Bidder Selection Problem in Position Auctions: A Fast and Simple Algorithm via Poisson Approximation
Nickolai Gravin, Yixuan Even Xu, Renfei Zhou
TL;DR
This work tackles the Bidder Selection Problem (BSP) in position auctions under real-time constraints, where a platform must choose at most $k$ bidders from $n$ with independent priors to maximize expected social welfare or revenue. The authors introduce a Poisson relaxation that yields a concave, polynomial-time solvable approximation to BSP and proves a rate of convergence $1 - O(k^{-1/4})$ to the true welfare with a rounding step giving a $(1 - O(k^{-1/4}))$-approximation; in the single-item case the bound improves to $1 - O\big(\sqrt{\log k / k}\big)$. This framework provides a PTAS by combining the relaxation with brute-force search, and an efficient EPTAS for $k \ge \log n$, while being implementable in practice. Extensive numerical experiments with realistic input sizes show the method not only matches or outperforms standard heuristics like Greedy in both solution quality and running time, but also scales to large instances where prior PTAS approaches are intractable.
Abstract
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. Previous PTAS results 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 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.