This paper analyses two-player nonzero-sum games of optimal stopping on a class of linear regular diffusions with not nonsingular boundary behaviour [in the sense of Itô and McKean (Diffusion Processes and Their Sample Paths (1974) Springer, page 108)]. We provide sufficient conditions under which Nash equilibria are realised by each player stopping the diffusion at one of the two boundary points of an interval. The boundaries of this interval solve a system of algebraic equations. We also provide conditions sufficient for the uniqueness of the equilibrium in this class.
"Nash equilibria of threshold type for two-player nonzero-sum games of stopping." Ann. Appl. Probab. 28 (1) 112 - 147, February 2018. https://doi.org/10.1214/17-AAP1301