On Revenue Maximization in Second-Price Ad Auctions

TL;DR

This paper studies the computational complexity of revenue maximization in second-price ad auctions, revealing significant hardness results and providing approximation algorithms for the problem.

Contribution

It introduces the Second-Price Ad Auctions problem, analyzes its complexity, and offers approximation algorithms contrasting with first-price auction models.

Findings

Second-Price Ad Auctions problem is NP-hard to approximate.

Offline 2PM is APX-hard, online 2PM has no non-trivial deterministic algorithms.

Provides a 2-approximation for offline 2PM and a 5.083-competitive online algorithm.

Abstract

Most recent papers addressing the algorithmic problem of allocating advertisement space for keywords in sponsored search auctions assume that pricing is done via a first-price auction, which does not realistically model the Generalized Second Price (GSP) auction used in practice. Towards the goal of more realistically modeling these auctions, we introduce the Second-Price Ad Auctions problem, in which bidders' payments are determined by the GSP mechanism. We show that the complexity of the Second-Price Ad Auctions problem is quite different than that of the more studied First-Price Ad Auctions problem. First, unlike the first-price variant, for which small constant-factor approximations are known, it is NP-hard to approximate the Second-Price Ad Auctions problem to any non-trivial factor. Second, this discrepancy extends even to the 0-1 special case that we call the Second-Price Matching problem (2PM). In particular, offline 2PM is APX-hard, and for online 2PM there is no deterministic algorithm achieving a non-trivial competitive ratio and no randomized algorithm achieving a competitive ratio better than 2. This stands in contrast to the results for the analogous special case in the first-price model, the standard bipartite matching problem, which is solvable in polynomial time and which has deterministic and randomized online algorithms achieving better competitive ratios. On the positive side, we provide a 2-approximation for offline 2PM and a 5.083-competitive randomized algorithm for online 2PM. The latter result makes use of a new generalization of a classic result on the performance of the "Ranking" algorithm for online bipartite matching.