{
  "abstract": "Pacing equilibria describe how budget-constrained advertisers clear an ad market, but computing one at production scale is hard when every request fills several slots: the allocation jumps whenever two campaigns' scores cross. We consider the first-price pacing equilibrium (FPPE) of auctions with $K$ homogeneous slots and unit-demand campaigns, whose Eisenberg-Gale dual reduces, via a TopK identity, to a convex program over bids. Regularizing each request's allocation with the binary (Fermi-Dirac) entropy gives a smooth, strictly convex dual whose gradient is exactly spend minus budget and whose evaluation splits into independent one-dimensional threshold solves, one per request. The smoothed equilibrium converges to the hard one at an explicit $O(\\sqrt{T})$ rate, and a duality gap certifies how far any bids are from the hard equilibrium. We implement an efficient multi-GPU solver and test it on a production ad workload. We use it to compute an equilibrium price anchor: each campaign's hindsight equilibrium cost per key result.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24323",
  "author_total": 5,
  "authors": [
    {
      "name": "Yuchen Xie"
    },
    {
      "name": "Shenyinying Tu"
    },
    {
      "name": "Yuan Gao"
    },
    {
      "name": "Yi Zhang"
    },
    {
      "name": "Deepak Kumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/88g9v0hf4mwhz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24323",
  "fulltext_url": null,
  "id": "88g9v0hf4mwhz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24323",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.GT"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1803|Auction Theory and Applications"
    ]
  ],
  "openalex": "W7222993753",
  "pdf_url": "https://arxiv.org/pdf/2610.24323",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/88g9v0hf4mwhz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24323",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24323"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Computing Multi-Slot First-Price Pacing Equilibria at Scale via Entropic Smoothing",
  "title_text": "Computing Multi-Slot First-Price Pacing Equilibria at Scale via Entropic Smoothing",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/88g9v0hf4mwhz",
  "venue": "arXiv",
  "versions": []
}