On a Class of Two Player Two State Differential Games

Abstract

Linear closed-loop no-memory (CLNM) strategies are considered for a class of linear-quadratic (LQ) differential games. The class of games considered has two players (each with a single control), two state variables and an infinite time horizon. Conditions are provided under which Nash equilibrium linear CLNM strategies are shown to exist. The derivation of these conditions turns on solving a particular cubic equation in one of the strategy paramenters. Two applications that fall within this class of games are considered. The first application is an analysis of the capacity investment strategies of two sellers in a duopolistic market. Each seller is subject to convex costs of investment and, as a consequence, will always make any adjustments to capacity gradually over time. Nash equilibrium linear CLNM strategies are shown to embody a form of preemptive investment; each firm's rate of investment is a decreasing function of its rival's capacity. The second application involves an analysis of Richardson's model of arms race between two nations. This model was formulated as a LQ differential game by Simaan and Cruz. This LQ model is shown to fall within the class of games for which the existence and stability results outlined above apply.