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Nash Convergence of Mean-Based Learning Algorithms in First Price Auctions

2021-10-08 06:01:27
Xiaotie Deng, Xinyan Hu, Tao Lin, Weiqiang Zheng

Abstract

We consider repeated first price auctions where each bidder, having a deterministic type, learns to bid using a mean-based learning algorithm. We completely characterize the Nash convergence property of the bidding dynamics in two senses: (1) time-average: the fraction of rounds where bidders play a Nash equilibrium approaches to 1 in the limit; (2) last-iterate: the mixed strategy profile of bidders approaches to a Nash equilibrium in the limit. Specifically, the results depend on the number of bidders with the highest value: - If the number is at least three, the bidding dynamics almost surely converges to a Nash equilibrium of the auction, both in time-average and in last-iterate. - If the number is two, the bidding dynamics almost surely converges to a Nash equilibrium in time-average but not necessarily in last-iterate. - If the number is one, the bidding dynamics may not converge to a Nash equilibrium in time-average nor in last-iterate. Our discovery opens up new possibilities in the study of convergence dynamics of learning algorithms.

Abstract (translated)

URL

https://arxiv.org/abs/2110.03906

PDF

https://arxiv.org/pdf/2110.03906.pdf


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