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Equilibrium in Functional Stochastic Games with Mean-Field Interaction

Eduardo Abi Jaber, Eyal Neuman, Moritz Voß

Abstract

We consider a general class of finite-player stochastic games with mean-field interaction, in which the linear-quadratic cost functional includes linear operators acting on controls in $L^2$. We propose a novel approach for deriving the Nash equilibrium of the game semi-explicitly in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations of the second kind and deriving their solution in semi-explicit form. Furthermore, by proving stability results for the system of stochastic Fredholm equations, we derive the convergence of the equilibrium of the $N$-player game to the corresponding mean-field equilibrium. As a by-product, we also derive an $\varepsilon$-Nash equilibrium for the mean-field game, which is valuable in this setting as we show that the conditions for existence of an equilibrium in the mean-field limit are less restrictive than in the finite-player game. Finally, we apply our general framework to solve various examples, such as stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact.

Equilibrium in Functional Stochastic Games with Mean-Field Interaction

Abstract

We consider a general class of finite-player stochastic games with mean-field interaction, in which the linear-quadratic cost functional includes linear operators acting on controls in . We propose a novel approach for deriving the Nash equilibrium of the game semi-explicitly in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations of the second kind and deriving their solution in semi-explicit form. Furthermore, by proving stability results for the system of stochastic Fredholm equations, we derive the convergence of the equilibrium of the -player game to the corresponding mean-field equilibrium. As a by-product, we also derive an -Nash equilibrium for the mean-field game, which is valuable in this setting as we show that the conditions for existence of an equilibrium in the mean-field limit are less restrictive than in the finite-player game. Finally, we apply our general framework to solve various examples, such as stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact.
Paper Structure (16 sections, 19 theorems, 196 equations)

This paper contains 16 sections, 19 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. The Finite-Player Game
  3. Function spaces, integral operators
  4. Definition of the $N$-player game
  5. Main results for the $N$-player game
  6. Illustrative Examples
  7. Stochastic Volterra Linear-Quadratic games
  8. Propagator models: optimal liquidation with transient impact and signals
  9. Delay in the control: an inter-bank lending and borrowing model
  10. Delay in the state: an advertising model with mean-field effect
  11. The Mean-Field Game
  12. The Infinite-Player Game Formulation
  13. Convergence and Approximation Results
  14. Stochastic Fredholm equations
  15. Proofs of Theorems \ref{['thm:opt_ubar']} and \ref{['thm:main-finite']}
  16. ...and 1 more sections

Key Result

Theorem 2.7

Under Assumption assum-op and ass:P any Nash equilibrium $\hat{u}^N \in \mathcal{U}^N$ to the game def:uNotation is such that $\bar{u}^N := \frac{1}{N} \sum_{j=1}^N \hat{u}^{j,N}$ is given by where and for all $t \in [0,T]$.

Theorems & Definitions (66)

  • Definition 2.1: Class of admissible kernels $\mathcal{G}$
  • Definition 2.2: Admissible Volterra operator
  • Remark 2.3
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Remark 2.9
  • Remark 2.10
  • Remark 2.11
  • ...and 56 more