On Solutions for the Maximum Revenue Multi-item Auction under Dominant-Strategy and Bayesian Implementations

Very few exact solutions are known for the monopolist's $k$-item $n$-buyer maximum revenue problem with additive valuation in which $k, n >1$ and the buyers $i$ have independent private distributions $F^j_i$ on items $j$. In this paper we derive exact formulas for the maximum revenue when $k=2$ and $F^j_i$ 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 implementations 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 implementations. For instance, if $k=n=2$ and $Pr\{X_F=1\}=Pr\{X_F=2\}=\frac{1}{2}$, 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 $X_F$ over $[a, a+1]\cup[2a, 2a+1]$ for all large $a$.