Numerical Methods for Differential Games: Capital Structure in an R&D Duopoly

Abstract

This paper compares two different numerical methods used to solve the same differential game. In differential games strategies of individual players are represented as continuous functions of time and are typically solutions to the optimal control problems of the players. The game is an R&D duopoly with two players: an upstream firm that is primarily engaged in research and development (the R&D firm) and whose value comes from the market valuation of these activities, and a downstream firm primarily engaged in distribution and marketing (the D&M firm). The first method is assumed to be the benchmark since it is based on discretizing the first order conditions of each player’s optimal control problem. The second method is based on making random guesses of the parameters of a second order polynomial and searching for optimal solutions. The results suggest that the second method, which is more automated and has the potential of being applied to games with higher dimensionality, can give approximate solutions to differential games similar to the one considered here. The results also provide an important theoretical outcome. They illustrate that unlike the tradeoff and pecking order models of capital structure there are many markets in which capital structure is not driven by a reversion to a target debt-to-equity ratio or a pecking order, but by maximizing firm value under strategic considerations.