We introduce a new technique for proving positivity of certain divisor classes on
M
¯
0
,
n
\overline {M}_{0,n}
and its weighted variants
M
¯
0
,
A
\overline {M}_{0,\mathcal {A}}
. Our methods give a complete description of the models arising in the Hassett’s log minimal model program for
M
¯
0
,
n
\overline {M}_{0,n}
.