  Let $P$ be a simple polytope with $n-d = 2$, where $d$ is the dimension and
$n$ is the number of facets. The graph of such a polytope is also called a
grid. It is known that the directed random walk along the edges of $P$
terminates after $O(\log^2 n)$ steps, if the edges are oriented in a
(pseudo-)linear fashion. We prove that the same bound holds for the more
general unique sink orientations.
