Skiba Phenomena in Markov Perfect Equilibria of Asymmetric Differential Games

Abstract

This paper examines the existence of Markov-Perfect-Equilibria that give rise to coexisting locally stable steady states in asymmetric differential games. The strategic interactions between an incumbent in a market and a potential competitor, which tries to enter the market through product innovation, are considered. Whereas the potential entrant invests in the build-up of a knowledge stock, which is essential for product innovation, the incumbent tries to reduce this stock through interference activities. It is shown that in the presence of upper bounds on investment activities of both firms a Markov-Perfect-Equilibrium exists under which, depending on the initial conditions, the knowledge stock converges either to a positive steady state, thereby inducing an entry probability of one, or to a steady state with zero knowledge of the potential entrant. In the later case the entry probability is close to zero. It is shown that this Markov-Perfect-Equilibrium is characterized by a discontinuous value function for the incumbent and it is discussed that this feature is closely related to the existence of upper bounds on the investments of the players. Removing these constraints in general jeopardizes the existence of a Markov-Perfect-Equilibrium with multiple locally stable steady states.