On the Role of Tie-Breaking Rules in the Convergence of Fictitious Play for Symmetric First-Price Auctions

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.