Mean-Field Games of Finite-Fuel Capacity Expansion with Singular Controls

Abstract

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 in action and inaction region. We then show that a solution of the MFG of capacity expansion induces approximate Nash equilibria for the N -player games with approximation error ε 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.

A unifying framework for submodular mean field games

“…[31] employs a relaxed approach in order to establish existence for a general class of MFGs involving singular controls, while the more recent [30] extends the analysis to MFGs in which interaction takes place both through states and controls. In [14] and [35] MFGs for finite-fuel follower problems are considered. By employing, respectively, the connection to problems of optimal stopping and PDE methods, the structure of the mean-field equilibrium as well as its connection to Nash equilibria for the corresponding N -player stochastic differential games is derived.…”

“…Remark 5.11 (Mean-field-dependent dynamics and relation to [14]). The approach from Subsection 5.1 also allows to cover problems, where the drift of the underlying state process depends in an increasing way (w.r.t.…”

“…This could be, for example, achieved if b in (5.2) is replaced by a bounded increasing function of (µ t ) t∈[0,T ] . Another example is given by the two-dimensional MFG of finite-fuel capacity expansion considered in [14]. Therein, the mean of a uniformly bounded purely controlled process affects in a nondecreasing way the drift of an uncontrolled Itô-diffusion and there is no mean field dependence in the profit functional.…”

“…Therein, the mean of a uniformly bounded purely controlled process affects in a nondecreasing way the drift of an uncontrolled Itô-diffusion and there is no mean field dependence in the profit functional. We refer to Remark 3.15 in [14] for additional details on how the existence of a mean field equilibrium for the problem considered in that paper can be indeed achieved via our lattice-theoretic techniques.…”

“…This phenomenon has been investigated mainly on a case by case basis, and specific examples with multiple solutions have been presented in the recent literature [3,22,23,55], among others. Interestingly, the submodularity assumption appears implicitly in a number of classical linear-quadratic models (see, e.g., [9]) and in [3,14,22,23], although this property is not exploited therein. The increasing interest in non-uniqueness of solutions together with the perspective of characterizing many models through a unique key structural property has been one of the main motivations for our study of submodular MFGs.…”