Efficiency of (Revenue-)Optimal Mechanisms

TL;DR

This paper compares the efficiency of revenue-maximizing and efficiency-maximizing mechanisms in selling items, showing that a logarithmic number of extra bidders allows revenue mechanisms to match efficiency, with tight bounds and extensions to multiple items.

Contribution

It establishes tight bounds on the number of extra bidders needed for revenue mechanisms to achieve efficiency comparable to efficiency-maximizing ones, extending results to multiple items and classifying MHR distributions.

Findings

Logarithmic extra bidders suffice for efficiency matching

Bound is tight within a small additive constant

Extension to multiple items with specific bidder requirements

Abstract

We compare the expected efficiency of revenue maximizing (or {\em optimal}) mechanisms with that of efficiency maximizing ones. We show that the efficiency of the revenue maximizing mechanism for selling a single item with k + log_{e/(e-1)} k + 1 bidders is at least as much as the efficiency of the efficiency maximizing mechanism with k bidders, when bidder valuations are drawn i.i.d. from a Monotone Hazard Rate distribution. Surprisingly, we also show that this bound is tight within a small additive constant of 5.7. In other words, Theta(log k) extra bidders suffice for the revenue maximizing mechanism to match the efficiency of the efficiency maximizing mechanism, while o(log k) do not. This is in contrast to the result of Bulow and Klemperer comparing the revenue of the two mechanisms, where only one extra bidder suffices. More precisely, they show that the revenue of the efficiency maximizing mechanism with k+1 bidders is no less than the revenue of the revenue maximizing mechanism with k bidders. We extend our result for the case of selling t identical items and show that 2.2 log k + t Theta(log log k) extra bidders suffice for the revenue maximizing mechanism to match the efficiency of the efficiency maximizing mechanism. In order to prove our results, we do a classification of Monotone Hazard Rate (MHR) distributions and identify a family of MHR distributions, such that for each class in our classification, there is a member of this family that is pointwise lower than every distribution in that class. This lets us prove interesting structural theorems about distributions with Monotone Hazard Rate.