[Paper Review] Optimal stopping in mean field games, an obstacle problem approach

C. Bertucci|arXiv (Cornell University)|Apr 21, 2017

Stochastic processes and financial applications7 references3 citations

TL;DR

This paper introduces the concept of mixed solutions to solve optimal stopping problems in mean field games (MFGs), using an obstacle problem framework to model Nash equilibria in mixed strategies. It establishes existence and uniqueness of solutions under general assumptions for both stationary and time-dependent settings, providing a rigorous analytical foundation for MFGs with exit decisions.

ABSTRACT

This paper is interested in the problem of optimal stopping in a mean field game context. The notion of mixed solution is introduced to solve the system of partial differential equations which models this kind of problem. This notion emphasizes the fact that Nash equilibria of the game are in mixed strategies. Existence and uniqueness of such solutions are proved under general assumptions for both stationary and evolutive problems.

Motivation & Objective

• To address the lack of a natural solution concept for mean field games with optimal stopping, where players may exit the game at a cost.
• To develop a mathematical framework that captures Nash equilibria in mixed strategies, reflecting realistic player exit behavior.
• To extend the classical MFG system to incorporate optimal stopping through a forward-backward obstacle problem structure.
• To prove existence and uniqueness of solutions for both stationary and time-evolving MFG systems with optimal stopping.
• To provide a variational and optimal control interpretation of the system, linking it to convex optimization problems.

Proposed method

• Introduces the notion of a 'mixed solution' to model Nash equilibria in mean field games with optimal stopping, where players randomize their exit times.
• Models the problem as a forward-backward system of partial differential equations involving an obstacle problem for the value function.
• Uses a penalized approximation approach to regularize the obstacle problem, introducing a parameter ε to handle the non-smoothness of the max operator.
• Applies integration by parts and weak convergence arguments to pass to the limit in the penalized system and derive the weak formulation of the mixed solution.
• Employs a duality argument via the Fenchel conjugate to interpret the system as a convex optimization problem in the control-theoretic framework.
• Relies on monotonicity and convexity assumptions on the Hamiltonian and cost functions to ensure uniqueness of solutions.

Experimental results

Research questions

• RQ1 What is the appropriate solution concept for mean field games with optimal stopping, where players may exit the game?
• RQ2 How can the system of forward-backward obstacle problems be rigorously solved under general assumptions?
• RQ3 What conditions ensure the existence and uniqueness of mixed strategy equilibria in this MFG framework?
• RQ4 How does the optimal control interpretation connect the MFG system to convex optimization?
• RQ5 What is the role of the obstacle problem in characterizing the value function and the optimal exit time?

Key findings

• The paper proves the existence of mixed solutions for both stationary and time-dependent mean field games with optimal stopping under general assumptions.
• Uniqueness of mixed solutions is established when the cost function f is strictly monotone, ensuring a unique equilibrium.
• The solution concept is analytically natural and consistent with the probabilistic interpretation of optimal stopping in MFGs.
• The system is shown to be equivalent to a convex optimization problem involving the Fenchel conjugate of the Hamiltonian, with a unique minimizer under strict convexity.
• The penalized approximation scheme converges to a weak solution, and the limit satisfies the system in a distributional sense.
• The analysis confirms that the value function satisfies an obstacle equation, and the measure evolves via a transport equation with a sign constraint on the drift.