Paper: arXiv 2306.05433

Authors: 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.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly sophisticated theoretical framework using stochastic Fredholm equations and operator resolvents to solve functional stochastic games with mean-field interactions. While mathematically rigorous and novel with clear applications to market impact and systemic risk models, it is purely theoretical and entirely lacks empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve N-Player & Mean-Field Games with Functional Controls] --> B[Input: Linear-Quadratic Cost Functional with L2 Linear Operators]
    B --> C[Reduce First-Order Conditions to Stochastic Fredholm Equations]
    C --> D[Solve Equations Semi-Explicitly via Operator Resolvents]
    D --> E[Prove Stability Results for Fredholm System]
    E --> F[Establish N-Player Equilibrium Convergence to Mean-Field Limit]
    F --> G[Outcomes: Derivation of ε-Nash Equilibrium & Applications to Price Impact/Delay Models]