Markov Perfect Equilibria in Multi-mode Differential Games with Endogenous Timing of Mode Transitions

In this paper we study Markov-perfect equilibria (MPE) of two-player multi-mode differential games with controlled state dynamics, where one player controls the transition between modes. Different types of MPE are characterized distinguishing between delay equilibria, inducing for some initial conditions mode switches after a positive finite delay, and now or never equilibria, under which, depending on the initial condition, a mode switch occurs immediately or never. These results are applied to analyze the MPE of a game capturing the dynamic interaction between two incumbent firms among which one has to decide when to extend its product range by introducing a new product. The market appeal of the new product can be (positively or negatively) influenced over time by the competing firms through costly investments. It is shown that under a wide range of market introduction costs a now or never equilibrium co-exists with a continuum of delay equilibria, with each of them inducing a different time of product introduction.