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Ergodic Non-zero Sum Differential Game with McKean-Vlasov Dynamics

Qingshuo Song, Gu Wang, Zuo Quan Xu, Chao Zhu

Abstract

We investigate a two-player ergodic game problem under McKean-Vlasov dynamics. Due to the ergodicity of the controlled process, the associated system of Hamiltonian-Jacobi-Bellman (HJB) equations exhibits non-uniqueness in its solutions. We establish a two-stage verification theorem that connects the differential game problem with the HJB equations. The first stage involves the characterization of the Nash equilibrium and ergodic constants. The second stage focuses on the non-unique solutions of the HJB equations, which are linked to the value function of an auxiliary control problem. At the end, we analyze the linear-quadratic-Gaussian (LQG) case, leading to an intriguing set of measure-dependent algebraic Riccati equations.

Ergodic Non-zero Sum Differential Game with McKean-Vlasov Dynamics

Abstract

We investigate a two-player ergodic game problem under McKean-Vlasov dynamics. Due to the ergodicity of the controlled process, the associated system of Hamiltonian-Jacobi-Bellman (HJB) equations exhibits non-uniqueness in its solutions. We establish a two-stage verification theorem that connects the differential game problem with the HJB equations. The first stage involves the characterization of the Nash equilibrium and ergodic constants. The second stage focuses on the non-unique solutions of the HJB equations, which are linked to the value function of an auxiliary control problem. At the end, we analyze the linear-quadratic-Gaussian (LQG) case, leading to an intriguing set of measure-dependent algebraic Riccati equations.
Paper Structure (9 sections, 6 theorems, 66 equations)

This paper contains 9 sections, 6 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Ergodic Non-zero Sum Differential Game
  3. Problem Setup
  4. Flat derivative and chain rule
  5. Hamilton-Jacobi-Bellman equations
  6. Auxiliary control problem
  7. Problem formulation
  8. Verification theorem
  9. Linear Quadratic Control

Key Result

Lemma 1

Consider the stochastic differential equation eq:X and denote $\alpha_t = {\bf a}(\mu_t, X_t)$ for all $t\ge 0$. Suppose that the function $u: \mathcal{P}_2(\mathbb R^2) \to \mathbb R$ is fully-$\mathcal{C}^2$-differentiable and satisfies then we have

Theorems & Definitions (15)

  • Definition 1
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • proof
  • ...and 5 more