Simplified Prophet Inequalities for Combinatorial Auctions

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 $2k + 2 \sqrt{k(k-1)} -1$.