Pareto Optima for a Class of Singular Control Games

Abstract

We study a class of N -player stochastic differential games of singular control, motivated by the study of a dynamic model of interbank lending with benchmark rates. We describe Pareto optima for this game and show how they may be achieved through the intervention of a regulator, whose policy is a solution to a singular stochastic control problem. Pareto optima are characterized in terms of the solution to a new class of Skorokhod problems with piecewise-continuous free boundary.Pareto optimal policies are shown to correspond to the enforcement of endogenous bounds on interbank lending rates. Analytical comparison between Pareto optima and Nash equilibria for the case of two players allows to quantify the impact of regulatory intervention on the stability of the interbank rate.

Optimal installation of renewable electricity sources: the case of Italy

“…Another peculiarity is that this equilibrium induces the players to install before than when they would have done under a Pareto optimum. This is the converse phenomenon of what observed, e.g., in Cont et al (2020), where instead players following a Nash equilibrium act later than players following a Pareto optimum.…”

“…Remark 5.9 Since, after Remark 4.1, we have c < x, this means that the search for a Nash equilibrium induces the agents to perform an earlier installation with respect to the cooperative behavior of the Pareto optimum seen in the previous section. This phenomenon is the converse of the one observed in Cont et al (2020), where instead the Nash equilibrium's action regions are contained in the Pareto optima's ones, i.e., agents wait more under the Nash equilibrium than under the Pareto optimum. By continuity, we expect a similar behavior also for the case β > 0, at least for low values of β: in other words, also in the case when price impact is present, competitive Nash equilibria will induce players to install earlier than when they would install under a cooperative Pareto optimum.…”