Dominant-Strategy versus Bayesian Multi-item Auctions: Maximum Revenue Determination and Comparison

We address two related unanswered questions in maximum revenue multi-item auctions. Is dominant-strategy implementation equivalent to the semantically less stringent Bayesian one (as in the case of Myerson's 1-item auction)? Can one find explicit solutions for non-trivial families of multi-item auctions (as in the 1-item case)? In this paper, we present such natural families whose explicit solutions exhibit a revenue gap between the two implementations. More precisely, consider the k-item n-buyer maximum revenue auction where k, n >1 with additive valuation in the independent setting (i.e., the buyers i have independent private distributions Fij on items j). We derive exact formulas for the maximum revenue when k=2 and Fij are any IID distributions on support of size 2, for both the dominant-strategy (DIC) and the Bayesian (BIC) implementations. The formulas lead to the simple characterization that, the two models have identical maximum revenue if and only if selling-separately is optimal for the distribution. Our results also give the first demonstration, in this setting, of revenue gaps between the two models. For instance, if k=n=2 and Pr{X{F = 1} = Pr{XF =2 } = 1/2, then the maximum revenue in the Bayesian implementation exceeds that in the dominant-strategy by exactly 2%; the same gap exists for the continuous uniform distribution XF over [a, a+1] ∪ [2a, 2a+1] for all large a.