New Method to Characterize Subgame Perfect Nash Equilibria in Differential Games

Abstract

Abstract. In this paper, we present a rnethod for computing Nash equilibria in feedback strategies. This method gives necessary and sufficient conditions to characterize subgame perfect equilibria by means of a system of quasilinear partial ditferential equations. This characterization allows one to know explicitly the solution of the game in sorne cases. In other cases, this approach rnakes a qualitative study easier, We apply this rnethod to nonrenewable resource garnes.

Identification of Efficient Subgame-Perfect Nash Equilibria in a Class of Differential Games1

“…2 for computing Nash equilibria in feedback strategies. This method gives necessary and sufficient conditions characterizing subgame-perfect equilibria by means of the following system of quasilinear partial differential equations:…”

“…The aim of this paper is to establish necessary and sufficient conditions giving an a priori characterization of the Pareto efficiency of feedback Nash equilibria for differential games with unidimensional state and control variables and where each control is a 1 The authors thank the anonymous referees for helpful comments. 2 Assistant Professor, Departamento de Economía Aplicada (Matemáticas), Facultad de Ciencias Econó micas, Universidad de Valladolid, Valladolid, Spain.…”

“…To this end, we propose a new approach based on the characterization of feedback Nash equilibria as solutions to a system of quasilinear partial differential equations as shown in Ref. 2. It is well known that, in general, the use of feedback strategies in noncooperative games prevents the attainment of efficient outcomes.…”

Euler–Lagrange equations of stochastic differential games: application to a game of a productive asset

“…Further concrete applications of the use of the EL equations system in differential game theory can be found in Shimomura (1991), Dockner and Sorger (1996), and Sorger (1998), all of them in the deterministic counterpart of the model we study in this paper. In Rincón-Zapatero et al (1998) and Rincón-Zapatero (2004), this approach has been made systematic. These two papers also provide sufficient conditions of optimality which are independent of the value function.…”

On a PDE Arising in One-Dimensional Stochastic Control Problems

“…Though the initial idea of obtaining a system of PDEs for the optimal control appears in [1] in connection with deterministic control problems, the main antecedents of this paper are: [2] and [3] in deterministic differential games; [4], in stochastic control problems, where the diffusion parameter of the state process is independent of the control variables; [5] in the Merton problem; and [6] in a model of optimal liquidation in illiquid markets. In all these papers, the use of the PDE for optimal control has proved to be useful.…”