Title: Selfish Cops and Active Robber: Multi-Player Pursuit Evasion on Graphs,
Abstract: We introduce and study the game of "Selfish Cops and Active Robber" (SCAR)
which can be seen as an multiplayer variant of the "classic" two-player Cops
and Robbers (CR) game. In classic CR all cops are controlled by a single
player, who has no preference over which cop captures the robber. In SCAR, on
the other hand, each of N-1 cops is controlled by a separate player, and a
single robber is controlled by the N-th player; and the capturing cop player
receives a higher reward than the non-capturing ones. Consequently, SCAR is an
N-player pursuit game on graphs, in which each cop player has an increased
motive to be the one who captures the robber. The focus of our study is the
existence and properties of SCAR Nash Equilibria (NE). In particular, we prove
that SCAR always has one NE in deterministic positional strategies and (for N
greater than two) another in deterministic nonpositional strategies.
Furthermore, we study conditions which, at equilibrium, guarantee either
capture or escape of the robber and show that (because of the antagonism
between the "selfish" cop players) the robber may, in certain SCAR
configurations, be captured later than he would be in classic CR, or even not
captured at all. Finally we define the selfish cop number of a graph and study
its connection to the classic cop number.