  Random Edge is the most natural randomized pivot rule for the simplex
algorithm. Considerable progress has been made recently towards fully
understanding its behavior. Back in 2001, Welzl introduced the concepts of
\emph{reachmaps} and \emph{niceness} of Unique Sink Orientations (USO), in an
effort to better understand the behavior of Random Edge. In this paper, we
initiate the systematic study of these concepts. We settle the questions that
were asked by Welzl about the niceness of (acyclic) USO. Niceness implies
natural upper bounds for Random Edge and we provide evidence that these are
tight or almost tight in many interesting cases. Moreover, we show that Random
Edge is polynomial on at least $n^{\Omega(2^n)}$ many (possibly cyclic) USO. As
a bonus, we describe a derandomization of Random Edge which achieves the same
asymptotic upper bounds with respect to niceness and discuss some algorithmic
properties of the reachmap.
