Simple mechanisms for subadditive buyers via duality

A central problem in mechanism design is how to design simple and approximately revenue-optimal auctions in multi-item multi-buyer settings. Prior to our work, all results only apply to cases where the buyers' valuations are linear over the items. We unify and improve all previous results, as well as generalize the results to accommodate non-linear valuations [Cai and Zhao 2017]. In particular, we prove that a simple, deterministic and Dominant Strategy Incentive Compatible (DSIC) mechanism, namely, the sequential posted price with entry fee mechanism, achieves a constant fraction of the optimal revenue among all randomized, Bayesian Incentive Compatible (BIC) mechanisms, when buyers' valuations are XOS (a superclass of submodular valuations) over independent items. If the buyers' valuations are subadditive over independent items, the approximation factor degrades to O(log m), where m is the number of items. We obtain our results by first extending the Cai-Devanur-Weinberg duality framework to derive an effective benchmark of the optimal revenue for subadditive buyers, and then developing new analytic tools that combine concentration inequality of subadditive functions, prophet-inequality type of arguments, and a novel decomposition of the benchmark.