{"ID":22952737,"CreatedAt":"2026-09-17T02:12:05.498442134Z","UpdatedAt":"2026-09-20T18:11:56.143995915Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18848","arxiv_id":"2609.18848","title":"On the Role of Tie-Breaking Rules in the Convergence of Fictitious Play for Symmetric First-Price Auctions","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 $ε$-equilibrium of the original auction in a broad range of settings.","short_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...","url_abs":"https://arxiv.org/abs/2609.18848","url_pdf":"https://arxiv.org/pdf/2609.18848v1","authors":"[\"Benjamin Heymann\"]","published":"2026-09-16T15:53:40Z","proceeding":"cs.GT","tasks":"[\"cs.GT\",\"math.OC\"]","methods":"[]","has_code":false}