[Paper Review] Linear-Quadratic Stochastic Differential Games on Directed Chain Networks

Yichen Feng, Jean‐Pierre Fouque+1 more|arXiv (Cornell University)|Mar 19, 2020

Stochastic processes and financial applications4 references4 citations

TL;DR

This paper studies linear-quadratic stochastic differential games on directed chain networks, deriving explicit open-loop Nash equilibria for finite and infinite-player settings. It introduces Catalan functions and a novel Catalan Markov chain to characterize equilibrium dynamics, showing finite long-time variance even in the infinite-player limit, contrasting with mean-field models where variance diverges.

ABSTRACT

We study linear-quadratic stochastic differential games on directed chains inspired by the directed chain stochastic differential equations introduced by Detering, Fouque, and Ichiba. We solve explicitly for Nash equilibria with a finite number of players and we study more general finite-player games with a mixture of both directed chain interaction and mean field interaction. We investigate and compare the corresponding games in the limit when the number of players tends to infinity. The limit is characterized by Catalan functions and the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain, with or without the presence of mean field interaction.

Motivation & Objective

• To analyze finite-player linear-quadratic stochastic differential games on directed chain networks with explicit Nash equilibrium construction.
• To extend the analysis to infinite-player games and characterize the equilibrium dynamics using Catalan functions.
• To investigate the interplay between directed chain interaction and mean-field interaction via a tuning parameter.
• To compare long-time variance behavior under directed chain versus mean-field dynamics.
• To generalize results to periodic and tree-structured networks, conjecturing convergence to the same infinite-player limit.

Proposed method

• Formulates a finite-player game on a directed chain with independent Brownian motions and quadratic cost functions.
• Derives open-loop Nash equilibria by solving a Riccati system of ODEs, with solutions expressed via Catalan functions.
• Introduces a Catalan Markov chain to describe the infinite-player equilibrium dynamics, derived from the limit of the Riccati system.
• Incorporates mean-field interaction via a parameter $ u \in [0,1] $, allowing interpolation between pure directed chain and pure mean-field models.
• Uses generating functions and series solutions to analyze the infinite-dimensional equilibrium process and its long-time behavior.
• Applies asymptotic analysis and special functions (e.g., modified Bessel functions) to compute long-time variance of the equilibrium state.

Experimental results

Research questions

• RQ1 How do Nash equilibria emerge in linear-quadratic stochastic differential games on finite directed chain networks?
• RQ2 What is the structure of the equilibrium dynamics in the infinite-player limit on a directed chain, and how does it differ from mean-field models?
• RQ3 How does the inclusion of mean-field interaction affect the long-time variance of the equilibrium process?
• RQ4 Can the infinite-player limit of a periodic directed chain be characterized similarly to the linear chain case?
• RQ5 What is the role of Catalan functions in describing the equilibrium dynamics and their connection to a novel Markov process?

Key findings

• The infinite-player equilibrium dynamics on a directed chain is described by a Catalan Markov chain, a new stochastic process introduced in the paper.
• The long-time asymptotic variance of the equilibrium state is finite, specifically bounded in $ (1/2, \sqrt{2}/2] $, and achieves its maximum at $ d=1 $.
• For the pure directed chain model, the long-time variance is finite, contrasting sharply with mean-field models where variance diverges.
• The equilibrium dynamics for the infinite-player game is characterized by a Riccati system whose solutions are Catalan functions.
• In the mixed model with tuning parameter $ u \in [0,1] $, the long-time variance remains finite when $ u=1 $ (pure directed chain), but diverges when $ u=0 $ (pure mean field).
• Numerical evidence supports the conjecture that the infinite-player limit of the periodic directed chain coincides with the linear chain limit.