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

TL;DR

This paper introduces the Generalized Cops and Robbers game (GCR), extending the classic pursuit game to multiple players on graphs, analyzing its equilibria and connection to graph properties.

Contribution

It formulates GCR as a stochastic game, proves the existence of multiple Nash Equilibria for three or more players, and links equilibria to the graph's cop-number.

Findings

GCR has at least two Nash Equilibria types.

Equilibria relate to the cop-number of the graph.

GCR extends classic pursuit games to multiple players.

Abstract

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.