Mean-Field Games with two-sided singular controls for Lévy processes
Facundo Oliú
TL;DR
We address mean-field games on the real line where each player controls a Lévy-driven state via two-sided singular controls and interacts through the mean-field term in the cost. The authors derive existence of $\epsilon$-discounted MFG equilibria using an adjoint Dynkin-game formulation and a Brouwer fixed-point argument, and show these equilibria converge (abelian limit) to an ergodic MFG equilibrium as $\epsilon\to0$. Equilibria are characterized by barrier strategies $(a^*,b^*)$ obtained from the Dynkin-game value, and a counterexample demonstrates non-uniqueness of equilibria in general. They also justify finite-N approximations by proving that symmetric $N$-player equilibria are approximate Nash equilibria for large $N$, illustrated with a compound Poisson example and a strictly stable Lévy-process example, thereby connecting the mean-field model to practical multi-agent settings.
Abstract
In a probabilistic mean field game driven by a Lévy process an individual player aims to minimize a long run discounted/ergodic cost by controlling the process through a pair of increasing and decreasing càdlàg processes, while he is interacting with an aggregate of players through the expectation of a controlled process by another pair of càdlàg processes. With the Brouwer fixed point theorem, we provide easy to check conditions for the existence of mean field game equilibrium controls for both the discounted and ergodic control problem, characterize them as the solution of an integro-differential equation and show with a counterexample that uniqueness does not always holds. Furthermore, we study the convergence of equilibrium controls in the abelian sense. Finally, we treat the convergence of a finite-player game to this problem to justify our approach.