A numerical iterative method for solving two-point BVPs in infinite-horizon nonzero-sum differential games: Economic applications

Abstract

In this work, the Nash equilibrium solution of nonlinear infinite-horizon nonzero-sum differential games with open-loop information is investigated numerically. For this class of games, some difficulties are involved in finding an open-loop Nash equilibrium. For instance, we must solve a nonlinear differential equations system with split boundary conditions, namely the two-point boundary value problem (TPBVP), in which some of the boundary conditions are specified at the initial time and some at the infinite final time. In the current study, we provide a combined numerical algorithm based on a new set of basis functions on the half-line, called the exponential Chelyshkov functions (ECFs), and quasilinearization (QL) method to solve TPBVPs. In the first step, we reduce the nonlinear TPBVP to a sequence of linear differential equations by using the QL method. Although we have now a linearized system, another difficulty is finding the approximate solution of this linear system such that the transversality conditions at the infinite final time are satisfied. So, in the second step, we apply a collocation method based on the ECFs to solve the obtained linear system in the semi-infinite domain. The convergence of the proposed method is discussed in detail. To confirm the validity and efficiency of the proposed scheme, we compute the approximate solution of TPBVP as well as the open-loop Nash equilibrium for four applications of differential games in economics and management science.