We consider the stochastic-game-analogue of McKean's optimal stopping problem when the underlying source of randomness is a spectrally negative Lévy process. Compared to the solution for linear Brownian motion given in Kyprianou (2004) one finds two new phenomena. Firstly the breakdown of smooth fit and secondly the stopping domain for one of the players `thickens' from a singleton to an interval, at least in the case that there is no Gaussian component.
Erik Baurdoux. Andreas Kyprianou. "The McKean stochastic game driven by a spectrally negative Lévy process." Electron. J. Probab. 13 173 - 197, 2008. https://doi.org/10.1214/EJP.v13-484