  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.
