  We consider prophet inequalities for XOS and MPH-$k$ combinatorial auctions
and give a simplified proof for the existence of static and anonymous item
prices which recover the state-of-the-art competitive ratios.
  Our proofs make use of a linear programming formulation which has a
non-negative objective value if there are prices which admit a given
competitive ratio $\alpha \geq 1$. Changing our perspective to dual space by an
application of strong LP duality, we use an interpretation of the dual
variables as probabilities to directly obtain our result. In contrast to
previous work, our proofs do not require to argue about specific values of
buyers for bundles, but only about the presence or absence of items.
  As a side remark, for any $k \geq 2$, this simplification also leads to a
tiny improvement in the best competitive ratio for MPH-$k$ combinatorial
auctions from $4k-2$ to $2k + 2 \sqrt{k(k-1)} -1$.
