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

Created by
  • Haebom

Author

Xiaotie Deng, Xinyan Hu, Tao Lin, Weiqiang Zheng

Outline

This paper analyzes the convergence properties of the learning dynamics in repeated first-price auctions where bidders with fixed values bid using average-based algorithms (including no-regret algorithms such as Multiplicative Weights Update and Follow the Perturbed Leader). We fully characterize the learning dynamics of average-based algorithms under two convergence concepts: time-average convergence (the fraction of time in which bidders play a Nash equilibrium converges to 1) and last-iteration convergence (the mixed strategy profiles of bidders converge to a Nash equilibrium). The convergence outcome depends on the number of highest-valued bidders. If there are three or more highest-valued bidders, the Nash equilibrium is almost certain to converge in both the time-average and last iteration; if there are two highest-valued bidders, the Nash equilibrium converges in the time-average but not necessarily in the last iteration; and if there is one highest-valued bidder, the Nash equilibrium may not converge in either the time-average or last iteration.

Takeaways, Limitations

Takeaways: We present a complete characterization of the convergence characteristics of mean-based learning algorithms in repetitive auctions, providing Takeaways potential for various applications, including online advertising markets. In particular, we clarify the differences in convergence results depending on the number of highest bidders, suggesting new possibilities for studying learning dynamics.
Limitations: The analysis was limited to average-based algorithms, limiting generalizability to other types of learning algorithms. Furthermore, the analysis assumed fixed bidder values, so the results may vary in situations where values fluctuate. Finally, the fact that convergence in the final iteration is not guaranteed when there are two highest bidders requires further research.
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