A stochastic interaction between two firms

Abstract In this paper we construct a dynamic entry deterrence game in which market demand follows the Chan-Karolyi-Longstaff-Sanders (CKLS) stochastic differential equation (SDE). The incumbent firm, whose true strength is privately known, uses advertising and promotional expenditures strategically to shape the entrant’s beliefs. On the other hand, the entrant, facing a costly and irreversible decision, delays entry until demand reaches a critical level, with outcomes depending on both the incumbent’s hidden type and uncertain demand. Under this dynamic Stackelberg environment by implementing a path integral control approach, we determine a Markovian Nash feedback equilibrium. We also performed an empirical study based on revenue data from Enterprise Products Partners and Targa Resources over the period 2010-2024, and the resulting empirical findings are consistent with the theoretical predictions of the proposed model. The empirical trajectories exhibit qualitative patterns that are consistent with the mechanisms emphasized by the theoretical model, particularly with respect to persistence, recovery following adverse shocks, and differences in the responses of firms occupying distinct competitive positions.