Generalizing Virtual Values to Multidimensional Auctions: a Non-Myersonian Approach

We consider the revenue maximization problem of a monopolist via a non-Myersonian approach that could generalize to multiple items and multiple buyers. Although such an approach does not lead to any closed-form solution of the problem, it does provide some insights into this problem from different angles. In particular, we consider both Bayesian (Bayesian Incentive Compatible + Bayesian Individually Rational) and Dominant-Strategy (Dominant-Strategy Incentive Compatible + ex-post Individually Rational) implementations, where all the buyers have additive valuations and quasi-linear utilities and all the valuations are independent across buyers (not necessarily independent across items). The main technique of our approach is to formulate the problem as an LP (probably with exponential size) and apply primal-dual analysis. We observe that any optimal solution of the dual program naturally defines the virtual value functions for the primal revenue maximization problem in the sense that any revenue-maximizing auction must be a virtual welfare maximizer (cf. Myerson's auction for a single item [Myerson, 1981]). Based on this observation, we characterize a sufficient and necessary condition for BIC = DSIC, i.e., the optimal revenue of Bayesian implementations equals to the optimal revenue of dominant-strategy implementations (BRev = DRev). The condition is if and only if the optimal DSIC revenue DRev can be achieved by a DSIC and ex-post IR virtual welfare maximizer with buyer-independent virtual value functions (buyer i's virtual value is independent of other buyers' valuations). In light of the characterization, we further show that when all the valuations are i.i.d., it is further equivalent to that separate-selling is optimal. In particular, it respects one result from the recent breakthrough work on the exact optimal solutions in the multi-item multi-buyer setting by Yao [2016].