Selfish Cops and Active Robber: Multi-Player Pursuit Evasion on Graphs

TL;DR

This paper introduces a multiplayer pursuit-evasion game on graphs where multiple selfish cops compete to capture a robber, analyzing Nash equilibria and the impact of selfish motives on capture outcomes.

Contribution

It extends the classic Cops and Robbers game to a multiplayer setting with selfish players, proving the existence of Nash equilibria and exploring their properties.

Findings

SCAR always has a Nash equilibrium in deterministic positional strategies.

For more than two cops, a nonpositional equilibrium also exists.

Selfish motives can delay or prevent the robber's capture compared to classic CR.

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.