CS364B: Frontiers in Mechanism Design Bonus Lecture: Gross Substitutes and Greedy Algorithms∗

where A = {j : q(j) > p(j)} is the of items whose prices have increased (in q relative to p). That is, whenever a bidder loses the items of S ∩ A because of being outbid, it still wants to retain the items S \ A it has, at the original prices. We saw several examples of GS valuations, including k-unit demand valuations, downward-sloping valuations for identical items, and so on. In this lecture, we insist that Definition 1.1 holds for all real-valued prices vectors, including those that with some negative prices. We also won’t need to assume that vi(∅) = 0. We only consider valuations that are monotone (S ⊆ T implies v(S) ≤ v(T )). ∗ c ©2014, Tim Roughgarden. †Department of Computer Science, Stanford University, 462 Gates Building, 353 Serra Mall, Stanford, CA 94305. Email: [email protected].