  Following Baurdoux and Kyprianou [2] we consider the McKean stochastic game,
a game version of the McKean optimal stopping problem (American put), driven by
a spectrally negative Levy process. We improve their characterisation of a
saddle point for this game when the driving process has a Gaussian component
and negative jumps. In particular we show that the exercise region of the
minimiser consists of a singleton when the penalty parameter is larger than
some threshold and 'thickens' to a full interval when the penalty parameter
drops below this threshold. Expressions in terms of scale functions for the
general case and in terms of polynomials for a specific jump-diffusion case are
provided.
