Cops and Invisible Robbers: the Cost of Drunkenness

TL;DR

This paper studies the cost of drunkenness in an invisible Cops and Robbers game, analyzing capture times for adversarial and random-walking robbers on various graph families using game theory and POMDPs.

Contribution

It introduces the concept of the invisible Cost of Drunkenness (iCOD), providing asymptotic values, bounds, and examples of graphs with various iCOD levels, and compares different game variants.

Findings

Exact asymptotic iCOD for d-regular trees

Bounds for grid graphs

iCOD can be arbitrarily close to any value in [2, ∞)

Abstract

We examine a version of the Cops and Robber (CR) game in which the robber is invisible, i.e., the cops do not know his location until they capture him. Apparently this game (CiR) has received little attention in the CR literature. We examine two variants: in the first the robber is adversarial (he actively tries to avoid capture); in the second he is drunk (he performs a random walk). Our goal in this paper is to study the invisible Cost of Drunkenness (iCOD), which is defined as the ratio ct_i(G)/dct_i(G), with ct_i(G) and dct_i(G) being the expected capture times in the adversarial and drunk CiR variants, respectively. We show that these capture times are well defined, using game theory for the adversarial case and partially observable Markov decision processes (POMDP) for the drunk case. We give exact asymptotic values of iCOD for several special graph families such as $d$-regular trees, give some bounds for grids, and provide general upper and lower bounds for general classes of graphs. We also give an infinite family of graphs showing that iCOD can be arbitrarily close to any value in [2,infinty). Finally, we briefly examine one more CiR variant, in which the robber is invisible and "infinitely fast"; we argue that this variant is significantly different from the Graph Search game, despite several similarities between the two games.