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We study Nash equilibria for a sequence of symmetric $N$-player stochastic games of finite-fuel capacity expansion with singular controls and their mean-field game (MFG) counterpart. We construct a solution of the MFG via a simple iterative scheme that produces an optimal control in terms of a Skorokhod reflection at a (state-dependent) surface that splits the state space into action and inaction regions. We then show that a solution of the MFG of capacity expansion induces approximate Nash equilibria for the $N$-player games with approximation error $varepsilon$ going to zero as $N$ tends to infinity. Our analysis relies entirely on probabilistic methods and extends the well-known connection between singular stochastic control and optimal stopping to a mean-field framework.
Forcing finite state mean field games by a relevant form of common noise is a subtle issue, which has been addressed only recently. Among others, one possible way is to subject the simplex valued dynamics of an equilibrium by a so-called Wright-Fishe
In this paper, we develop a PDE approach to consider the optimal strategy of mean field controlled stochastic system. Firstly, we discuss mean field SDEs and associated Fokker-Plank eqautions. Secondly, we consider a fully-coupled system of forward-b
Mean-field games with absorption is a class of games, that have been introduced in Campi and Fischer [7] and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mea
A theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise. The concepts of strong and weak solutions are introduced in analogy with the theory of stochastic differential equations, and ex
The purpose of this paper is to provide a complete probabilistic analysis of a large class of stochastic differential games for which the interaction between the players is of mean-field type. We implement the Mean-Field Games strategy developed anal