The role of information in the cop-robber game

Abstract

We investigate the role of the information available to the players on the outcome of the cops and robbers game. This game takes place on a graph and players move along the edges in turns. The cops win the game if they can move onto the robber's vertex. In the standard formulation, it is assumed that the players can "see" each other at all times. A graph G is called cop-win if a single cop can capture the robber on G. We study the effect of reducing the cop's visibility. On the positive side, with a simple argument, we show that a cop with small or no visibility can capture the robber on any cop-win graph (even if the robber still has global visibility). On the negative side, we show that the reduction in cop's visibility can result in an exponential increase in the capture time. Finally, we start the investigation of the variant where the visibility powers of the two players are symmetrical. We show that the cop can establish eye contact with the robber on any graph and present a sufficient condition for capture. In establishing this condition, we present a characterization of graphs on which a natural greedy pursuit strategy suffices for capturing the robber.

Search and pursuit-evasion in mobile robotics

“…More recently, Isler and Karnad (2008), studied the effect of reducing the pursuer's (i.e., the cop's) visibility against an evader with global visibility. Let G be the class of graphs where a single pursuer with global visibility can capture the evader.…”

The Role of Visibility in Pursuit/Evasion Games

“…We use the following pairs of (M, N ) values: (1,30), (2,15), (3,10), (4,7), (5,6). Four of these pairs give a total of 30 nodes and the pair (M = 4, N = 7) gives n = 28 nodes; as M/N increases, we progress from a path to a nearly square grid.…”

“…Both the visible and invisible CR variants are natural models for discretized robotic PE problems; the connection has been noted and exploited relatively recently [2,8,11]. If it is further assumed that the robber is not actively trying to avoid capture (the case of drunk robber) we obtain a one-player graph game; this model has been used quite often in mobile robotics [12][13][14][15][16] and especially (when assuming random robber movement) in publications such as [17][18][19][20][21], which utilize partially observable Markov decision processes (POMDP, [22][23][24]).…”

Multiagent Pursuit Evasion, or Playing Kabaddi

“…Indeed, there is a rich literature on these games under various names [13], such as man-and-the-lion [12,16,9], cops-and-robber [6,1,8,3,4], robot-and-rabbit [6], and pursuit-evasion [14,5,7], just to name a few.…”

“…Finally, in the visibility-based pursuit-evasion games, the evader is often assumed to have infinite speed, and the capture is defined as being "seen" by some defenderboth infinite visibility or limited-range visibility models have been considered [5,8]. By contrast, kabaddi involves equal speed agents and requires a physical capture that Fig.…”