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.
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BibTeX
@misc{heymann2026role,
title = {{On the Role of Tie-Breaking Rules in the Convergence of Fictitious Play for Symmetric First-Price Auctions}},
author = {Benjamin Heymann},
howpublished = {HAL},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2609.18848},
eprint = {2609.18848},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05693390v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On the Role of Tie-Breaking Rules in the Convergence of Fictitious Play for Symmetric First-Price Auctions
AU - Benjamin Heymann
PY - 2026
DA - 2026/07/15
PB - HAL
DO - 10.48550/arxiv.2609.18848
UR - https://hal.science/hal-05693390v1
AB - 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.
ER -