Tight Revenue Gaps among Multi-Unit Mechanisms

TL;DR

This paper analyzes revenue gaps among four key multi-unit auction mechanisms in a Bayesian setting, revealing tight bounds within and across groups of mechanisms as the number of units grows.

Contribution

It establishes tight asymptotic bounds on revenue gaps both within groups and between groups of auction mechanisms in multi-unit settings.

Findings

Within-group revenue gap is asymptotically 1 + Θ(1/√k).

Cross-group revenue gap is asymptotically Θ(log k).

Results hold for non-identical value distributions.

Abstract

This paper considers Bayesian revenue maximization in the $k$-unit setting, where a monopolist seller has $k$ copies of an indivisible item and faces $n$ unit-demand buyers (whose value distributions can be non-identical). Four basic mechanisms among others have been widely employed in practice and widely studied in the literature: {\sf Myerson Auction}, {\sf Sequential Posted-Pricing}, {\sf $(k + 1)$-th Price Auction with Anonymous Reserve}, and {\sf Anonymous Pricing}. Regarding a pair of mechanisms, we investigate the largest possible ratio between the two revenues (a.k.a.\ the revenue gap), over all possible value distributions of the buyers. Divide these four mechanisms into two groups: (i)~the discriminating mechanism group, {\sf Myerson Auction} and {\sf Sequential Posted-Pricing}, and (ii)~the anonymous mechanism group, {\sf Anonymous Reserve} and {\sf Anonymous Pricing}. Within one group, the involved two mechanisms have an asymptotically tight revenue gap of $1 + \Theta(1 / \sqrt{k})$. In contrast, any two mechanisms from the different groups have an asymptotically tight revenue gap of $\Theta(\log k)$.