In this project we look at interpretation of the game of cops and robbers. In the original game, both a cop and a robber alternate turns moving from vertex to adjacent vertex on a connected graph G with the cop trying to catch the robber and the robber trying to evade the cop. Harris et al. introduced a variant of the game-the tipsy cop and drunken robber. We generalize the work of Harris et al. using a slightly different interpretation to model the our game. We let the cop and robber be any amount of tipsy and use a spinner wheel to determine the probability of whether the next move will be a sober cop move, a sober robber move, or a tipsy move by either player.
The questions we consider (given a specified set of initial conditions on the players' distance and tipsiness) are:
What is the probability P(i,j,M) that the game, beginning in state i, will be in state j after exactly M rounds?
What is the probability GM(d) that the game lasts at least M rounds and the expected number E(d) of rounds the game should last if they start distance d away?
We analyze the game on cycle, Petersen, friendship graphs, and toroidal grids using the theory of Markov chains. We provide an answer to questions from Harris et al. presenting a model for the game where players start off drunk and sober up as the game progresses and a model for the game where the players' tipsiness increases as a function of the distance between them.
