green). It holds that $d_{G'}(H'_1,H'_2)=4$ and thus $k=2$ in \ref{['eq:graph-distance']}.
  • Figure 2: Left: the infinite lattice $G\cong\mathbb{Z}^2$ with a specified vertex $v\in G$ (red), the neighborhood $H=B_3(G,v)$ (red/orange/green), its boundary $\partial H$ (green), and its interior $H^\circ$ (red/orange). Right: a tree $G'$ with a specified vertex $v\in G'$ (red), a neighborhood $H'\subset G'$ of $v$ (red/orange/green), its boundary $\partial H'$ (green), and its interior $H'^\circ$ (red/orange).

Theorems & Definitions (60)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 50 more