On Symmetries in Multi-dimensional Mechanism Design

We consider a revenue-maximizing seller with multiple items for sale to a single population of buyers. Our main result shows that for a single population of additive buyers with independent (but not necessarily identically distributed) item values, bundling all items together achieves a constant-factor approximation to the revenue-optimal item-symmetric mechanism. We further motivate this direction via fairness in ad auctions. In ad auction domains the items correspond to views from particular demographics, and recent works have therefore identified a novel fairness constraint: equally-qualified users from different demographics should be shown the same desired ad at equal rates. Prior work abstracts this to the following fairness guarantee: if an advertiser places an identical bid on two users, those two users should view the ad with the same probability [27, 34]. We first propose a relaxation of this guarantee from worst-case to Bayesian settings, which circumvents strong impossibility results from these works, and then study this guarantee through the lens of symmetries, as any item-symmetric auction is also fair (by this definition). Observe that in this domain, bundling all items together corresponds to concealing all demographic data [23].