Algorithms as Mechanisms: The Price of Anarchy of Relax and Round

Many algorithms that are originally designed without explicitly considering incentive properties are later combined with simple pricing rules and used as mechanisms. A key question is therefore to understand which algorithms, or, more generally, which algorithm design principles, when combined with simple payment rules such as pay your bid, yield mechanisms with a small price of anarchy. Our main result concerns mechanisms that are based on the relax-and-round paradigm. It shows that oblivious rounding schemes approximately preserve price of anarchy guarantees provable via smoothness. By virtue of our smoothness proofs, our price of anarchy bounds extend to Bayes–Nash equilibria and learning outcomes. In fact, they even apply out of equilibrium, requiring only that agents have no regret for deviations to half their value. We demonstrate the broad applicability of our main result by instantiating it for a wide range of optimization problems ranging from sparse packing integer programs, over single-source unsplittable flow problems and combinatorial auctions with fractionally subadditive valuations, to a maximization variant of the traveling salesman problem.