{
  "abstract": "We study continuous-time fictitious play in 2-bidder, symmetric first-price auctions with independently distributed discrete values and a discrete bid set. We first exhibit a minimal instance --- two bidders, two values, three positive bids --- on which fictitious play with the standard uniform-split tie-breaking rule does \\emph{not} converge to the symmetric Bayes--Nash equilibrium: the equilibrium is unstable and the dynamics converge to a stable limit cycle far from the Nash equilibrium. We then show that a small modification of the tie-breaking rule --- awarding a payoff of zero to every bidder in case of a tie --- restores convergence: fictitious play converges to a Nash equilibrium of the modified game. This limit is an $\\epsilon$-equilibrium of the original auction in a broad range of settings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18848",
  "author_total": 1,
  "authors": [
    {
      "name": "Benjamin Heymann"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/19cbfxn313x5r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18848",
  "fulltext_url": null,
  "id": "19cbfxn313x5r",
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  "landing_url": "https://hal.science/hal-05693390v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
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      "math.OC"
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      "1803|Auction Theory and Applications"
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  "openalex": "W7213524068",
  "pdf_url": "https://hal.science/hal-05693390/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/19cbfxn313x5r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:21Z",
      "native_id": "hal-05693390",
      "source": "hal",
      "url": "https://hal.science/hal-05693390v1"
    },
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18848",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18848"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "On the Role of Tie-Breaking Rules in the Convergence of Fictitious Play for Symmetric First-Price Auctions",
  "title_text": "On the Role of Tie-Breaking Rules in the Convergence of Fictitious Play for Symmetric First-Price Auctions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/19cbfxn313x5r",
  "venue": "HAL",
  "versions": []
}