  We study a quantum game played by two players with restricted multiple
strategies. It is found that in this restricted quantum game Nash equilibrium
does not always exist when the initial state is entangled. At the same time, we
find that when Nash equilibrium exists the pay off function is usually
different from that in the classical counterpart except in some special cases.
This presents an explicit example where quantum game and classical game may
differ. When designing a quantum game with limited strategies, the allowed
strategy should be carefully chosen according to the type of initial state.
