  In this paper, by using permutation matrices as a representation of symmetric
group $S_N$ and Fourier matrix, we investigate quantum roulette with an
arbitrary $N$-state. This strategy, which we introduce, is general method that
allows us to solve quantum game for an arbitrary $N$-state. We consider the
interaction between the system and its environment and study the effect of the
depolarizing channel on this strategy. Finally, as an example we employ this
strategy for quantum roulette with N=3.
