This paper presents a numerical method for the characterization of Markov-perfect

equilibria of symmetric differential games exhibiting coexisting stable steady states.

The method relying on the calculation of ’local value functions’ through collocation

in overlapping parts of the state space, is applicable for games with multiple state

variables. It is applied to analyze a piecewise deterministic game capturing the dynamic

competition between two oligopolistic firms, which are active in an established

market and invest in R&D. Both R&D investment and an evolving public knowledge

stock positively influence a breakthrough probability, where the breakthrough generates

the option to introduce an innovative product on the market. Additionally, firms

engage in activities influencing the appeal of the established and new product to consumers.

Markov-perfect equilibrium profiles are numerically determined for different

parameter settings and it is shown that for certain constellations the new product

is introduced with probability one if the initial strength of the established market is

below a threshold, which depends on the initial level of public knowledge. In case the

initial strength of the established market is above this threshold, the R&D effort of

both firms quickly goes to zero and with a high probability the new product is never

introduced. Furthermore, it is shown that after the introduction of the new product

the innovator engages in activities weakening the established market, although it is

still producing positive quantities of that product.