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

Ruimeng Hu, Jihao Long, Haosheng Zhou

TL;DR

A comprehensive framework is proposed that extends the existing literature by incorporating heterogeneous and interpretable player interactions and offers a more realistic depiction of strategic decision-making processes in finite-agent linear-quadratic games on graphs.

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.

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

TL;DR

A comprehensive framework is proposed that extends the existing literature by incorporating heterogeneous and interpretable player interactions and offers a more realistic depiction of strategic decision-making processes in finite-agent linear-quadratic games on graphs.

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.
Paper Structure (25 sections, 13 theorems, 167 equations, 3 figures, 5 tables)

This paper contains 25 sections, 13 theorems, 167 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. A Linear-Quadratic Game on Graphs
  3. The game setup
  4. The associated HJB system
  5. The Equilibrium via Fictitious Play
  6. The mathematical formulation
  7. Convergence results
  8. Semi-Explicit Equilibrium under Vertex-Transitive Graph
  9. Main results
  10. Generalizations to $l-$neighborhood interactions
  11. Proof roadmap
  12. The well-posedness of equation and the existence of
  13. Verification
  14. The alternative governing equation for
  15. Verification through the alternative governing equation for
  16. ...and 10 more sections

Key Result

Lemma 3.4

Under Assumption assu:feedback, the optimal strategy $\phi^{i,k}$ for player $i$ at FP stage $k$, can always be represented as $\phi^{i,k}(t,x) = (\varphi^{i,k}_t)^{\operatorname{T}} x$ for some $\varphi^{i,k}:[0,T]\to\mathbb{R}^N$, for any $k\in\mathbb{N}$.

Figures (3)

  • Figure 1: Comparisons of equilibrium state (left panel) and strategy (right panel) trajectories for $N = 50$ players in the linear-quadratic game on the complete graph $G = K_{50}$. In both panels, the colored solid lines represent the baseline solution (by numerically solving the Riccati system \ref{['eqn:Riccati']}), the colored circles are obtained by numerically solving the semi-explicit solution in Theorem \ref{['thm:NE_closed_form']}, and the black dots are computed by the closed-form solution given in carmona2013mean. For the sake of clarity, only trajectories of four randomly selected players are plotted.
  • Figure 2: Comparisons of equilibrium state (left panel) and strategy (right panel) trajectories for $N = 50$ players in the linear-quadratic game on the cycle graph $G = C_{50}$. In both panels, the colored solid lines represent the baseline solution (by numerically solving the Riccati system \ref{['eqn:Riccati']}), and the colored circles are obtained by numerically solving the semi-explicit solution in Theorem \ref{['thm:NE_closed_form']}. For the sake of clarity, only trajectories of four randomly selected players are plotted
  • Figure 3: Comparisons of log-MAE curves for the linear-quadratic game with $N = 50$ players on different graphs. The behavior at the tail of the log-MAE curves for the complete graphs is due to the numerical round-off error

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2: Normalized graph Laplacian
  • Remark 2.3: Model interpretation
  • Remark 2.4
  • Definition 2.5: Nash equilibrium
  • Remark 3.2: Variants of FP algorithms
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 26 more