[Paper Review] Improved Approximations for Free-Order Prophets and Second-Price Auctions.

Hedyeh Beyhaghi, Negin Golrezaei+3 more|arXiv (Cornell University)|Jul 10, 2018

Auction Theory and Applications18 citations

TL;DR

This paper improves revenue guarantees for two key mechanisms in single-item auctions: the eager second-price auction with personalized reserves and sequential posted price mechanisms. Using a novel analytical approach, it establishes that both mechanisms achieve at least 66.20% and 65.43% of optimal revenue, respectively, with tighter bounds for small n, and introduces a polynomial-time algorithm for optimizing reserve prices when their order is fixed, validated on real Google ad auction data.

ABSTRACT

We study the fundamental problem of selling a single indivisible item to one of $n$ buyers with independent and potentially nonidentical value distributions. We focus on two simple and widely used selling mechanisms: the second price auction with \emph{eager} personalized reserve prices and the sequential posted price mechanism. Using a new approach, we improve the best-known performance guarantees for these mechanisms. We show that for every value of the number of buyers $n$, the eager second price (ESP) auction and sequential posted price mechanisms respectively earn at least $0.6620$ and $0.6543$ fractions of the optimal revenue. We also provide improved performance guarantees for these mechanisms when the number of buyers is small, which is the more relevant regime for many applications of interest. This in particular implies an improved bound of $0.6543$ for free-order prophet inequalities. Motivated by our improved revenue bounds, we further study the problem of optimizing reserve prices in the ESP auctions when the sorted order of personalized reserve prices among bidders is exogenous. We show that this problem can be solved polynomially. In addition, by analyzing a real auction dataset from Google's advertising exchange, we demonstrate the effectiveness of order-based pricing.

Motivation & Objective

• To improve performance guarantees for second-price auctions with eager personalized reserves and sequential posted price mechanisms in single-item, single-good auctions.
• To provide tighter revenue approximation ratios, especially for small numbers of bidders, which are more relevant in practical applications.
• To study the problem of optimizing reserve prices when the sorted order of personalized reserves is exogenous, aiming for efficient computation.
• To validate the effectiveness of order-based reserve pricing using real auction data from Google's advertising exchange.

Proposed method

• Introduces a new analytical framework to bound the expected revenue of the eager second-price auction and sequential posted price mechanisms under independent, potentially nonidentical value distributions.
• Applies stochastic dominance and conditional expectation techniques to derive improved approximation ratios for finite n, particularly for small n.
• Reformulates the reserve price optimization problem under a fixed order of reserves as a polynomial-time solvable problem using dynamic programming or convex optimization.
• Employs a real dataset from Google's ad exchange to empirically validate the performance gains from order-based reserve pricing.
• Leverages structural properties of free-order prophet inequalities to derive improved bounds for the sequential posted price mechanism.

Experimental results

Research questions

• RQ1 What is the best possible revenue approximation ratio achievable by the eager second-price auction with personalized reserves for any fixed n?
• RQ2 How do the performance guarantees of sequential posted price mechanisms compare to optimal revenue, especially when n is small?
• RQ3 Can the problem of optimizing personalized reserve prices be solved efficiently when the relative order of reserves is fixed?
• RQ4 To what extent can order-based reserve pricing improve revenue in real-world auction systems?

Key findings

• The eager second-price auction with personalized reserves achieves at least 66.20% of the optimal revenue for any number of bidders n.
• The sequential posted price mechanism guarantees at least 65.43% of the optimal revenue, improving upon previous bounds.
• For small values of n, the paper provides even tighter performance guarantees, which are more relevant in practical auction design.
• The problem of optimizing personalized reserve prices, given a fixed order of reserves, can be solved in polynomial time.
• Empirical analysis on a real Google ad exchange dataset confirms the effectiveness of order-based reserve pricing in practice.