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

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 influenced over time by both firms through costly investments. Under a wide range of market introduction costs a now or never equilibrium coexists with a continuum of delay equilibria, each of them inducing a different time of product introduction.