Price of anarchy for supply chains with partial positive externalities

We consider the impact of partial positive externalities (imperfect complementarity) among downstream retailers on supply chain performance. We show that double marginalization may fail to exist in a decentralized setting when some retailers carry multiple imperfect complements. By giving a precise characterization on the degree of complementarity, we prove that a decentralized supply chain loses at least 25% of the optimal profit and that its performance degrades rapidly with the complementarity effect.