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

TL;DR

This paper derives exact formulas for the maximum revenue in a two-item, multi-buyer auction setting with independent private distributions, revealing revenue gaps between dominant-strategy and Bayesian implementations and characterizing when they coincide.

Contribution

It provides the first exact formulas for maximum revenue in this setting and characterizes when the two auction implementations yield the same revenue.

Findings

Exact formulas for k=2, IID distributions with support size 2.

Revenue gaps identified between DIC and BIC implementations.

Conditions under which selling separately is optimal.

Abstract

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 $a$.