  We consider generalisations of tree width to directed graphs, that attracted
much attention in the last fifteen years. About their relative strength with
respect to "bounded width in one measure implies bounded width in the other"
many problems remain unsolved. Only some results separating directed width
measures are known. We give an almost complete picture of this relation. For
this, we consider the cops and robber games characterising DAG-width and
directed tree width (up to a constant factor). For DAG-width games, it is an
open question whether the robber-monotonicity cost (the difference between the
minimal numbers of cops capturing the robber in the general and in the monotone
case) can be bounded by any function. Examples show that this function (if it
exists) is at least $f(k) > 4k/3$ (Kreutzer, Ordyniak 2008). We approach a
solution by defining weak monotonicity and showing that if $k$ cops win weakly
monotonically, then $O(k^2)$ cops win monotonically. It follows that bounded
Kelly-width implies bounded DAG-width, which has been open since the definition
of Kelly-width by Hunter and Kreutzer in 2008. For directed tree width games we
show that, unexpectedly, the cop-monotonicity cost (no cop revisits any vertex)
is not bounded by any function. This separates directed tree width from D-width
defined by Safari in 2005, refuting his conjecture.
