Bidder Selection Problem in Position Auctions: A Fast and Simple Algorithm via Poisson Approximation

In the Bidder Selection Problem (BSP) there is a large pool of n potential advertisers competing for ad slots on the user's web page. Due to strict computational restrictions, the advertising platform can run a proper auction only for a fraction k<n of advertisers. We consider the basic optimization problem underlying BSP: given n independent prior distributions, how to efficiently find a subset of k with the objective of either maximizing expected social welfare or revenue of the platform. We study BSP in the classic multi-winner model of position auctions for welfare and revenue objectives using the optimal (respectively, VCG mechanism, or Myerson's auction) format for the selected set of bidders. This is a natural generalization of the fundamental problem of selecting k out of n random variables in a way that the expected highest value is maximized. Previous PTAS results ([Chen, Hu, Li, Li, Liu, Lu, NIPS 2016], [Mehta, Nadav, Psomas, Rubinstein, NIPS 2020], [Segev and Singla, EC 2021]) for BSP optimization were only known for single-item auctions and in case of [Segev and Singla 2021] for l-unit auctions. More importantly, all of these PTASes were computational complexity results with impractically large running times, which defeats the purpose of using these algorithms under severe computational constraints.

We propose a novel Poisson relaxation of BSP for position auctions that immediately implies that 1) BSP is polynomial-time solvable up to a vanishingly small error as the problem size k grows; 2) there is a PTAS for position auctions after combining our relaxation with the trivial brute force algorithm. Unlike all previous PTASes, we implemented our algorithm and did extensive numerical experiments on practically relevant input sizes. First, our experiments corroborate the previous experimental findings of Mehta et al. that a few simple heuristics used in practice (e.g., Greedy for general submodular maximization) perform surprisingly well in terms of approximation factor. Furthermore, our algorithm outperforms Greedy both in running time and approximation on medium and large-sized instances, i.e., its running time scales better with the instance size.