  We introduce and study the Generalized Cops and Robbers game (GCR), an
N-player pursuit game in graphs. The two-player version is essentially
equivalent to the classic Cops and Robbers (CR) game. The three-player version
can be understood as two CR games played simultaneously on the same graph; a
player can be at the same time both pursuer and evader. The same is true for
four or more players. We formulate GCR as a discounted stochastic game of
perfect information and prove that, for three or more players, it has at least
two Nash Equilibria: one in positional deterministic strategies and another in
non-positional ones. We also study the capturing properties of GCR Nash
Equilibria in connection to the cop-number of a graph. Finally, we briefly
discuss GCR as a member of a wider family of multi-player graph pursuit games
with rather interesting properties.
