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Stationary Mean-Field Games of Singular Control under Knightian Uncertainty

Giorgio Ferrari, Ioannis Tzouanas

TL;DR

The paper addresses stationary mean-field games of singular stochastic control under Knightian uncertainty by formulating a one-dimensional controlled diffusion with a worst-case probability measure and a scalar mean-field interaction through a stationary distribution. It develops a robust, ergodic framework where the inner zero-sum game is solved via a free-boundary problem using a shooting method, and a verification theorem identifies a saddle point for the robust objective. Existence and uniqueness of a stationary mean-field equilibrium are established through continuity and compactness arguments, specifically using a Schauder–Tychonoff fixed-point approach, with additional regularity results for the free boundary. A numerical case study on optimal natural-resource extraction under uncertainty demonstrates how ambiguity and volatility affect the equilibrium barrier, stationary distribution, and prices, and a policy-iteration algorithm is proposed to approximate the equilibrium. The work advances robust MFGs with singular controls by integrating running profits, state-dependent costs, and mean-field coupling under model uncertainty, with potential applications to resource management and macroeconomic policy under ambiguity.

Abstract

In this work, we study a class of stationary mean-field games of singular stochastic control under model uncertainty. The representative agent adjusts the dynamics of an Itô diffusion via one-sided singular stochastic control, aiming to maximize a long-term average expected profit criterion. The mean-field interaction is of scalar type through the stationary distribution of the population. Due to the presence of uncertainty, the problem involves the study of a stochastic (zero-sum) game, where the decision maker chooses the "best" singular control policy, while the adversarial player selects the "worst" probability measure. Using a constructive approach, we prove existence and uniqueness of a stationary mean-field equilibrium. Finally, we present an example of mean-field optimal extraction of natural resources under uncertainty and we analyze the impact of uncertainty on the mean-field equilibrium.

Stationary Mean-Field Games of Singular Control under Knightian Uncertainty

TL;DR

The paper addresses stationary mean-field games of singular stochastic control under Knightian uncertainty by formulating a one-dimensional controlled diffusion with a worst-case probability measure and a scalar mean-field interaction through a stationary distribution. It develops a robust, ergodic framework where the inner zero-sum game is solved via a free-boundary problem using a shooting method, and a verification theorem identifies a saddle point for the robust objective. Existence and uniqueness of a stationary mean-field equilibrium are established through continuity and compactness arguments, specifically using a Schauder–Tychonoff fixed-point approach, with additional regularity results for the free boundary. A numerical case study on optimal natural-resource extraction under uncertainty demonstrates how ambiguity and volatility affect the equilibrium barrier, stationary distribution, and prices, and a policy-iteration algorithm is proposed to approximate the equilibrium. The work advances robust MFGs with singular controls by integrating running profits, state-dependent costs, and mean-field coupling under model uncertainty, with potential applications to resource management and macroeconomic policy under ambiguity.

Abstract

In this work, we study a class of stationary mean-field games of singular stochastic control under model uncertainty. The representative agent adjusts the dynamics of an Itô diffusion via one-sided singular stochastic control, aiming to maximize a long-term average expected profit criterion. The mean-field interaction is of scalar type through the stationary distribution of the population. Due to the presence of uncertainty, the problem involves the study of a stochastic (zero-sum) game, where the decision maker chooses the "best" singular control policy, while the adversarial player selects the "worst" probability measure. Using a constructive approach, we prove existence and uniqueness of a stationary mean-field equilibrium. Finally, we present an example of mean-field optimal extraction of natural resources under uncertainty and we analyze the impact of uncertainty on the mean-field equilibrium.
Paper Structure (19 sections, 20 theorems, 140 equations, 3 figures, 1 algorithm)

This paper contains 19 sections, 20 theorems, 140 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Literature
  4. Organization of the Paper
  5. Problem Formulation
  6. Probabilistic setting
  7. The ergodic mean-field game
  8. Solving the Ergodic Zero-Sum Game
  9. Existence of a solution to (\ref{['free-boundary problem']})
  10. Verification Result
  11. Mean-Field Equilibrium
  12. Continuity and boundedness of the free-boundary with respect to $\theta$
  13. Existence and Uniqueness of the Ergodic MFG Equilibrium
  14. A Case Study: Optimal Extraction of a Natural Resource
  15. Sensitivity Analysis
  16. ...and 4 more sections

Key Result

Proposition 3.1

For fixed $\beta\in \mathbb{R}_{+}$ and $\gamma\in\mathbb{R}$, the boundary-value problem (Auxiliary Second order ODE) has a unique solution $\phi_{\beta}^{\gamma}(\cdot,\theta)\in C^{1}(\mathbb{R}_{+})$, for any $\theta\in\mathbb{R}_{+}$.

Figures (3)

  • Figure 1: Comparative statics of equilibrium w.r.t. level of ambiguity $\epsilon$.
  • Figure 2: Comparative statics of equilibria w.r.t. level of volatility $\sigma$.
  • Figure :

Theorems & Definitions (48)

  • Remark 2.1
  • Definition 2.1: Admissible controls
  • Definition 2.2: Admissible measures
  • Remark 2.2
  • Definition 2.3: Ergodic MFG Equilibrium
  • Remark 2.3
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.1: Regular solution to (\ref{['Auxiliary Second order ODE']})
  • proof
  • ...and 38 more