  We study pricing and superhedging strategies for game options in an imperfect
market with default. We extend the results obtained by Kifer in \cite{Kifer} in
the case of a perfect market model to the case of an imperfect market with
default, when the imperfections are taken into account via the nonlinearity of
the wealth dynamics. We introduce the {\em seller's price} of the game option
as the infimum of the initial wealths which allow the seller to be superhedged.
We {prove} that this price coincides with the value function of an associated
{\em generalized} Dynkin game, recently introduced in \cite{DQS2}, expressed
with a nonlinear expectation induced by a nonlinear BSDE with default jump. We
moreover study the existence of superhedging strategies.
  We then address the case of ambiguity on the model, - for example ambiguity
on the default probability - and characterize the robust seller's price of a
game option as the value function of a {\em mixed generalized} Dynkin game.
  We study the existence of a cancellation time and a trading strategy which
allow the seller to be super-hedged, whatever the model is.
