Generalized Cops and Robbers: A Multi-player Pursuit Game on Graphs

We introduce and study the Generalized Cops and Robbers (GCR) game, 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 nonpositional ones. We also study the capturing properties of GCR Nash equilibria in connection with 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.