  In this paper we study the existence of an optimal hedging strategy for the
shortfall risk measure in the game options setup. We consider the continuous
time Black--Scholes (BS) model. Our first result says that in the case where
the game contingent claim (GCC) can be exercised only on a finite set of times,
there exists an optimal strategy. Our second and main result is an example
which demonstrates that for the case where the GCC can be stopped on the all
time interval, optimal portfolio strategies need not always exist.
