CS364B: Frontiers in Mechanism Design Lecture #11: Undominated Implementations and the Shrinking Auction∗

Part I of the course focused on special cases where we can achieve all three properties — gross substitutes (GS) valuations and special cases thereof. In Part II of the course, we focused on more general valuation classes in which the exact version of the second property is already incompatible with the third (assuming P 6= NP ). We focused further on special cases where, ignoring incentive constraints, a constant-factor approximation to the optimal welfare can be computed in polynomial time. We’ve seem some pleasing positive results: when there is a logarithmic supply of every item or with coverage valuations, we designed MIDR allocation rules (and hence DSIC mechanisms) that have best-possible approximation guarantees (assuming P 6= NP ), albeit through quite complex designs. For general submod∗ c ©2014, Tim Roughgarden. †Department of Computer Science, Stanford University, 462 Gates Building, 353 Serra Mall, Stanford, CA 94305. Email: [email protected]. Let’s recall the technical fine print. The result for logarithmic supply assumes that each bidder only wants one copy of each item and that the valuations are given as black boxes that support demand queries. The approximation guarantee is (1− ) provided the supply is at least c −2 logm for some (modest) constant c. The approximation guarantee for coverage valuations is 1− 1e ≈ 0.63. This guarantee extends to valuations that are convex combinations of gross substitutes valuations, assuming that such valuations are given as black boxes that support a randomized version of a value oracle.