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

2026-09-16

AI Summary

研究了在对称首价拍卖中，通过修改平局规则使虚构博弈收敛到近似纳什均衡的方法。

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.

Problem

• Fictitious Play
• Tie-Breaking Rules
• First-Price Auctions
• Bayes--Nash Equilibrium

Innovation

• tie-breaking rule
• fictitious play
• symmetric first-price auctions
• Bayes-Nash equilibrium
• convergence