Separation between Second Price Auctions with Personalized Reserves and the Revenue Optimal Auction

TL;DR

This paper establishes the first known upper bound on the revenue approximation of the second price auction with personalized reserves, showing it cannot surpass 77.8% of the optimal revenue in general, nearly closing the gap with known lower bounds.

Contribution

It provides the first separation results for eager personalized reserves, demonstrating their revenue cannot exceed 77.8% of the optimal, even with i.i.d. distributions not satisfying regularity.

Findings

ESP cannot achieve more than 0.778 of MyeRev in general.

ESP can achieve at least 0.745 of MyeRev, nearly bridging the bounds.

The results hold even for non-regular, i.i.d. distributions.

Abstract

What fraction of the single item $n$ buyers setting's expected optimal revenue MyeRev can the second price auction with reserves achieve? In the special case where the buyers' valuation distributions are all drawn i.i.d. and the distributions satisfy the regularity condition, the second price auction with an anonymous reserve (ASP) is the optimal auction itself. As the setting gets more complex, there are established upper bounds on the fraction of MyeRev that ASP can achieve. On the contrary, no such upper bounds are known for the fraction of MyeRev achievable by the second price auction with eager personalized reserves (ESP). In particular, no separation was earlier known between ESP's revenue and MyeRev even in the most general setting of non-identical product distributions that don't satisfy the regularity condition. In this paper we establish the first separation results for ESP: we show that even in the case of distributions drawn i.i.d., but not necessarily satisfying the regularity condition, the ESP cannot achieve more than a $0.778$ fraction of MyeRev in general. Combined with Correa et al.'s result (EC 2017) that ESP can achieve at least a $0.745$ fraction of MyeRev, this nearly bridges the gap between upper and lower bounds on ESP's approximation factor.