Certainty equivalence principle in stochastic differential games: An inverse problem approach

Producción CientíficaThis paper aims to characterize a class of stochastic differential games, which satisfy the certainty equivalence principle beyond the cases with quadratic, linear, or logarithmic value functions. We focus on scalar games with linear dynamics in the players' strategies and with separable payoff functionals. Our results are based on the resolution of an inverse problem that determines strictly concave utility functions of the players so that the game satisfies the certainty equivalence principle. Besides establishing necessary and sufficient conditions, the results obtained in this paper are also a tool for discovering new closed-form solutions, as we show in two specific applications: in a generalization of a dynamic advertising model and in a game of noncooperative exploitation of a productive asset.Este trabajo se ha hecho con ayuda de los proyectos del Ministerio de Economía, Industria y Competitividad, Grant/Award Number: ECO2017-86261-P, ECO2014-56384-P, y MDM 2014-0431, de la Consejería de Educación, Juventud y Deporte de la Comunidad de Madrid, Grant/Award Number: MadEco-CM S2015/HUM-3444, y de la Consejería de Educación de la Junta de Castilla y León VA148G18

Infinite Horizon Noncooperative Differential Games

For a non-cooperative differential game, the value functions of the various players satisfy a system of Hamilton-Jacobi equations. In the present paper, we consider a class of infinite-horizon games with nonlinear costs exponentially discounted in time. By the analysis of the value functions, we establish the existence of Nash equilibrium solutions in feedback form and provide results and counterexamples on their uniqueness and stability.Comment: 25 pages, 7 figure

On the convergence problem in Mean Field Games: a two state model without uniqueness

We consider N-player and mean field games in continuous time over a finite horizon, where the position of each agent belongs to {-1,1}. If there is uniqueness of mean field game solutions, e.g. under monotonicity assumptions, then the master equation possesses a smooth solution which can be used to prove convergence of the value functions and of the feedback Nash equilibria of the N-player game, as well as a propagation of chaos property for the associated optimal trajectories. We study here an example with anti-monotonous costs, and show that the mean field game has exactly three solutions. We prove that the value functions converge to the entropy solution of the master equation, which in this case can be written as a scalar conservation law in one space dimension, and that the optimal trajectories admit a limit: they select one mean field game soution, so there is propagation of chaos. Moreover, viewing the mean field game system as the necessary conditions for optimality of a deterministic control problem, we show that the N-player game selects the optimizer of this problem

Linear-Quadratic NN-person and Mean-Field Games with Ergodic Cost

We consider stochastic differential games with $N$ players, linear-Gaussian dynamics in arbitrary state-space dimension, and long-time-average cost with quadratic running cost. Admissible controls are feedbacks for which the system is ergodic. We first study the existence of affine Nash equilibria by means of an associated system of $N$ Hamilton-Jacobi-Bellman and $N$ Kolmogorov-Fokker-Planck partial differential equations. We give necessary and sufficient conditions for the existence and uniqueness of quadratic-Gaussian solutions in terms of the solvability of suitable algebraic Riccati and Sylvester equations. Under a symmetry condition on the running costs and for nearly identical players we study the large population limit, $N$ tending to infinity, and find a unique quadratic-Gaussian solution of the pair of Mean Field Game HJB-KFP equations. Examples of explicit solutions are given, in particular for consensus problems.Comment: 31 page

Markov Perfect Nash Equilibrium in stochastic differential games as solution of a generalized Euler Equations System

This paper gives a new method to characterize Markov Perfect Nash Equilibrium in stochastic differential games by means of a set of Generalized Euler Equations. Necessary and sufficient conditions are given

On one-dimensional stochastic control problems: applications to investment models

The paper provides a systematic way for finding a partial differential equation that characterize directly the optimal control, in the framework of one?dimensional stochastic control problems of Mayer, with no constraints on the controls. The results obtained are applied to some significative models in financial economics