Nonzero-sum stochastic games and mean-field games with impulse controls

Abstract

We consider a general class of nonzero-sum N -player stochastic games with impulse controls, where players control the underlying dynamics with discrete interventions. We adopt a verification approach and provide sufficient conditions for the Nash equilibria (NEs) of the game. We then consider the limit situation of N → ∞, that is, a suitable mean-field game (MFG) with impulse controls. We show that under appropriate technical conditions, the MFG is an -NE approximation to the N -player game, with = O 1 √ N . As an example, we analyze in details a class of stochastic games which extends the classical cash management problem to the game setting. In particular, we characterize the NEs for its two-player case and compare the results to the single-player case, showing the impact of competition on the player's optimal strategy, with sensitivity analysis of the model parameters.

Optimal market making under partial information and numerical methods for impulse control games with applications

“…Restricting attention to the one-dimensional infinite horizon two-player case, this chapter (based on [Zab19]) puts the focus on certain nonzero-sum impulse games which display a symmetric structure between the players. This class is broad enough to include many interesting applications; no less than the competing central banks problem (whether in its linear form [ABC + 19] or others considered in the single bank formulation [AF16,CZ99,JP93, MØ98]), the cash management problem [BCG19] (reducing its dimension by a simple change of variables) and the generalization of many impulse control problems to the two-player case.…”

“…Using this result, they provide the first example of an (almost) fully analytically solvable game, motivated by central banks competition over the exchange rate. The result is generalized to arbitrary N players in [BCG19], which also gives a semianalytical solution (i.e., depending on several parameters found numerically) to a concrete cash management problem. 2 A different, more probabilistic, approach is taken in [FK19] to find a semi-analytical solution of a strategic pollution control problem and to prove another Verification Theorem.…”

“…In Chapter 3, we specialize our study by putting the focus on symmetric games. This subclass is broad enough to include plenty of interesting applications, such as competition between central banks [ABC + 19, AF16, CZ99, JP93, MØ98], cash management problems [BCG19] and the generalization of many impulse control problems to the two-player instance.…”

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

“…A class of controls closely related to singular controls is that of impulses, which has also received attention recently within MFG theory. We mention the two papers Basei et al (2019) and Zhou and Huang (2017), where MFGs with impulse controls are considered and solved using an approach based on quasi-variational inequalities and exploiting the stationarity properties of their settings. Finally, the article Bertucci (2020) provides a variational characterization of the density of jumping particles coming from an impulse control problem.…”

Nonzero-sum stochastic differential games between an impulse controller and a stopper

“…Thereafter, Ferrari and Koch [15] produced a model of pollution control where the two players, the regulator and the energy producer, are assumed to face proportional and fixed costs and, as such, play an impulse nonzero-sum game which admits an equilibrium under some suitable conditions. Lastly, Basei et al [4] studied the mean field game version of the nonzero-sum impulse game in [1] and proved the existence of -Nash equilibrium for the corresponding N -player game. Regarding the zero-sum case, here we quote Cosso [14], who examined a finite time horizon two-player game where both players act via impulse control strategies and showed that such games have a value which is the unique viscosity solution of the double-obstacle quasi-variational inequality.…”